## Quantum Computers Could Solve Countless Problems—And Create a Lot of New Ones

O ne of the secrets to building the world’s most powerful computer is probably perched by your bathroom sink.

At IBM’s Thomas J. Watson Research Center in New York State’s Westchester County, scientists always keep a box of dental floss—Reach is the preferred brand—close by in case they need to tinker with their oil-drum-size quantum computers , the latest of which can complete certain tasks millions of times as fast as your laptop.

Inside the shimmering aluminum canister of IBM’s System One, which sits shielded by the same kind of protective glass as the Mona Lisa, are three cylinders of diminishing circumference, rather like a set of Russian dolls. Together, these encase a chandelier of looping silver wires that cascade through chunky gold plates to a quantum chip in the base. To work properly, this chip requires super-cooling to 0.015 kelvins—a smidgen above absolute zero and colder than outer space. Most materials contract or grow brittle and snap under such intense chill. But ordinary dental floss, it turns out, maintains its integrity remarkably well if you need to secure wayward wires.

“But only the unwaxed, unflavored kind,” says Jay Gambetta, IBM’s vice president of quantum. “Otherwise, released vapors mess everything up.”

Tech giants from Google to Amazon and Alibaba —not to mention nation-states vying for technological supremacy—are racing to dominate this space. The global quantum-computing industry is projected to grow from $412 million in 2020 to $8.6 billion in 2027, according to an International Data Corp. analysis .

Whereas traditional computers rely on binary “bits”—switches either on or off, denoted as 1s and 0s—to process information, the “qubits” that underpin quantum computing are tiny subatomic particles that can exist in some percentage of both states simultaneously, rather like a coin spinning in midair. This leap from dual to multivariate processing exponentially boosts computing power. Complex problems that currently take the most powerful supercomputer several years could potentially be solved in seconds. Future quantum computers could open hitherto unfathomable frontiers in mathematics and science, helping to solve existential challenges like climate change and food security. A flurry of recent breakthroughs and government investment means we now sit on the cusp of a quantum revolution. “I believe we will do more in the next five years in quantum innovation than we did in the last 30,” says Gambetta.

But any disrupter comes with risks, and quantum has become a national-security migraine. Its problem-solving capacity will soon render all existing cryptography obsolete, jeopardizing communications, financial transactions, and even military defenses. “People describe quantum as a new space race,” says Dan O’Shea, operations manager for Inside Quantum Technology, an industry publication. In October, U.S. President Joe Biden toured IBM’s quantum data center in Poughkeepsie, N.Y., calling quantum “vital to our economy and equally important to our national security.” In this new era of great-power competition, China and the U.S. are particularly hell-bent on conquering the technology lest they lose vital ground. “This technology is going to be the next industrial revolution,” says Tony Uttley, president and COO for Quantinuum, a Colorado-based firm that offers commercial quantum applications. “It’s like the beginning of the internet, or the beginning of classical computing.”

Quantum computing is a rapidly-emerging technology that harnesses the laws of quantum mechanics to solve problems too complex for classical computers.

Today, IBM Quantum makes real quantum hardware — a tool scientists only began to imagine three decades ago — available to hundreds of thousands of developers. Our engineers deliver ever-more-powerful superconducting quantum processors at regular intervals, alongside crucial advances in software and quantum-classical orchestration. This work drives toward the quantum computing speed and capacity necessary to change the world.

These machines are very different from the classical computers that have been around for more than half a century. Here's a primer on this transformative technology.

When scientists and engineers encounter difficult problems, they turn to supercomputers. These are very large classical computers, often with thousands of classical CPU and GPU cores capable of running very large calculations and advanced artificial intelligence. However, even supercomputers are binary code-based machines reliant on 20th-century transistor technology. They struggle to solve certain kinds of problems.

If a supercomputer gets stumped, that's probably because the big classical machine was asked to solve a problem with a high degree of complexity. When classical computers fail, it's often due to complexity.

Complex problems are problems with lots of variables interacting in complicated ways. Modeling the behavior of individual atoms in a molecule is a complex problem, because of all the different electrons interacting with one another. Identifying subtle patterns of fraud in financial transactions or new physics in a supercollider are also complex problems. There are some complex problems that we do not know how to solve with classical computers at any scale.

The real world runs on quantum physics. Computers that make calculations using the quantum states of quantum bits should in many situations be our best tools for understanding it.

Let's look a an example that shows how quantum computers can succeed where classical computers fail:

A classical computer might be great at difficult tasks like sorting through a big database of molecules. But it will struggle to solve more complex problems, like simulating how those molecules behave.

Today, for the most part, if scientists want to know how a molecule will behave they have to synthesize it and experiment with it in the real world. If they want to know how a slight tweak would impact its behavior, they usually need to synthesize the new version and run their experiment all over again. This is an expensive, time-consuming process that impedes progress in fields as diverse as medicine and semiconductor design.

A classical supercomputer might try to simulate molecular behavior with brute force, leveraging its many processors to explore every possible way every part of the molecule might behave. But as it moves past the simplest, most straightforward molecules available, the supercomputer stalls. No computer has the working memory to handle all the possible permutations of molecular behavior using any known methods.

Quantum algorithms take a new approach to these sorts of complex problems — creating multidimensional computational spaces. This turns out to be a much more efficient way of solving complex problems like chemical simulations.

We do not have a good way to create these computational spaces with classical computers, which limits their usefulness without quantum computation. Industrial chemists are already exploring ways to integrate quantum methods into their work. This is just one example. Engineering firms, financial institutions, global shipping companies — among others — are exploring use cases where quantum computers could solve important problems in their fields. An explosion of benefits from quantum research and development is taking shape on the horizon. As quantum hardware scales and quantum algorithms advance, many big, important problems like molecular simulation should find solutions.

An IBM Quantum processor is a wafer not much bigger than the one found in a laptop. And a quantum hardware system is about the size of a car, made up mostly of cooling systems to keep the superconducting processor at its ultra-cold operational temperature.

A classical processor uses classical bits to perform its operations. A quantum computer uses qubits (CUE-bits) to run multidimensional quantum algorithms.

Superconductors

Your desktop computer likely uses a fan to get cold enough to work. Our quantum processors need to be very cold – about a hundredth of a degree above absolute zero — to avoid “decoherence,” or retain their quantum states. To achieve this, we use super-cooled superfluids. At these ultra-low temperatures certain materials exhibit an important quantum mechanical effect: electrons move through them without resistance. This makes them "superconductors."

When electrons pass through superconductors they match up, forming "Cooper pairs." These pairs can carry a charge across barriers, or insulators, through a process known as quantum tunneling. Two superconductors placed on either side of an insulator form a Josephson junction.

Our quantum computers use Josephson junctions as superconducting qubits. By firing microwave photons at these qubits, we can control their behavior and get them to hold, change, and read out individual units of quantum information.

Superposition

A qubit itself isn't very useful. But it can perform an important trick: placing the quantum information it holds into a state of superposition, which represents a combination of all possible configurations of the qubit. Groups of qubits in superposition can create complex, multidimensional computational spaces. Complex problems can be represented in new ways in these spaces.

Entanglement

Quantum entanglement is an effect that correlates the behavior of two separate things. Physicists have found that when two qubits are entangled, changes to one qubit directly impact the other.

Interference

In an environment of entanged qubits placed into a state of superposition, there are waves of probabilities. These are the probabilities of the outcomes of a measurement of the system. These waves can build on each other when many of them peak at a particular outcome, or cancel each other out when peaks and troughs interact. These are both forms of interference.

A computation on a quantum computer works by preparing a superposition of all possibile computational states. A quantum circuit, prepared by the user, uses interference selectively on the components of the superposition according to an algorithm. Many possible outcomes are cancelled out through interference, while others are amplified. The amplified outcomes are the solutions to the computation.

Right now, IBM Quantum leads the world in quantum computing hardware and software. Our Roadmap is a clear, detailed plan to scale quantum processors, overcome the scaling problem, and build the hardware necessary for quantum advantage in the era of noisy quantum machines.

Today, a great deal of the work in the field of quantum computing is devoted to realizing error correction — a technqiue that would enable noise-free quantum computation on very large quantum computers.

Recent work from IBM and elsewhere has shown that noisy quantum computers may be able to do useful work in the near future, even before the advent of error correction, using techniques known as error mitigation.

IBM has spent years advancing the software that will be necessary to do that useful work. We introduced the Qiskit quantum SDK. It is open-source, Python-based, and by far the most widely-used quantum SDK in the world — useful for executions both on IBM’s fleet of superconducting quantum computers and on systems that use alternative technologies like ions trapped in magnetic fields.

We developed Qiskit Runtime, the most powerful quantum programming model in the world. (Learn more about both Qiskit and Qiskit Runtime, and how to get started, in the next section.)

Achieving quantum advantage will require new methods of suppressing errors, increasing speed, and orchestrating quantum and classical resources. The foundations of that work are being laid today in Qiskit Runtime by IBM and our partners in industry, academia, and startups.

IBM's quantum computers are programmed using Qiskit (link resides outside ibm.com), our open-source, python-based quantum SDK. Qiskit has modules that cover applications in finance, chemistry, optimization, and machine learning.

- Check out the documentation (link resides outside ibm.com) to get started quickly and learn more about our suite of developer tools .
- Build research and development-level code to run on simulators or real hardware.
- Join our growing community of 400,000+ users

Ready for larger workloads? Execute at scale with Qiskit Runtime , our quantum programming model for efficiently building and scaling workloads. Qiskit Runtime enables users to deploy custom quantum-classical applications with easy access to HPC hybrid computations on the highest performing quantum systems in the world. Qiskit Runtime provides an execution environment for weaving together quantum circuits with classical processing, natively accelerating the execution of certain quantum programs. This means faster iteration, reduced latency, and more uninhibited compute time on the world's leading quantum systems: Qiskit Runtime's cloud-based execution model demonstrated a 120x speedup in simulating molecular behavior

Global businesses are readying themselves today for the era of quantum computing. See how our industry experts prepare our clients to use this technology for competitive advantage.

Build programs that solve problems in new ways on IBM Quantum systems—the most popular and powerful quantum hardware in the world.

Securing the world’s digital infrastructure for the era of quantum computing.

IBM's quantum computers are programmed using Qiskit, our open-source, Python-based quantum SDK. Qiskit has modules that cover applications in finance, chemistry, optimization, and machine learning. Execute at scale with Qiskit Runtime, our quantum programming model for efficiently building and scaling workloads with primitives and quantum middleware for easy optimization. Qiskit Runtime enables users to deploy custom quantum-classical applications with easy access to HPC hybrid computations on the highest performing quantum systems in the world.

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National Academy of Engineering. Frontiers of Engineering: Reports on Leading-Edge Engineering from the 2018 Symposium. Washington (DC): National Academies Press (US); 2019 Jan 28.

## Frontiers of Engineering: Reports on Leading-Edge Engineering from the 2018 Symposium.

- Hardcopy Version at National Academies Press

## Quantum Computing: What It Is, Why We Want It, and How We're Trying to Get It

SARA GAMBLE .

## Affiliations

Quantum mechanics emerged as a branch of physics in the early 1900s to explain nature on the scale of atoms and led to advances such as transistors, lasers, and magnetic resonance imaging. The idea to merge quantum mechanics and information theory arose in the 1970s but garnered little attention until 1982, when physicist Richard Feynman gave a talk in which he reasoned that computing based on classical logic could not tractably process calculations describing quantum phenomena. Computing based on quantum phenomena configured to simulate other quantum phenomena, however, would not be subject to the same bottlenecks. Although this application eventually became the field of quantum simulation, it didn't spark much research activity at the time.

In 1994, however, interest in quantum computing rose dramatically when mathematician Peter Shor developed a quantum algorithm, which could find the prime factors of large numbers efficiently. Here, “efficiently” means in a time of practical relevance, which is beyond the capability of state-of-the-art classical algorithms. Although this may seem simply like an oddity, it is impossible to overstate the importance of Shor's insight. The security of nearly every online transaction today relies on an RSA cryptosystem that hinges on the intractability of the factoring problem to classical algorithms.

- WHAT IS QUANTUM COMPUTING?

Quantum and classical computers both try to solve problems, but the way they manipulate data to get answers is fundamentally different. This section provides an explanation of what makes quantum computers unique by introducing two principles of quantum mechanics crucial for their operation, superposition and entanglement.

Superposition is the counterintuitive ability of a quantum object, like an electron, to simultaneously exist in multiple “states.” With an electron, one of these states may be the lowest energy level in an atom while another may be the first excited level. If an electron is prepared in a superposition of these two states it has some probability of being in the lower state and some probability of being in the upper. A measurement will destroy this superposition, and only then can it be said that it is in the lower or upper state.

Understanding superposition makes it possible to understand the basic component of information in quantum computing, the qubit. In classical computing, bits are transistors that can be off or on, corresponding to the states 0 and 1. In qubits such as electrons, 0 and 1 simply correspond to states like the lower and upper energy levels discussed above. Qubits are distinguished from classical bits, which must always be in the 0 or 1 state, by their ability to be in superpositions with varying probabilities that can be manipulated by quantum operations during computations.

Entanglement is a phenomenon in which quantum entities are created and/or manipulated such that none of them can be described without referencing the others. Individual identities are lost. This concept is exceedingly difficult to conceptualize when one considers how entanglement can persist over long distances. A measurement on one member of an entangled pair will immediately determine measurements on its partner, making it appear as if information can travel faster than the speed of light. This apparent action at a distance was so disturbing that even Einstein dubbed it “spooky” ( Born 1971 , p. 158).

The popular press often writes that quantum computers obtain their speedup by trying every possible answer to a problem in parallel. In reality a quantum computer leverages entanglement between qubits and the probabilities associated with superpositions to carry out a series of operations (a quantum algorithm) such that certain probabilities are enhanced (i.e., those of the right answers) and others depressed, even to zero (i.e., those of the wrong answers). When a measurement is made at the end of a computation, the probability of measuring the correct answer should be maximized. The way quantum computers leverage probabilities and entanglement is what makes them so different from classical computers.

- WHY DO WE WANT IT?

The promise of developing a quantum computer sophisticated enough to execute Shor's algorithm for large numbers has been a primary motivator for advancing the field of quantum computation. To develop a broader view of quantum computers, however, it is important to understand that they will likely deliver tremendous speed-ups for only specific types of problems. Researchers are working to both understand which problems are suited for quantum speed-ups and develop algorithms to demonstrate them. In general, it is believed that quantum computers will help immensely with problems related to optimization, which play key roles in everything from defense to financial trading.

Multiple additional applications for qubit systems that are not related to computing or simulation also exist and are active areas of research, but they are beyond the scope of this overview. Two of the most prominent areas are (1) quantum sensing and metrology, which leverage the extreme sensitivity of qubits to the environment to realize sensing beyond the classical shot noise limit, and (2) quantum networks and communications, which may lead to revolutionary ways to share information.

- HOW ARE WE TRYING TO GET IT?

Building quantum computers is incredibly difficult. Many candidate qubit systems exist on the scale of single atoms, and the physicists, engineers, and materials scientists who are trying to execute quantum operations on these systems constantly deal with two competing requirements. First, qubits need to be protected from the environment because it can destroy the delicate quantum states needed for computation. The longer a qubit survives in its desired state the longer its “coherence time.” From this perspective, isolation is prized. Second, however, for algorithm execution qubits need to be entangled, shuffled around physical architectures, and controllable on demand. The better these operations can be carried out the higher their “fidelity.” Balancing the required isolation and interaction is difficult, but after decades of research a few systems are emerging as top candidates for large-scale quantum information processing.

Superconducting systems, trapped atomic ions, and semiconductors are some of the leading platforms for building a quantum computer. Each has advantages and disadvantages related to coherence, fidelity, and ultimate scalability to large systems. It is clear, however, that all of these platforms will need some type of error correction protocols to be robust enough to carry out meaningful calculations, and how to design and implement these protocols is itself a large area of research. For an overview of quantum computing, with more detail regarding experimental implementations, see Ladd et al. (2010) .

In this article, “quantum computing” has so far been used as a blanket term describing all computations that utilize quantum phenomena. There are actually multiple types of operational frameworks. Logical, gate-based quantum computing is probably the best recognized. In it, qubits are prepared in initial states and then subject to a series of “gate operations,” like current or laser pulses depending on qubit type. Through these gates the qubits are put in superpositions, entangled, and subjected to logic operations like the AND, OR, and NOT gates of traditional computation. The qubits are then measured and a result obtained.

Another framework is measurement-based computation, in which highly entangled qubits serve as the starting point. Then, instead of performing manipulation operations on qubits, single qubit measurements are performed, leaving the targeted single qubit in a definitive state. Based on the result, further measurements are carried out on other qubits and eventually an answer is reached.

A third framework is topological computation, in which qubits and operations are based on quasiparticles and their braiding operations. While nascent implementations of the components of topological quantum computers have yet to be demonstrated, the approach is attractive because these systems are theoretically protected against noise, which destroys the coherence of other qubits.

Finally, there are the analog quantum computers or quantum simulators envisioned by Feynman. Quantum simulators can be thought of as special purpose quantum computers that can be programmed to model quantum systems. With this ability they can target questions such as how high-temperature superconductors work, or how certain chemicals react, or how to design materials with certain properties.

- CONCLUSIONS AND OUTLOOK

Quantum computers have the potential to revolutionize computation by making certain types of classically intractable problems solvable. While no quantum computer is yet sophisticated enough to carry out calculations that a classical computer can't, great progress is under way. A few large companies and small start-ups now have functioning non-error-corrected quantum computers composed of several tens of qubits, and some of these are even accessible to the public through the cloud. Additionally, quantum simulators are making strides in fields varying from molecular energetics to many-body physics.

As small systems come online a field focused on near-term applications of quantum computers is starting to burgeon. This progress may make it possible to actualize some of the benefits and insights of quantum computation long before the quest for a large-scale, error-corrected quantum computer is complete.

- Born M. The Born-Einstein Letters. London: Walker; 1971.
- Feynman RP. Simulating physics with computers. International Journal of Theoretical Physics. 1982; 21 (6-7):467–488.
- Ladd TD, Jelezko F, Laflamme R, Nakamura Y, Monroe C, O'Brien JL. Quantum computers. Nature. 2010; 464 (7285):45–53. [ PubMed : 20203602 ]
- Shor PW. Algorithms for quantum computation: Discrete logarithms and factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science; 1994. pp. 124–134. https://arxiv .org/abs/quant-ph/9508027 .
- Cite this Page GAMBLE S. Quantum Computing: What It Is, Why We Want It, and How We're Trying to Get It. In: National Academy of Engineering. Frontiers of Engineering: Reports on Leading-Edge Engineering from the 2018 Symposium. Washington (DC): National Academies Press (US); 2019 Jan 28.
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## Quantum computers in 2023: how they work, what they do, and where they’re heading

Senior Lecturer, UTS Chancellor's Postdoctoral Research and ARC DECRA Fellow, University of Technology Sydney

## Disclosure statement

Christopher Ferrie receives funding from the Australian Research Council. He is a co-founder of quantum startup Eigensystems.

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In June, an IBM computing executive claimed quantum computers were entering the “utility” phase , in which high-tech experimental devices become useful. In September, Australia’s Chief Scientist Cathy Foley went so far as to declare “ the dawn of the quantum era ”.

This week, Australian physicist Michelle Simmons won the nation’s top science award for her work on developing silicon-based quantum computers.

Obviously, quantum computers are having a moment. But – to step back a little – what exactly are they?

## What is a quantum computer?

One way to think about computers is in terms of the kinds of numbers they work with.

The digital computers we use every day rely on whole numbers (or integers ), representing information as strings of zeroes and ones which they rearrange according to complicated rules. There are also analogue computers, which represent information as continuously varying numbers (or real numbers ), manipulated via electrical circuits or spinning rotors or moving fluids.

Read more: There's a way to turn almost any object into a computer – and it could cause shockwaves in AI

In the 16th century, the Italian mathematician Girolamo Cardano invented another kind of number called complex numbers to solve seemingly impossible tasks such as finding the square root of a negative number. In the 20th century, with the advent of quantum physics, it turned out complex numbers also naturally describe the fine details of light and matter.

In the 1990s, physics and computer science collided when it was discovered that some problems could be solved much faster with algorithms that work directly with complex numbers as encoded in quantum physics.

The next logical step was to build devices that work with light and matter to do those calculations for us automatically. This was the birth of quantum computing.

## Why does quantum computing matter?

We usually think of the things our computers do in terms that mean something to us — balance my spreadsheet, transmit my live video, find my ride to the airport. However, all of these are ultimately computational problems, phrased in mathematical language.

As quantum computing is still a nascent field, most of the problems we know quantum computers will solve are phrased in abstract mathematics. Some of these will have “real world” applications we can’t yet foresee, but others will find a more immediate impact.

One early application will be cryptography. Quantum computers will be able to crack today’s internet encryption algorithms, so we will need quantum-resistant cryptographic technology. Provably secure cryptography and a fully quantum internet would use quantum computing technology.

In materials science, quantum computers will be able to simulate molecular structures at the atomic scale, making it faster and easier to discover new and interesting materials. This may have significant applications in batteries, pharmaceuticals, fertilisers and other chemistry-based domains.

Quantum computers will also speed up many difficult optimisation problems, where we want to find the “best” way to do something. This will allow us to tackle larger-scale problems in areas such as logistics, finance, and weather forecasting.

Machine learning is another area where quantum computers may accelerate progress. This could happen indirectly, by speeding up subroutines in digital computers, or directly if quantum computers can be reimagined as learning machines.

## What is the current landscape?

In 2023, quantum computing is moving out of the basement laboratories of university physics departments and into industrial research and development facilities. The move is backed by the chequebooks of multinational corporations and venture capitalists.

Contemporary quantum computing prototypes – built by IBM , Google , IonQ , Rigetti and others – are still some way from perfection.

Read more: Error correcting the things that go wrong at the quantum computing scale

Today’s machines are of modest size and susceptible to errors, in what has been called the “ noisy intermediate-scale quantum ” phase of development. The delicate nature of tiny quantum systems means they are prone to many sources of error, and correcting these errors is a major technical hurdle.

The holy grail is a large-scale quantum computer which can correct its own errors. A whole ecosystem of research factions and commercial enterprises are pursuing this goal via diverse technological approaches.

## Superconductors, ions, silicon, photons

The current leading approach uses loops of electric current inside superconducting circuits to store and manipulate information. This is the technology adopted by Google , IBM , Rigetti and others.

Another method, the “trapped ion” technology, works with groups of electrically charged atomic particles, using the inherent stability of the particles to reduce errors. This approach has been spearheaded by IonQ and Honeywell .

A third route of exploration is to confine electrons within tiny particles of semiconductor material, which could then be melded into the well-established silicon technology of classical computing. Silicon Quantum Computing is pursuing this angle.

Yet another direction is to use individual particles of light (photons), which can be manipulated with high fidelity. A company called PsiQuantum is designing intricate “guided light” circuits to perform quantum computations.

There is no clear winner yet from among these technologies, and it may well be a hybrid approach that ultimately prevails.

## Where will the quantum future take us?

Attempting to forecast the future of quantum computing today is akin to predicting flying cars and ending up with cameras in our phones instead. Nevertheless, there are a few milestones that many researchers would agree are likely to be reached in the next decade.

Better error correction is a big one. We expect to see a transition from the era of noisy devices to small devices that can sustain computation through active error correction.

Another is the advent of post-quantum cryptography. This means the establishment and adoption of cryptographic standards that can’t easily be broken by quantum computers.

Read more: Quantum computers threaten our whole cybersecurity infrastructure: here's how scientists can bulletproof it

Commercial spin-offs of technology such as quantum sensing are also on the horizon.

The demonstration of a genuine “quantum advantage” will also be a likely development. This means a compelling application where a quantum device is unarguably superior to the digital alternative.

And a stretch goal for the coming decade is the creation of a large-scale quantum computer free of errors (with active error correction).

When this has been achieved, we can be confident the 21st century will be the “quantum era”.

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## Understanding quantum computing

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Quantum computing holds the promise of solving some of our planet's biggest challenges - in the areas of environment, agriculture, health, energy, climate, materials science, and more. For some of these problems, classical computing is increasingly challenged as the size of the system grows. When designed to scale, quantum systems will likely have capabilities that exceed those of today's most powerful supercomputers. As the global community of quantum researchers, scientists, engineers, and business leaders collaborate to advance the quantum ecosystem, we expect to see quantum impact accelerate across every industry.

Azure Quantum now the ability to mix classical and quantum computation and unlock a new generation of hybrid algorithms, bringing research and experimentation with the current generation of quantum computers into a new and exciting phase. The Integrated hybrid model allows you to write your quantum program to make real-time decisions based on mid-circuit measurements while the qubits remain alive. For more information, see Hybrid quantum computing .

For more information about the beginnings and motivation of quantum computing, see quantum computing history and background .

Azure Quantum is an open ecosystem to build quantum computing solutions on a diverse selection of today’s quantum hardware, and it offers flexibility to use your preferred development tools with support for Cirq, Qiskit, and Q#. You can use the familiar and trusted Azure platform to learn how to develop quantum algorithms and how to program and run them on real hardware from multiple providers.

If you want to accelerate your quantum computing journey, check out the Copilot in Azure Quantum , a unique feature of the Azure Quantum website . With the Copilot you can run your Q# programs, generate new Q# code from your prompts, and ask any questions about quantum computing.

The Azure Quantum website offers the Copilot in Azure Quantum, an AI that can help you create and run quantum programs, as well as chat with you about quantum concepts. You can also find blogs, videos, and articles to learn more about quantum computing and the Azure Quantum service.

Learn how to create an Azure Quantum workspace and start submitting your quantum programs on real quantum hardware. First-time users automatically get free Azure Quantum Credits for use with each participating quantum hardware provider (500 USD each) when creating your workspace. If you need more credits, you can apply to the Azure Quantum Credits program .

Free trial. If you don’t have an Azure subscription, you can create an Azure free account (check out free Azure accounts for students ).

## What can quantum computing and Azure Quantum be used for?

A quantum computer isn't a supercomputer that can do everything faster. In fact, one of the goals of quantum computing research is to study which problems can be solved by a quantum computer faster than a classical computer and how large the speedup can be.

Quantum computers do exceptionally well with problems that require calculating a large number of possible combinations. These types of problems can be found in many areas, such as quantum simulation, cryptography, quantum machine learning, and search problems.

For the latest information about Microsoft's quantum computing research, see the Microsoft Research Quantum Computing page.

## Resource estimation

The quantum computers available today are enabling interesting experimentation and research but they are unable to accelerate computations necessary to solve real-world problems. While the industry awaits hardware advances, quantum software innovators are eager to make progress and prepare for a quantum future. Creating algorithms today that will eventually run on tomorrow's fault-tolerant scaled quantum computers is a daunting task. These innovators are faced with questions such as what hardware resources are required? How many physical and logical qubits are needed and what type? How long is the run time?

You can use the Azure Quantum Resource Estimator to help answer these questions. As a result, you'll be able to refine your algorithms and build solutions that take advantage of scaled quantum computers when they become available.

To get started, see Run your first resource estimate .

Learn more about assessing requirements to scale to practical quantum advantage using the Azure Quantum Resource Estimator in arXiv:2211.07629 .

## Quantum simulation

Quantum mechanics is the underlying "operating system" of our universe. It describes how the fundamental building blocks of nature behave. Nature's behaviors, such as chemical reactions, biological reactions, and material formations, often involve many-body quantum interactions. For simulating intrinsically quantum mechanical systems, such as molecules, quantum computing is promising, because qubits (quantum bits) can be used to represent the natural states in question. Examples of quantum systems that we can model include photosynthesis, superconductivity, and complex molecular formations.

The Quantum Development Kit (QDK) comes with the quantum chemistry library to simulate electronic structure problems and quantum dynamics on a quantum computer. An example of such simulations is the simple molecular energy estimation of the ground state of a molecule . This and more QDK and Azure Quantum samples can be found in the code samples .

Azure Quantum Elements is purpose-built to accelerate scientific discovery. Reinvent your research and development productivity with simulation workflows optimized for scaling on Azure High-Performance Computing (HPC) clusters, AI-accelerated computing, augmented reasoning using AI, integration with quantum tools to start experimenting with existing quantum hardware, and access in the future to Microsoft’s quantum supercomputer. For more information, see Unlocking the power of Azure for Molecular Dynamics .

## Quantum speedups

One of the goals of quantum computing research is to study which problems can be solved by a quantum computer faster than a classical computer and how large the speedup can be. Two well-known examples are Grover's algorithm and Shor's algorithm, which yield a polynomial and an exponential speedup, respectively, over their classical counterparts.

Shor's algorithm running on a quantum computer could break classical cryptographic schemes such as the Rivest–Shamir–Adleman (RSA) scheme, which is widely used in e-commerce for secure data transmission. This scheme is based on the practical difficulty of factoring prime numbers by using classical algorithms. Quantum cryptography promises information security by harnessing basic physics rather than complexity assumptions.

Like Shor's algorithm for factoring, the hidden shift problem is a natural source of problems for which a quantum computer has an exponential advantage over the best known classical algorithms. This may eventually help in solving deconvolution problems and enable us to efficiently find patterns in complex data sets. It turns out that a quantum computer can in principle compute convolutions at high speed, which in turn is based on the quantum computer's ability to compute Fourier transforms extremely rapidly. In the sample gallery of your Azure Quantum workspace you will find a Hidden Shifts Jupyter notebook sample (an Azure account is required).

Grover's algorithm speeds up the solution to unstructured data searches, running the search in fewer steps than any classical algorithm could. Indeed, any problem that allows you to check whether a given value $x$ is a valid solution (a "yes or no problem") can be formulated in terms of the search problem. The following are some examples:

- Boolean satisfiability problem: Is the set of Boolean values $x$ an interpretation (an assignment of values to variables) that satisfies the given Boolean formula?
- Traveling salesman problem: Does $x$ describe the shortest possible loop that connects all cities?
- Database search problem: Does the database table contain a record $x$?
- Integer factorization problem: Is the fixed number $N$ divisible by the number $x$?

For a practical implementation of Grover's algorithm to solve mathematical problems, take a look at the Grover's Search Jupyter notebook in the Sample gallery of your Azure Quantum workspace (an Azure account is required). For more information on setting up a workspace, see Create an Azure Quantum workspace . For a more in-depth examination of Grover's algorithm, see the tutorial Implement Grover's search algorithm in Q# .

For more quantum algorithm samples, see the code samples .

## Quantum machine learning

Machine learning on classical computers is revolutionizing the world of science and business. However, the high computational cost of training the models hinders the development and scope of the field. The area of quantum machine learning explores how to devise and implement quantum software that enables machine learning that runs faster than classical computers.

The Quantum Development Kit (QDK) comes with the quantum machine learning library that gives you the ability to run hybrid quantum-classical machine learning experiments. The library includes samples and tutorials, and provides the necessary tools to implement a new hybrid quantum–classical algorithm, the circuit-centric quantum classifier, to solve supervised classification problems.

## How does quantum computing solve problems?

Quantum computers are controllable quantum mechanical devices that exploit the properties of quantum physics to perform computations. For some computational tasks, quantum computing provides exponential speedups. These speedups are possible thanks to three phenomena from quantum mechanics: superposition, interference, and entanglement.

## Superposition

Imagine that you are exercising in your living room. You turn all the way to your left and then all the way to your right. Now turn to your left and your right at the same time. You can’t do it (not without splitting yourself in two, at least). Obviously, you can’t be in both of those states at once – you can’t be facing left and facing right at the same time.

However, if you are a quantum particle, then you can have a certain probability of facing left AND a certain probability of facing right due to a phenomenon known as superposition (also known as coherence ).

Just as bits are the fundamental units of information in classical computing, qubits are the fundamental units of information in quantum computing. While a bit, or binary digit, can have a value either 0 or 1, a qubit can have a value that is either 0, 1 or a quantum superposition of 0 and 1.

Unlike classical particles, if two states $A$ and $B$ are valid quantum states of a quantum particle, then any linear combination of the states is also a valid quantum state: $\text{qubit state}=\alpha A + \beta B$. This linear combination of quantum states $A$ and $B$ is called superposition. Here, $\alpha$ and $\beta$ are the probability amplitudes of $A$ and $B$, respectively, such that $|\alpha|^{2} + |\beta|^{2} = 1$.

Only quantum systems like ions, electrons or superconducting circuits can exist in the superposition states that enable the power of quantum computing. A quantum particle such as an electron has its own “facing left or facing right” property, namely spin , referred to as either up or down, so the quantum state of an electron is a superposition of "spin up" and "spin down".

Generally, and to make it more relatable to classical binary computing, if a quantum system can be in two quantum states, these states are referred as 0 state and 1 state.

## Qubits and probability

Classical computers store and process information in bits, which can have a state of either 1 or 0, but never both. The equivalent in quantum computing is the qubit . A qubit is any quantum system that can be in a superposition of two quantum states, 0 and 1. Each possible quantum state has an associated probability amplitude. Only after measuring a qubit, its state collapses to either the 0 state or the 1 state depending on the associated probability, thus, one of the possible states is obtained with a certain probability.

The qubit's probability of collapsing one way or the other is determined by quantum interference . Quantum interference affects the state of a qubit in order to influence the probability of a certain outcome during measurement, and this probabilistic state is where the power of quantum computing excels.

For example, with two bits in a classical computer, each bit can store 1 or 0, so together you can store four possible values – 00 , 01 , 10 , and 11 – but only one of those at a time. With two qubits in superposition, however, each qubit can be 1 or 0 or both , so you can represent the same four values simultaneously. With three qubits, you can represent eight values, with four qubits, you can represent 16 values, and so on.

For more information, see The qubit in quantum computing .

## Entanglement

One of the most interesting phenomenon of quantum mechanics is the ability of two or more quantum systems to become entangled with each other. Entanglement is a quantum correlation between quantum systems. When qubits become entangled, they form a global system such that the quantum state of individual subsystems cannot be described independently. Two systems are entangled when the state of the global system cannot be written as a combination of the state of the subsystems, in particular, two systems are entangled when the state of the global system cannot be written as the tensor product of states of the subsystems. A product state contains no correlations.

Entangled quantum systems maintain this correlation even when separated over large distances. This means that whatever operation or process you apply to one subsystem correlates to the other subsystem as well. Because there is a correlation between the entangled qubits, measuring the state of one qubit provides information about the state of the other qubit – this particular property is very helpful in quantum computing.

Not every correlation between the measurements of two qubits means that the two qubits are entangled. Besides quantum correlations, there exist also classical correlations. The difference between classical and quantum correlations is subtle, but it's essential for the speedup provided by quantum computers. For more information, see Understanding classical correlations .

If you want to learn more, see the tutorial Exploring quantum entanglement with Q# .

## Quantum computers vs quantum simulators

A quantum computer is a machine that combines the power of classical and quantum computing. The current quantum computers correspond to a hybrid model: a classical computer that controls a quantum processor.

The development of quantum computers is still in its infancy. Quantum hardware is expensive and most systems are located in universities and research labs. Where classical computers use familiar silicon-based chips, quantum computers use quantum systems such as atoms, ions, photons, or electrons. The technology is advancing, though, and limited public cloud access to quantum systems is available.

Azure Quantum allows you to create quantum algorithms for multiple platforms at once, while preserving flexibility to tune the same algorithms for specific systems. You can pick from many programming languages such as Qiskit, Cirq, and Q# and run your algorithms on multiple quantum systems. On Azure Quantum, it’s easy to simultaneously explore today’s quantum systems and be ready for the scaled quantum systems of the future.

First-time users automatically get free $500 (USD) Azure Quantum Credits for use with each participating quantum hardware provider. If you have consumed all the credits and you need more, you can apply to the Azure Quantum Credits program .

## Azure Quantum hardware

A quantum computer has three primary parts:

- A device that houses the qubits
- A method for performing quantum operations (also known as quantum gates) on the qubits and measuring them
- A classical computer to run a program and send instructions

Qubits are fragile and highly sensitive to environmental interference. For some methods of qubit storage, the unit that houses the qubits is kept at a temperature just above absolute zero to maximize their coherence. Other types of qubit housing use a vacuum chamber to help minimize vibrations and stabilize the qubits. Operations can be performed using a variety of methods including microwaves, laser, and voltage, depending on the type of qubit.

Quantum computers face a multitude of challenges to operate correctly. Error correction in quantum computers is a significant issue, and scaling up (adding more qubits) increases the error rate. Because of these limitations, a quantum PC for your desktop is far in the future, but a commercially-viable lab-based quantum computer is closer.

Microsoft is partnering with quantum hardware companies to provide cloud access to quantum hardware. With the Azure Quantum platform and the QDK, you can explore and run quantum programs on different types of quantum hardware. These are the currently available quantum targets:

- Quantinuum : Trapped-ion system with high-fidelity, fully connected qubits, and the ability to perform mid-circuit measurements.
- IonQ : Dynamically reconfigurable trapped-ion quantum computers for up to 23 fully connected qubits, that let you run a two-qubit gate between any pair.
- Rigetti : Gate-based superconducting processors that utilize Quantum Intermediate Representation (QIR) to enable low latency and parallel execution.

For more information, see the full quantum computing target list .

## Azure Quantum simulators

For the moment, the use of real quantum hardware is limited due to resources and budget. Quantum simulators serve to the purpose of running quantum algorithms, making it easy to test and debug an algorithm and then run it on real hardware with confidence that the result will match the expectations.

Quantum simulators are software programs that run on classical computers and make it possible to run and test quantum programs in an environment that predicts how qubits react to different operations, making it easy to test and debug an algorithm and then run it on real hardware with confidence that the result will match the expectations.

The Quantum Development Kit (QDK) includes different classes of quantum simulators representing different ways of simulating the same quantum algorithm, such as a sparse simulator for simulating large systems, a noise simulator for simulating quantum algorithms under the presence of noise. For more information, see in-memory simulators .

In addition, each of Microsoft's quantum hardware partners offers a quantum simulator as a target.

If you have an Azure account, in the sample gallery of your Azure Quantum workspace you will find several Jupyter Notebook samples that use quantum simulators. See Get started with Q# and an Azure Quantum notebook .

- Explore Azure Quantum
- Quantum computing history and background
- What are the Q# programming language and Quantum Development Kit (QDK)?
- Set up the Quantum Development Kit
- Tutorial: Implement a quantum random number generator in Q#
- In-memory quantum simulators
- The Q# libraries

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- Published: 31 October 2023

## An optimization case study for solving a transport robot scheduling problem on quantum-hybrid and quantum-inspired hardware

- Dominik Leib 1 ,
- Tobias Seidel 1 ,
- Sven Jäger 1 ,
- Raoul Heese 1 ,
- Caitlin Jones 2 ,
- Abhishek Awasthi 2 ,
- Astrid Niederle 3 &
- Michael Bortz 1

Scientific Reports volume 13 , Article number: 18743 ( 2023 ) Cite this article

Metrics details

- Applied mathematics
- Mathematics and computing
- Quantum information

We present a comprehensive case study comparing the performance of D-Waves’ quantum-classical hybrid framework, Fujitsu’s quantum-inspired digital annealer, and Gurobi’s state-of-the-art classical solver in solving a transport robot scheduling problem. This problem originates from an industrially relevant real-world scenario. We provide three different models for our problem following different design philosophies. In our benchmark, we focus on the solution quality and end-to-end runtime of different model and solver combinations. We find promising results for the digital annealer and some opportunities for the hybrid quantum annealer in direct comparison with Gurobi. Our study provides insights into the workflow for solving an application-oriented optimization problem with different strategies, and can be useful for evaluating the strengths and weaknesses of different approaches.

## Introduction

Quantum computing (QC) is a field that has witnessed a rapid increase in interest and development over the past few decades since it was theoretically shown that quantum computers can provide an exponential speedup for certain tasks 1 , 2 , 3 . Translating this potential into a practically relevant quantum advantage, however, has proven to be a very challenging endeavor. Nevertheless, the emerging field is considered to have a highly disruptive potential for many domains, for example in machine learning 4 , chemical simulations 5 and optimization 6 , the domain of this work. Due to the fact that optimization problems are of utmost importance also for industrial applications, we investigated a potential advantage of quantum and quantum-inspired technology for the so-called transport robot scheduling problem (TRSP), a real-world use-case in optimization that is derived from an industrial application of an automatized robot in a high-throughput laboratory. The optimization task is to plan a time-efficient schedule for the robot’s movements as it transports chemical samples between a rack and multiple machines to conduct experiments. This is an NP-hard problem which for certain instances can be challenging to solve using classical computing techniques, and hence is an attractive candidate to search for an advantage with non-classical techniques.

In our study, we compared the solution quality and runtime of different solvers on a large set of instances of the problem. As solvers, we considered D-Wave’s hybrid Leap framework (LBQM) that makes use of the D-Wave quantum annealer 7 , Fujitsu’s digital annealer (FDA) 8 , Fujitsu’s digital annealer hybrid framework (FDAh), as well as the industry-grade Gurobi solver 9 . As a key element of this work, we provide three different models for the TRSP that follow different design philosophies. This is justified by the different ways in which the problem task can be modelled and the inherent differences in the problem formulations that the solvers addressed can accept. LBQM, FDA and FDAh are restricted to a formulation as a quadratic unconstrained binary optimization (QUBO), whereas a mixed integer program (MIP) with integer and float variables can be used by Gurobi, which makes a comparison of multiple formulations meaningful.

The TRSP considered in this paper is a special combination of different scheduling problems that, to our knowledge, has not been considered before. Scheduling problems have been studied intensively for several decades and classical algorithms exist for numerous variants 10 , 11 . Since most of the industry-relevant scheduling problems are NP-hard, these classical algorithms mainly consist of meta-heuristics or use general-purpose MIP solvers, which basically solve the problem using a branch and bound approach with several additional improvements like cutting planes. In addition to classical algorithmic developments, a considerable amount of research has also been done in hardware-based parallel computing, especially in general purpose computation on graphics processing unit GPGPU parallelization 12 , 13 . The problem discussed in this work is an extension of the typical job shop scheduling problem JSSP, where the inclusion of a robot adds additional restrictions. More specifically, the studied scheduling problem falls into the category of robotic cell scheduling and automated guided vehicles AGV scheduling problems. Most work on robotic cell scheduling deals with infinite cyclic schedules 14 . This comprises polynomial-time algorithms and hardness results 15 , MIP techniques 16 , 17 , 18 and heuristic approaches 19 . Many efficiently solvable and hard special cases have been identified 20 and heuristics have been proposed for some of the hard cases 21 . Those problems differ from our use case in one way or another. The problems considered by the above-cited papers allow, unlike our use case, that the jobs can wait at a machine after their completion before being picked up by the robot. Robotic cell scheduling problems without this possibility have been studied by Ref. 22 , 23 , whose problems differ from our, among others, in the considered objective function. Our objective function, the total job completion time, has been extensively studied for flow shop scheduling problems without a robot 11 , 24 , 25 , 26 , the latter of which shows that the no-wait variant is strongly NP-hard on two machines. Apart from the no-wait constraint, the problem considered in our work is characterized by the fact that jobs have to go to the last machine several times. Such settings are known as a re-entrant flow shops, for which Ref. 27 developed a heuristic algorithm.

We are mainly interested in the performance of non-standard solution approaches using quantum or quantum-inspired solvers in this study. Because these solvers rely on heuristics, benchmarks for real-world applications are a highly relevant research topic. Most quantum optimization approaches fall into two major groups, one for gate-based hardware and one for annealing-based hardware 28 . The majority of gate-based approaches to optimization use parameterized gates to find the ground state of a Hamiltonian related to the cost function of the optimization problem in a quantum-classical hybrid fashion, for example via the quantum approximate optimization algorithm (QAOA) 29 , 30 .

Approaches based on quantum annealing also seek to find the ground state of a Hamiltonian, but by aiming for an adiabatic change from an initial state that can be easily prepared. In contrast to actual quantum computing devices, other classical software and hardware components are merely inspired by quantum computing, for example FDA 31 and Toshiba’s Simulated Bifurcation (TSB) 32 . Typically, optimization tasks for quantum solvers and the aforementioned quantum-inspired technologies are modeled as QUBO problems 33 . An in-depth analysis of pure QUBO comparison on four quantum and quantum-inspired solvers can be found in Ref. 34 . In their work, the authors compare the solutions of a library of quadratic benchmark problems on the D-Wave quantum annealer, FDA, and TSB against each other.

QC has already been successfully used for optimization in various fields. For example, in Ref. 35 , chemical reaction networks are optimized with quantum computing. In Ref. 36 , it is shown that using the QAOA, it is possible to beat some classical heuristic algorithms on the binary paint shop problem. However, some work has shown that the current circuit model algorithms are not always adequate enough to reach significant convergence required for a good solution 37 . Quantum annealing has proven to offer some advantage against the classical simulated annealing algorithm for a spin-glass problem, using D-Wave hardware 38 , but this is no conclusive evidence. In one of the more recent works on quantum annealing 39 , the authors suggest a nature inspired hybrid quantum algorithm for robot trajectory optimization for PVC sealing in a real industrial setting. In Ref. 40 , the authors present a solution to the maximum independent set (MIS) problem using a Rydberg atom device, along with a claim of a possible super-linear quantum speed-up against classical simulated annealing. Other classical algorithms might still be superior to a quantum approach on current devices 41 . Several works consider scheduling problems 42 , 43 , 44 . In Ref. 45 , an AGV transportation problem using different classical and quantum approaches is studied and Ref. 46 investigates a nurse scheduling problem with the usage of a quantum annealer.

The remaining manuscript is structured as follows. We provide a detailed description of the TRSP and its mathematical modeling in Sect. "Transport robot scheduling problem" . In Sect. "Benchmark setup" , we describe the design of our numerical study and list the problem instances and solvers that we use. The results of this study are presented in Sect. "Benchmark results" . Finally, we conclude our study in Sect. "Conclusion and outlook" . Detailed model descriptions, solver information, further information on the benchmark setup and instance lists are contained in the supplementary material (referenced by a preceding “S” to the label it is referring to).

## Transport robot scheduling problem

In this section, we present a detailed explanation of the TRSP, which is a real-world use case derived from one of BASF’s high-throughput laboratories. This optimization problem is about finding the most time-efficient route of a transport robot tasked with moving chemical samples from one processing machine to another. In the following, we first provide a general description of the problem setup and then present different modeling approaches. These models build the foundation of the subsequent benchmarks.

## Problem description

The laboratory we are modeling consists of a sample rack and three different processing machines: a water mixer , a sample shaker and a photo booth . And, finally, the robot itself that is tasked with carrying chemical samples from one place to another with the goal to conduct chemical experiments. Only the experimental plan (i. e., how each sample has to be processed in the laboratory) is predefined in advance, but not the specific order of the experiments. Initially, a certain number of samples is stored on the rack. Each of these samples needs to be first taken to the water mixer, then to the sample shaker. Once the sample shaking is completed, one or more photos have to be taken of each sample at the photo booth. Consecutive photos need to be taken after specific (i. e., predefined) time intervals, where the first photo of each sample has to be taken immediately after the shaking process. Finally, each sample has to be brought back to the rack. The processing times for different samples on the same machine can be different as specified by the experimental plan. We assume that each machine can only hold (and process) one single sample at any given time, or remain idle, and the processing steps cannot be interrupted before their completion. It is required that a machine starts processing a sample as soon as the sample is brought by the robot. Moreover, we assume that a sample has to be moved by the robot in-between two processing steps. Hence, a sample has to be lifted from a machine (and the machine is made available) as soon as it finishes processing.

By definition, the robot requires exactly one time unit to move from any place to any other, with or without a sample, and picking up or dropping a sample does not require extra time. Like the machines, the robot can transport only a single sample at any given time or drive empty or remain idle. In particular, it is not possible that the robot places a sample at a machine and picks up another at the same time.

The objective of this scheduling task is to minimize the sum of sample completion times , i. e., the sum of the times when the samples arrive at the the rack after their last photo has been taken. The solution of this optimization problem is a sequence of tasks for the robot that yields an efficient laboratory operation.

## Mathematical modeling

In our benchmark, we test three modeling approaches against each other. On the quantum and quantum-inspired side we consider a QUBO formulation, whereas on the classical side we use two MIP formulations. First, a so-called sequence model and second, a so-called time-indexed model . In the following, we first introduce the common terminology for all modeling approaches. Next, we shortly sketch the main features of each model. For a more detailed description, we refer to Section S1. The motivation for the development of multiple models is to carry out a comparison between the solutions obtained by the most suitable problem encoding for quantum and classical solvers. This ensures that we are comparing the best of both worlds (classical and quantum), and do not restrict ourselves to a model which is more suitable for quantum over classical computing.

## Common terminology

The processing machines are addressed by \(M_1\) for the water mixer, \(M_2\) for the sample shaker and \(M_3\) for the photo booth. The scheduling time is discretized into time slots which all have length of one time unit. The transport robot takes one time unit for each operation that is either transportation or empty traversal between the machines and the rack. In this way, each transport robot scheduling problem is uniquely determined by the number of samples to be scheduled \(N \ge 1\) , the number of photos \(K \ge 1\) , which agrees for each sample \(j \in \{1,\ldots ,N\}\) , the processing times \(p_{j,1},p_{j,2}, p_{j,3} \in \mathbb {N}_{>0}\) for machines \(M_1, M_2\) and \(M_3\) , which can vary for each sample \(j \in \{1,\ldots ,N\}\) and the time gaps \(g_{j, k} \in \mathbb {N}_{\ge 2}\) to be kept between consecutive photos k and \(k+1\) for \(k \in \{1,\ldots ,K-1\}\) , which also can vary for each sample \(j \in \{1,\ldots ,N\}\) . As an example, Fig. 1 provides a feasible schedule in form of a Gantt chart to visualize these parameters.

An example Gantt chart of a robot transport scheduling problem with \(N=2\) samples and \(K=2\) photos.Tasks associated with sample one (two) are colored blue (red). When a sample is processed on one of the machines or carried by the robot in the time-frame \([t,t']\) , a bar is drawn from t to \(t'\) in the respective row in a corresponding color. Empty movements of the robot are not drawn explicitly. For example, at time \(t=13\) the robot is at the rack as sample 1 has been brought to the sample rack from \(t=12\) to 13. It takes one unit of time for the robot to travel from the rack to the water mixer to pick up sample 2 at \(t=14\) . From \(t=22\) to \(t=23\) , the sample is brought from the photo booth to the rack and back, which is a consequence of the assumption that a sample has to be moved by the robot in-between two processing steps. The objective value of the depicted schedule is \(19+26 = 45\) .

A general QUBO reads

for some matrix \(Q \in \mathbb R^{n \times n}\) , where x represents a vector of n binary optimization variables. Two challenging properties of QUBOs must be taken into account in the modeling. Since only binary variables are allowed, this implies that other types of variables must be avoided, i. e. a reformulation into a binary form is necessary. Second, the problem is unconstrained. This restriction can be overcome by using penalty terms , which are quadratic functions in the model variables that evaluate to a positive value when the current assignment of values to the variables leads to an infeasible solution. Typically, the penalty terms are designed to yield 0 if the corresponding solution is feasible, so that they do not contribute to the objective values of feasible solutions. More general information about QUBOs and their properties can be found, e. g. , in Ref. 33 , 47 , 48 .

Our proposed QUBO model for the TRSP is based on the well-known starting time formulation (see e. g. Ref. 43 ) and can be written as

where F is the objective function and \(P_1, \ldots , P_7\) denote the penalty functions and \(\rho _0,\ldots ,\rho _7 \in \mathbb {R}_{>0}\) are tunable parameters that have to be chosen such that the objective and penalty terms are suitably balanced. As in Eq. 1 , n represents the total number of binary optimization variables. These have a distinct meaning that can be identified with three indices. Specifically,

for all \(j \in \{1,\ldots ,N\}\) , \(m \in \{1,2,3\}\) and \(t \in \{1,\ldots ,T-1\}\) . Here, T denotes the time horizon, which is chosen in such a way that there is enough time to schedule all samples sequentially, implying that there is at least one feasible solution. It can be explicitly computed for each instance as described in Section S1.1. In terms of Fig. 1 , one has, for example, \(x_{1,1,1} = 1\) and \(x_{1,2,8} = 1\) .

The penalty terms for the QUBO model have to be formulated using the binary optimization variables. This section only provides an example for such a term, a complete description can be found in Section S1.1. Specifically, we consider here the constraint that each sample must access the machines \(M_1\) and \(M_2\) exactly once, which can be achieved by

This term evaluates to zero if and only if for each pair of sample j and machine \(M_m\) , the variable \(x_{j,m,t}\) is 1 for precisely one time slot t . Since \(P_1\) is bounded below by 0 due to its quadratic nature, each local minimum of \(P_1\) is a feasible solution w.r.t. the rule of machine access to \(M_1\) and \(M_2\) . The other penalty terms can be formulated similarly.

Finally, the objective function F sums up for each sample the time when the sample arrives at the rack after the entire scheduling process (“sum of sample completion times”). For example, the objective function in the case of Fig. 1 evaluates to 45 time units.

MIPs have been used since the late 1950s as a tool for solving scheduling problems. It is not possible to model the disjunctive constraints resulting from the discrete ordering decisions only by means of starting time variables. Different types of binary variables have been proposed to achieve this. The main types are position variables \(x_{ijk}\) indicating if job j is the k th job on machine i 49 , linear ordering variables \(\delta _{ijk}\) deciding if job j is processed before job k on machine i 50 and time-indexed variables \(x_{ijt}\) specifying that job j is started (or processed or completed) on machine i at time t 51 , 52 . Ref. 53 compared these three approaches experimentally for a job shop scheduling problem.

Due to the powerful nature of (mixed) integer programming in contrast to the restrictive nature of the QUBO models, we provide two MIP models to be solved using Gurobi, where we follow two state-of-the-art approaches for formulating scheduling problems as MIPs 11 . The first one, in the following named sequence model , makes use of continuous start time and binary linear ordering variables. The second model, called the time-indexed model , is restricted to a binary formulation comparable to the QUBO model, where we make use of time-indexed variables. The latter provides a model with a natural vicinity to the QUBO formulation whereas the sequence model exploits the features of MIP formulations. In this sense we provide a baseline from two different angles, one for each solution approach.

## MIP: sequence model

In the sequence model, we model sequences of events that affect the behavior of the transport robot with respect to the machines and the photos of a sample. We define the set of events as

An event \(e = (j,i,0)\) represents either that a sample j is placed on machine \(M_i\) for \(i \in \{1,2\}\) or to the \((i-2)\) th photo shoot for \(i > 2\) , an event ( j , i , 1) corresponds to picking it up again. For each event \(e \in E\) we define an optimization variable \(\tau _e \in \mathbb {R}_{\ge 0}\) to model the time for event e to happen. In terms of Fig. 1 , we have, for example, \(\tau _{(1,1,0)} = 1\) and \(\tau _{(1,1,1)}=4\) . A simple formulation can be achieved by additionally introducing a binary variable for each pair \(e,f \in E\) , \(e \ne f\) of events that indicates if e occurs before f . We reduce the size of the model by exploiting the fact that the ordering of some events is fixed or coupled. For example, we do not need a variable that specifies the order in which a given sample is brought to the water mixer and to the sample shaker. This leads to three sets of linear ordering variables that can be found in Section S1.2 , as well as the various constraints to ensure feasibility. The objective function (i. e., the sum of the sample completion times) can be easily expressed using the variables \(\tau _e\) corresponding to events when a sample is picked up from the last photo.

## MIP: time-indexed model

The second constrained model makes use of discrete time-indexed variables similar to the QUBO model from Section S1.1 . In this formulation, we model the behavior of the transport robot by defining certain routes a sample can be transported along, which include those from the rack to all machines and back or movements between subsequent machines. The numbering of the moves is shown in Fig. S1.

As the model name implies, we have, given a discrete time horizon \(T \in \mathbb {N}_{>0}\) , binary variables to model when each sample takes which route as

for all \(j \in \{1,\ldots ,N\}\) , \(r \in \{1,\ldots , 8\}\) and \(t \in \{0,\ldots ,T-1\}\) . In terms of the Gantt chart from Fig. 1 , this would imply \(y_{1,1,0} = 1\) , \(y_{1,2,4} = 1\) , \(y_{2,1,5} =1\) and so on. The time horizon T is defined as for the QUBO model, see Eq. (S2).

The constraints of the model are similar to the penalty terms of the QUBO Model and are listed in Section S1.3. The objective function (i. e., the sum of the sample completion times) is defined in terms of the ancilla optimization variables \(z_j\) for \(j\in \{1,\dots ,N\}\) , that are bounded below by the arrival time of sample j at the rack after the schedule has finished.

## Benchmark setup

In the present section, we describe the design of the benchmark. We start with an outline of the considered problem instances that are listed in more detail in Section S2. Subsequently, we describe the three different commercial technologies that we use.

To set the stage for our benchmark, we specify 260 test instances of our optimization problem of interest, each defined by a different set of parameters. Specifically, each instance is uniquely determined by the number of samples N , the number of photos K , the gaps \(g_{j,k}\) between subsequent photos k and \(k+1\) for \(k \in \{1, \ldots , K-1\}\) and \(j \in \{1, \ldots , N\}\) , and, finally, the processing times \(p_{j,1}, p_{j,2}, p_{j,3}\) of the water mixer, sample shaker and photo booth, respectively, as explained in Sect. "Mathematical modeling" . For the sake of simplicity, the processing time of the photo booth agrees for all samples of the same instance, that is \(p_{j,3}:= p_3\) for all \(j \in \{1,\ldots ,N\}\) .

In Section S2, we describe the algorithm that was used to generate parameter sets for the benchmark instances. Since the resulting instances span a wide range of complexity, we divide the resulting benchmark library into two parts, where each part is defined by the number of binary variables in the corresponding QUBO formulation from Sect. "Mathematical modeling" as explained in Section S1.1 in more detail. The first part, which we call library of minor instances , contains all 161 instances that have at least 2071 and at most 8080 binary variables. The second part, which we call library of major instances , contains the remaining 99 instances with at least 10822 and at most 22692 binary variables. The reason for that specific division is that 8192 is the maximal amount of variables that can be solved directly on Fujitsu’s digital annealer.

We collect groups of instances ( N , K ) that have the same number of samples and photos as shown in Fig. S2, i. e., within those groups the leftover parameters \(p_{j,m}\) and \(g_{j, k}\) for \(j \in \{1,\ldots ,N\}, m \in \{1,2,3\}\) and \(k \in \{1,\ldots ,K-1\}\) may vary. These groups can be understood as a collection of “similar” TRSPs in the sense that the complexity of the tasks to be solved is comparable. However, some instances may still be easier or more difficult to solve than others in practice. This grouping approach allows us to consider statistical metrics over several instances when we compare models and solvers. Moreover, it allows us to estimate the scaling behavior of different solution approaches. In Section S2, we list how many instances each group contains.

## Quantum and classical solvers

In our benchmark, we solve the generated instances with a selection of model and solver combinations with the main goal to assess the performance of quantum and quantum-inspired technology. Specifically, we consider three solver candidates:

Gurobi: As a baseline, we use the branch and bound algorithm of Gurobi, which is a state-of-the-art mathematical programming solver running on classical hardware 9 . In summary, it relies on an implicit enumeration that allows the original problem to be split into smaller sub-problems using a decision tree. The use of lower bounds derived from linear programming (LP) relaxations allows for a reduction of the search space. Gurobi is an all-purpose solver that can in principle solve the proposed optimization problems to a guaranteed optimality in a deterministic fashion (given sufficient time). In this work we utilized the cloud based service of Gurobi solver, which ran on a Intel(R) Xeon(R) Platinum 8275CL CPU (3.00 GHz with 8 physical cores).

D-Wave’s hybrid Leap framework (LBQM): D-Wave provides cloud-based access to their adiabatic quantum computers with over 5000 qubits 7 . By design, their hardware is specifically tailored to solve QUBOs. To this end, the QUBO is encoded in an Hamiltonian such that each optimization variable is represented by one qubit 54 and the ground state corresponds to the optimal solution. The quantum annealing mechanism aims to find the ground state by performing a suitable time evolution of the quantum system with a subsequent measurement of all qubits to reveal the optimal solution. The D-Wave hardware has only limited connectivity, which means that each qubit can only interact with a certain number of other qubits. This limitation restricts the correlations between optimization variables that can be represented by the Hamiltonian. Finding a suitable representation with these constraints is an NP-hard problem 55 that has to be solved classically to configure the quantum annealer for a certain problem. In practice, the quantum annealer can typically only be used for QUBOs with much less than 5000 optimization variables. For this reason, D-Wave also provides a hybrid software framework LBQM, which is a black-box algorithm for binary quadratic models (BQMs) that runs on both classical and quantum annealing hardware. It allows larger optimization problems that are too big for the quantum hardware to be handled by presenting only parts of the original problem to the quantum annealer. However, the exact mode of operation of LBQM is not publicly available. In this study, we use only the quantum annealer in a hybrid fashion via LBQM. The quantum machine used in the hybrid framework is the D-Wave Advantage System 4.1 and the region na-west-1 . We choose to use a constant number of 1000 samples (or readouts) for all evaluations and use default settings for all parameters.

Fujitsu’s digital annealer (FDA) and Fujitsu’s digital annealer hybrid framework FDAh: The digital annealer from Fujitsu can be considered as a quantum-inspired algorithm that runs on dedicated (classical) hardware 31 and can be accessed using a cloud service. It is based on simulated annealing 56 , 57 with two major differences. Firstly, the utilization of an efficient parallel-trial scheme to exploit the parallelization capabilities of the hardware and, secondly, a dynamic escape mechanism to avoid locally optimal solutions. The detailed hardware specifications are confidential. The solver supports QUBOs with up to 8192 variables. In addition, the hybrid solver FDAh is provided to solver bigger problem instances by utilizing both dedicated and classical hardware 8 similar to D-Wave’s LBQM. In this study, we use both FDA and FDAh. Both solvers require a set of parameters that specify how the annealing is done, which also include the number of repetitions and parallel runs on the chip. The specific parameters we used for FDA and FDAh are provided in Section S3.

In a small pre-study, we excluded a few other solvers; see Section S4. The main scope of the paper is to benchmark the performance of quantum-hybrid and quantum-inspired technologies on the TRSP on a high level against an all-purpose solver with an out-of-the-box performance. In this sense, we also exclude meta-heuristics that are tailor-made to the problem as well.

Each instance can be modelled with each of the three modeling approaches from Sect. "Transport robot scheduling problem" . However, not all solvers are applicable to all problem formulations and all instances. The MIP sequence model is solved with Gurobi for all instances. The time-indexed model is solved with Gurobi only for the minor instances. The QUBO model is solved with LBQM and FDA for minor instances. For major instances, the QUBO model is only solved with FDAh.

We call each valid model and solver combination an approach and use a unique name to refer to it. Summarized, we consider Gurobi with the sequence model (SE-GU), Gurobi with the time-indexed model (TI-GU), LBQM with the QUBO model (QU-LBQM), FDA with the QUBO model (QU-FDA) and FDAh with the QUBO model (QU-FDAh). An overview over all approaches is shown in Fig. 2 .

Summary of model (see Section "Mathematical modeling" ) and solver (see Section "Quantum and classical solvers" ) combinations for the benchmarks.

For all problems, we prescribe a runtime limit of 3600 seconds for Gurobi. This limit was determined on a heuristic basis, since initial experiments have shown that Gurobi can solve the considered problem instances on this time scale with a practically relevant quality. This time limit exceeds the runtimes of LBQM, FDA and FDAh by far to provide Gurobi enough time to return solutions that are suitable for a relative comparison (see Fig. 4 ).

Both LBQM and FDA also require a time limit for each run, which scales with the problem size in the QUBO formulation as follows. The time limit for LBQM is set to be \(\min \{100, 1.5 \cdot \frac{n}{100}\}\) seconds, where n is the number of variables in the QUBO formulation for the minor instances. The runtime of the digital annealer is implicitly set with the steps parameter, where each step taken in the annealing process takes a constant amount of time. We set the number of steps to be 1 e 7 for the instances with \({2071} \le n \le {4096}\) , 5 e 7 for the ones with \({4096} < n \le {6000}\) and 1 e 8 for the instances with \({6000} < n \le {8080}\) variables in the QUBO formulation. Lastly the major instances computed with the hybrid framework FDAh based on the digital annealer require a time limit as well. For this we distributed the available time of 5 hours to the instances, correspondingly to their number of variables. This computes approximatively as \(n \cdot {0.0117}\) seconds where n is the number of variables in the QUBO formulation.

The benchmark setup is summarized in Table 1 , where we recall the approaches from Fig. 2 . The table also contains the values of the QUBO parameters \(\rho _0,\ldots , \rho _7\) from Eq. (S16) that were chosen for LBQM, FDA and FDAh, respectively. The choice was made according to previous experiments with smaller problem instances. For this purpose, a typical strategy is to iteratively increase the parameter \(\rho _i\) if the corresponding penalty term \(P_i\) is non-vanishing. Additionally, one needs to make sure that the parameter \(\rho _0\) for the target function is set such that it is not in favor to violate penalty terms and a good optimization is achieved.

Some solutions of the library of minor instances have not been solved to feasibility by LBQM, i. e., the solution vector returned does not translate to a feasible schedule of the TRSP. Those instances can be identified by having an objective value of at least \(10^4\) , which is the minimum of the penalty parameters chosen for the QUBO model according to Table Table 1 . This can be seen as follows: the parameters of the library of minor instances are bounded as \(N \le 9\) , \(K \le 4\) , \(p_{j,3} \le 3, p_{j,1} \le 8, p_{j,2} \le 4, g_{j,1} \le 5, g_{j,2} \le 12\) and \(g_{j,3} \le 24\) for \(j=1,\ldots ,N\) . Using those upper bounds we compute a maximal time horizon of \(T = 648\) time units for those instances. It follows that the sum of sample completion times is bounded above by \(9 \cdot 648 = 5832 < 10^4\) , i. e., a solution to an instance of the library of minor instances is feasible if and only if it has an objective value below \(10^4\) . Of course this does neither apply to the library of major instances nor to the solutions of FDA or FDAh as they have lower penalty parameters due to prestudies with the smallest instances. In a general setup a way to identify infeasible solutions is to store the penalty term \(\sum _{i=1}^7 P_i(x)\) and evaluate the solution with it. The solution is feasible in this case if and only if the penalty term evaluates to 0 on it.

## Benchmark results

In the current section, we present the results of our previously described benchmark, which is summarized in Table 1 . For this purpose, we first show the results for the minor instances and subsequently the results for the major instances.

## Results for Minor Instances

In Fig. 3 , we show the objective values and runtimes of several approaches as scatter plots. All runtimes are end-to-end runtimes, that is, we consider the entire evaluation pipeline, beginning with the submission of the problem to the solver and ending with the return of a solution, including potential network delays. The programmatic construction of the optimization problem for the application programming interface (API) of the solver based on the instance data is not part of the runtime.

From Fig. 3 a, we can observe that both the SE-GU and TI-GU solutions reach a better objective value than the solutions from QU-LBQM and QU-FDA. When comparing objective values, it has to be taken into account that the QUBO model objective, Eq. 2 , also includes penalty terms, which become positive for infeasible solutions and therefore increase the objective value accordingly. Specifically, we find that only QU-LBQM yields infeasible solutions for some instances, whereas all other approaches yield feasible solutions (SE-GU and TI-GU solutions are by definition always feasible). For our analysis, we include both feasible and infeasible solutions. By performing a Welch t-test 58 , we find that the means of the results from both SE-GU and TI-GU are lower than the means of the QU-FDA and QU-LBQM results with a statistical significance of over \(99 \%\) , respectively. The same holds for the QU-FDA objective values in comparison to QU-LBQM.

Benchmark results for minor instances as scatter plots. The results are grouped into sets of instances ( N , K ) with the same number of samples N and photos K . A horizontal line marks the upper time limit of 3600s for Gurobi in Fig. 3 b. Some instances have not been solved to feasibility by QU-LBQM, as indicated by the peaks above \(10^4\) in Fig. 3 a. Abbreviations according to Fig. 2 .

On the other hand, according to Fig. 3 b, the computation time for TI-GU and for some instances of SE-GU exceed the computation time of QU-LBQM and QU-FDA. Since MIP solvers typically spend a lot of time proving that a solution is optimal, we are also interested in the time taken by Gurobi (for both SE-GU and TI-GU) to find solutions of the same quality as those obtained from QU-LBQM or QU-FDA. Hence, we perform an additional analysis of the iterative solver progress of each Gurobi run and look for the earliest computation time at which Gurobi has reached an objective value that is less than or equal to the corresponding objective value returned by the competing solvers for the same instance. We call this earliest computation time the relative runtime . Specifically, we consider the relative runtime of TI-GU w.r.t. QU-LBQM (TI-GU@QU-LBQM), the relative runtime of SE-GU w.r.t. QU-LBQM (SE-GU@QU-LBQM), the relative runtime of TI-GU w.r.t. QU-FDA (TI-GU@QU-FDA) and the relative runtime of SE-GU w.r.t. QU-FDA (SE-GU@QU-FDA). In the special case that Gurobi is not able to find an objective value of the desired quality within its limit of 3600 seconds (which only occurs for some major instances), this time limit is used in place of the earliest computation time. Exemplarily, we consider a specific instance to visualize TI-GU@QU-LBQM and TI-GU@QU-FDA in Fig. 4 .

Visualization of the relative runtime of TI-GU w.r.t. QU-LBQM and QU-FDA, denoted by TI-GU@QU-LBQM and TI-GU@QU-FDA, respectively. Here, we consider the example instance (7, 4, 3)(3); see supplementary material. The orange dots (connected by lines for better visualization) mark the resulting objective values of TI-GU at the corresponding time steps. The horizontal upper, blue and lower, green line mark the final objective value of QU-FDA and QU-LBQM, respectively, on the same instance. The blue and green lines intersect with the orange lines at some point. The time coordinate of the next lower TI-GU objective value after this intersection represents the relative runtime of TI-GU w.r.t. the solver, which is marked as a vertical line in the corresponding color. In other words, the relative runtime represents how long TI-GU has to run until it reaches an objective value that is at least as good as the result from QU-LBQM or QU-FDA, respectively.

The results of this analysis are shown in Fig. 5 . This plot shows that QU-LBQM is not able to compete with SE-GU. All problems from the first 4 out of 9 instance groups have been solved with SE-GU in under 1 second while the remaining instances in less than 10 seconds, whereas the QU-LBQM runtimes range between 50 and 100 seconds. However, LBQM finds a comparable solution faster than TI-GU for most problems with 6 or more samples and remains competitive for smaller problems. A Welch-t test confirms that the mean of TI-GU runtime is larger than the one of QU-LBQM runtime with a significance over \(99 \%\) .

Furthermore, Fig. 5 b shows that QU-FDA is outperformed by SE-GU as well. Analogous to Fig. 5 a, the instances in groups (4, 4), (5, 3), (5, 4) and (6, 3) have been solved by SE-GU in 1 second or less. But in contrast to Fig. 5 a, the other groups have their median between 1 second and 10 seconds, i. e., which reflects that the target objectives from QU-FDA are lower than those from QU-LBQM (see Fig. 3 a). Nonetheless, the time taken for SE-GU to reach the solution quality of QU-FDA is 10 to 100 times smaller. Regarding TI-GU, QU-FDA finds a comparable solution almost always faster with a few exceptions.

Benchmark results for minor instances as scatter plots. We show the relative runtimes of TI-GU and SE-GU w.r.t. QU-LBQM and QU-FDA, denoted by TI-GU@QU-LBQM, TI-GU@QU-FDA, SE-GU@QU-LBQM and SE-GU@QU-FDA, respectively. The results are grouped into sets of instances ( N , K ) in analogy to Fig. 3 . See Fig. 4 for an example of the relative runtime computation. Abbreviations according to Fig. 2 .

## Results for Major Instances

The results for major instances are presented in analogy to the results for minor instances from the previous section. In Fig. 6 , we show the runtime and the target value of the solvers on the corresponding models as scatter plots.

Benchmark results for major instances as scatter plots. The results are grouped into sets of instances ( N , K ) as for previous the plots. Abbreviations according to Fig. 2 .

The objective values of QU-FDAh are worse than the ones of SE-GU with a significance of over \(97\%\) , but Fig. 6 b shows that the runtime of SE-GU increases strictly until it reaches the upper bound for the computation time of 3600 seconds, which happens for ca. 15 samples. On the other hand, the computation time of QU-FDAh ranges between 120 and 300 seconds, where only a slight increase can be seen.

Analogously to Fig. 5 b, we evaluate the earliest computation times of SE-GU model to reach objective values equal to or lower than the objective values obtained from QU-FDAh, denoted by the relative runtime of SE-GUw.r.t. QU-FDAh (SE-GU@QU-FDAh). The results are shown in Fig. 7 .

Benchmark results for major instances as scatter plots. We show the relative runtime of of SE-GU w.r.t. QU-FDAh, denoted by SE-GU@QU-FDAh, in analogy to Fig. 5 . We also show the runtime of QU-FDAh from Fig. 6 b. The results are grouped into sets of instances ( N , K ) as for previous plots. Abbreviations according to Fig. 2 .

In Fig. 7 , a strictly increasing computation time can be seen for SE-GU, whereas the QU-FDAh runtime remains almost constant. For the biggest instances with \(N=20\) samples, QU-FDAh has a clear advantage with respect to the computation time, whereas it is competitive to SE-GU for the instances with 15 samples. In this sense QU-FDAh finds a solution of comparable quality much faster for problems with 20 samples than SE-GU and the latter was not able to prove optimality for some of the instances with 20 samples. A Welch t-test confirms with a significance of over \(99 \%\) that the QU-FDAh mean is lower than the SE-GU@QU-FDAh mean.

## Conclusion and outlook

This paper presents a thorough benchmarking of an industrially relevant use case of combinatorial optimization, the transport robot scheduling problem (TRSP) with the goal to achieve a time-optimal robot schedule, as motivated by a BASF high-throughput laboratory. We solve a large set of instances for this optimization problem with varying difficulty using three commercially available solvers: (i) the D-Wave’s hybrid Leap framework, (ii) the quantum-inspired Fujitsu digital annealer and (iii) the classical state-of-the-art solver Gurobi. To this end, we develop several mathematical models: a (QUBO) model for the quantum and digital annealer and two different MIP models for Gurobi, which we call time-indexed and sequence model, respectively. Modeling the same problem in different, solver-specific forms helps us to optimally assess the capabilities of each solver. In total, we compare five different approaches (i. e., model and solver combinations as sketched in Fig. 2 ): (i) Gurobi with the time-indexed model (TI-GU), (ii) Gurobi with the sequence model (SE-GU), (iii) D-Wave’s hybrid Leap framework (LBQM) with the QUBO model (QU-LBQM), (iv) Fujitsu’s digital annealer (FDA) with the QUBO model (QU-FDA) and (v) Fujitsu’s digital annealer hybrid framework (FDAh) with the QUBO model (QU-FDAh). For our performance study, we separated all problem instances into two groups. First, the minor instances with problems less than 10000 binary variables in the QUBO formulation and, second, the major instances with problems with more than 10000 and up to 22000 variables. For practical reasons, we only solve the minor instances with SE-GU, TI-GU, QU-LBQM and QU-FDA, whereas the major instances are only solved with SE-GU and QU-FDAh, respectively.

Our benchmark reveals insights both regarding the objective values of the optimization problem (i. e., the sum of sample completion times) as well as the end-to-end runtimes for the considered approaches. Regarding the objective values, we observe for minor instances that SE-GU and TI-GU give similar results, outperforming QU-FDA, which in turn outperforms QU-LBQM (cf. Fig. 3 a). For major instances, SE-GU outperforms QU-FDAh (cf. Fig. 6 a). Regarding the runtime, we find that for smaller instances TI-GU takes the highest time and SE-GU takes mostly the lowest. Between these two extremes, QU-FDA and QU-LBQM take about the same amount of time (cf. Fig. 3 b). However, the runtime of SE-GU significantly increases with increasing instance complexity. This same observation continues for the large instances, for which the runtime of SE-GU is mostly larger than that of QU-FDAh (cf. Fig. 6 b).

To get further insights into the relationship between objective value and runtime, we also studied the relative runtime of Gurobi, that is the time that Gurobi took to find an objective value that is at least as good as the final result from another approach. For minor instances, we find that the relative runtimes of SE-GU w.r.t. QU-LBQM and QU-FDA, respectively, are strictly lower than the runtimes of QU-LBQM and QU-FDA, i. e., Gurobi found solutions of comparable quality faster than the quantum and quantum-inspired approaches (cf. Fig. 5 a and 5 b). This is not surprising since SE-GU tended to find better objectives in shorter time. For major instances, the relative runtimes of SE-GU w.r.t. QU-FDAh increase significantly with increasing instance complexity and clearly exceed the runtime of QU-FDAh for the biggest instances (cf. Fig. 7 ). Thus, QU-FDAh shows an advantage on some bigger instances. Although the resulting objective values of QU-FDAh were not optimal, the approach shows a clear advantage on some bigger instances when compared to SE-GU on a similar time scale.

Our benchmark spans instances of different scales and therefore allows qualitative estimation of the scaling behavior of different approaches. Specifically, we observe that TI-GU and SE-GU show a runtime that scales exponentially with the instance complexity (as estimated by the number of samples and photos), whereas the runtime of QU-LBQM, QU-FDA and QU-FDAh remains almost constant. The quality of the solutions is not significantly determined by the instance complexity. Further research is needed to investigate and quantify these observations in more detail.

Summarized, no general advantage of the quantum and quantum-inspired solvers was found. However, for certain instances the quantum-inspired hybrid usage of the Fujitsu digital annealer turned out to be a very promising alternative to Gurobi and was clearly superior to the usage of D-Wave’s hybrid Leap framework. Our study is not a conclusive result but rather an application-oriented case study that provides a snapshot of the current technology and leaves room for performance improvements on the modeling as well as the solver side. For example, an improvement of the quantum annealer inside the hybrid framework might be possible with additional problem-specific fine-tuning of the annealing schedule or other hardware-related parameters. Moreover, the recently released constrained quadratic model (CQM) solver from D-Wave also promises to provide much better performance compared to the solver used in this work. Especially in an agile field such as quantum computing, a technology snapshot such as ours can hardly provide any forecasts about future developments. Therefore, in order to preserve an up-to-date assessment, further practical evaluations for real-world use cases will be necessary. The methods and results from this project can serve as a blueprint or at least point of reference for this kind of ongoing research.

## Data availability

Data of the problem instances and solver configurations are presented within the paper. The code is available upon reasonable request.

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## Acknowledgements

We would like to thank Behrang Shafei, Jens Meissner and Horst Weiss for their invaluable input and support throughout the research process. Without their ongoing contributions, the work would not have been accomplished. This work was partly funded by the German Federal Ministry of Education and Research (Bundesministerium für Bildung und Forschung, BMBF) within the project “Rymax One”.

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## Contributions

All authors contributed to the writing and overall research process. D.L., T.S., S.J. and A.A. built the models, designed and ran the benchmarks and analyzed the results. A.N. provided the considered use case. R.H. and M.B. were responsible for the conceptualization. R.H. was responsible for project administration. All authors reviewed the manuscript.

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Correspondence to Dominik Leib .

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Leib, D., Seidel, T., Jäger, S. et al. An optimization case study for solving a transport robot scheduling problem on quantum-hybrid and quantum-inspired hardware. Sci Rep 13 , 18743 (2023). https://doi.org/10.1038/s41598-023-45668-1

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## What is quantum computing?

Flip a coin. Heads or tails, right? Sure, once we see how the coin lands. But while the coin is still spinning in the air, it’s neither heads nor tails. It’s some probability of both.

This grey area is the simplified foundation of quantum computing.

## Get to know and directly engage with senior McKinsey experts on quantum computing

Ondrej Burkacky is a senior partner in McKinsey’s Munich office, Miklós Gábor Dietz is a senior partner in the Vancouver office, Dieter Kiewell and Jared Moon are senior partners in the London office, Alexandre Ménard is a senior partner in the Paris office, Mark Patel is a senior partner in the Bay Area office, and Rodney Zemmel is a senior partner in the New York office.

Digital computers have been making it easier for us to process information for decades. But quantum computers are poised to take computing to a whole new level. Quantum computers represent a completely new approach to computing. And while they won’t replace today’s computers, by using the principles of quantum physics, they will be able to solve very complex statistical problems that today’s computers can’t. Quantum computing has so much potential and momentum that McKinsey has identified it as one of the next big trends in tech . Quantum computing alone—just one of three main areas of emerging quantum technology—could account for nearly $1.3 trillion in value by 2035.

Here’s how it works: classical computing, the technology that powers your laptop and smartphone, is built on bits. A bit is a unit of information that can store either a zero or a one. By contrast, quantum computing is built on quantum bits, or qubits, which can store zeros and ones. Qubits can represent any combination of both zero and one simultaneously—this is called a superposition.

When classical computers solve a problem with multiple variables, they must conduct a new calculation every time a variable changes. Each calculation is a single path to a single result. Quantum computers, however, have a larger working space, which means they can explore a massive number of paths simultaneously. This possibility means that quantum computers can be much, much faster than classical computers.

But the first real proof that quantum computers could handle problems too complicated for classical computers didn’t arrive until 2019, when Google announced that its quantum computer had made a major breakthrough: it solved a problem in 200 seconds that would have taken a classical computer 10,000 years.

Although this was an important milestone in computing, it was more of a theoretical leap forward rather than a practical one, since the problem the quantum computer solved had no real-world use at all. But we’re rapidly approaching a time when quantum computers will have a real impact on our lives. Read on to find out how.

Learn more about McKinsey Digital.

## How do quantum computers solve problems?

Today’s classical computers are relatively straightforward. They work with a limited set of inputs and use an algorithm and spit out an answer—and the bits that encode the inputs do not share information about one another. Quantum computers are different. For one thing, when data are input into the qubits, the qubits interact with other qubits, allowing for many different calculations to be done simultaneously. This is why quantum computers are able to work so much faster than classical computers. But that’s not the end of the story: quantum computers don’t deliver one clear answer like classical computers do; rather, they deliver a range of possible answers.

For calculations that are limited in scope, classical computers are still the preferred tools. But for very complex problems, quantum computers can save time by narrowing down the range of possible answers.

## When will quantum computers arrive?

Quantum computers aren’t like your average desktop computer. It’s unlikely that you will be able to wander down to a store and pick one up. The kind of quantum computers that are capable of solving major problems will be expensive, complicated machines operated by just a few key players.

Over the next few years, the major players in quantum computing, as well as a small cohort of start-ups, will steadily increase the number of qubits that their computers can handle. Progress is expected to be slow: McKinsey estimates that by 2030, only about 5,000 quantum computers will be operational. The hardware and software required to handle the most complex problems may not exist until 2035 or later.

## Introducing McKinsey Explainers : Direct answers to complex questions

But some businesses will begin to derive value from quantum well before then. At first, businesses will receive quantum services via the cloud, from the same providers they use now. Several major computing companies have already announced their quantum cloud offerings.

## What are some obstacles that impede the development of quantum computing?

One major obstacle to the advancement of quantum computing is that qubits are volatile . Whereas a bit in today’s computers can be in a state of either one or zero, a qubit can be any possible combination of the two. When a qubit changes its status, inputs can be lost or altered, throwing off the accuracy of the results. Another obstacle to development is that a quantum computer operating at the scale needed to deliver significant breakthroughs will require potentially millions of qubits to be connected. The few quantum computers that exist today are nowhere near that number.

## How can classical computers and quantum computers work together?

Slowly, at first. For the time being, quantum computing will be used alongside classical computing to solve multivariable problems. One example? Quantum computers can narrow the range of possible solutions to a finance or logistics problem, helping a company reach the best solution a little bit faster. This kind of slower progress will be the norm until quantum computing advances enough to deliver massive breakthroughs.

Quantum computers can narrow the range of possible solutions to a finance or logistics problem, helping a company reach the best solution a little bit faster.

## What are some potential business use cases for quantum computers?

Quantum computers have four fundamental capabilities that differentiate them from today’s classical computers:

- Quantum simulation . Quantum computers are able to model complex molecules, which may eventually help reduce development time for chemical and pharmaceutical companies. Scientists looking to develop new drugs need to examine the structure of a molecule to understand how it will interact with other molecules. It’s almost impossible for today’s computers to provide accurate simulations, because each atom interacts with other atoms in complex ways. But experts believe that quantum computers are powerful enough to eventually be able to model even the most complex molecules in the human body. This opens up the possibility for faster development of new drugs and transformative new cures.
- Optimization and search . Every industry relies in one way or another on optimization. Where should I place robots on the factory floor? What’s the shortest route for my delivery truck? There are almost infinite questions that need to be answered to optimize for efficiency and value creation. With classical computing, companies must make one complicated calculation after another, which is a time-consuming and costly process given the many variables of any situation. Since quantum computers can work with multiple variables simultaneously, they can be used to quickly narrow the range of possible answers . From there, classical computing can be used to zero in on one precise answer.
- Quantum AI . Quantum computers have the potential to work with better algorithms that could transform machine learning across industries as diverse as pharmaceuticals and automotive. In particular, quantum computers could accelerate the arrival of self-driving vehicles. Companies like Ford, GM, Volkswagen, and numerous mobility start-ups are running video and image data through complex neural networks. Their goal? To use AI to teach a car to make crucial driving decisions. Quantum computers’ ability to perform multiple complex calculations with many variables simultaneously allows for faster training of such AI systems.
- Prime factorization . Businesses today use large, complex prime numbers as the basis for their encryption efforts, numbers too large for classical computers to process. Quantum computing will be able to use algorithms to solve these complex prime numbers easily, a process called prime factorization. (In fact, a quantum algorithm known as Shor’s algorithm theoretically already can; there’s just not a computer powerful enough to run it.) Once quantum computers have advanced enough, new quantum-encryption technologies will be needed to protect our online services . Scientists are already at work on quantum cryptography to prepare for this eventuality. McKinsey estimates quantum computers will be powerful enough for prime factorization by the late 2020s at the very earliest.

As these capabilities develop at pace with quantum computing power, use cases will proliferate.

Experts believe that quantum computers are powerful enough to eventually be able to model even the most complex molecules in the human body.

## What industries stand to benefit the most from quantum computing?

Research suggests that four industries stand to reap the greatest short-term benefits from quantum computing based on the use cases discussed in the previous section. Collectively—and conservatively—the value at stake for these industries could be as much as $1.3 trillion.

- Pharmaceuticals . Quantum computing has the potential to revolutionize the research and development of molecular structures in the biopharmaceuticals industry. With quantum technologies, research and development for drugs will become less reliant on trial and error, and therefore more efficient. ( Read more on how quantum computing stands to affect the pharmaceutical industry .)
- Chemicals . Quantum computing could be used to improve catalyst design, which could enable savings on existing production processes. Innovative catalysts could also enable the replacement of petrochemicals with more sustainable feedstock or the breakdown of carbon for CO 2 usage. ( Read more on how quantum computing might affect the chemicals industry .)
- Automotive . The automotive industry could benefit from quantum computing in its R&D, product design, supply chain management, production, and mobility and traffic management. For example, quantum computing could be applied to decrease manufacturing costs by optimizing complex multirobot processes including welding, gluing, and painting. ( Read more about how quantum technologies could affect the automotive industry .)
- Finance . Quantum-computing use cases in finance are slightly further in the future. The long-term promise of quantum computing in finance lies in portfolio and risk management. One example could be quantum-optimized loan portfolios that focus on collateral to allow lenders to improve their offerings. ( Read more about how quantum computing could affect financial services .)

These four industries likely stand to gain the most from quantum computing. But leaders in every sector can—and should—prepare for the inevitable quantum advancements of the next few years.

## What are some other quantum technologies aside from computing?

According to McKinsey’s analysis, quantum computing is still years away from widespread commercial application. Other quantum technologies such as quantum communication (QComms) and quantum sensing (QS) could become available much earlier . Quantum communication will enable strong encryption protocols that could greatly increase the security of sensitive information. QComms enables the following functions:

- Full security when information is transferred between locations. Quantum-encryption protocols are more secure than classical protocols, most of which will likely be able to be broken once quantum computers attain more computing power or can work with more efficient algorithms.
- Enhanced quantum-computing power in two important types of quantum processing: parallel quantum processing (where multiple processors are connected and simultaneously execute different calculations from the same problem) and blind quantum computing (where quantum communications provide access to remote, large-scale quantum computers in the cloud). Both types of processing are made possible by the entanglement of quantum particles. Entanglement is when quantum particles like qubits have connected properties, which means one particle’s properties can be manipulated by actions done to another.

Quantum sensing allows for more accurate measurements than ever before, including of physical properties like temperature, magnetic fields, and rotation. Plus, once optimized and decreased in size, quantum sensors will be able to measure data that can’t be captured by current sensors.

The markets for QComms and QS are currently smaller than those for quantum computing, which has so far attracted most of the headlines and funding. But McKinsey expects both Qcomms and QS to attract serious interest and funding in the future. The risks are significant, but the potential payoff is high: by 2030, QS and QComms could generate $13 billion in revenues.

Learn more about quantum sensors and quantum communications.

## How can organizations ensure that they have the quantum-computing talent they need?

A wide talent gap exists between the business need for quantum computing and the number of quantum professionals available to meet that need. This skill gap could jeopardize potential value creation, which McKinsey estimates to be as much as $1.3 trillion.

McKinsey research has found that there is only one qualified quantum candidate for every three quantum job openings. By 2025, McKinsey predicts that less than 50 percent of quantum jobs will be filled, unless there are significant changes to the talent pool or predicted rate of quantum-job creation.

Here are five lessons derived from the AI talent journey that can help organizations build the quantum talent they need to capture value:

- Define your talent needs clearly . In the early days of AI, some organizations hired data scientists without a clear understanding of what skills were needed. To avoid making the same error with quantum, organizations should first identify possible fields of applications that a quantum-computing team would work on and then ensure that new hires come from diverse backgrounds (reflecting best practices).
- Invest early in translators. As buzz built up around AI, the role of analytics translators became crucial to helping leaders identify and prioritize challenges best suited for AI to solve. With quantum, there’s a similar need: for translators with engineering, application, and scientific backgrounds who can help organizations understand the opportunities and players in the rapidly expanding ecosystem.
- Create pathways for a diverse talent pipeline . Many of the first AI models reflected the same biases that were present in the information that was used to train them. There often was also a lack of people with diverse perspectives and experience building and testing the models, which contributed to the bias issue. While it’s too early to know what risks will emerge from quantum technologies, we can expect similar challenges if we don’t build and empower a diverse quantum workforce. Efforts are needed at the university level, as well as in K–12 education.
- Build technology literacy for all. In order for employees at all levels of an organization to understand the potential of a new technology, they need a basic understanding of how it works and what it can do. With quantum, business leaders as well as workers up and down the supply chain, in marketing, IT infrastructure, finance, and more will require basic fluency in quantum topics.
- Don’t forget talent development strategies. Companies focus heavily on talent attraction during times of technological foment—but that’s just one piece of the talent puzzle. In order to retain specialists, companies need to carve out clear paths for talent development. One pharmaceutical company leans into both the purpose of its work—developing use cases that will help save lives—and the freedom it offers its team.

Learn more about McKinsey Digital and check out quantum-computing job opportunities if you’re interested in working at McKinsey.

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## Want to know more about quantum computing?

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## The Rise of Quantum Computing

## How quantum computing can help tackle global warming

## A quantum computer just solved a decades-old problem three million times faster than a classical computer

Scientists from quantum computing company D-Wave have demonstrated that, using a method called quantum annealing, they could simulate some materials up to three million times faster than it would take with corresponding classical methods.

Together with researchers from Google, the scientists set out to measure the speed of simulation in one of D-Wave's quantum annealing processors, and found that performance increased with both simulation size and problem difficulty, to reach a million-fold speedup over what could be achieved with a classical CPU.

## Quantum Computing

- Nvidia's new platform could usher in the era of hybrid quantum-classical computing
- White House: Quantum computers could crack encryption, so here's what to do
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The calculation that D-Wave and Google's teams tackled is a real-world problem; in fact, it has already been resolved by the 2016 winners of the Nobel Prize in Physics, Vadim Berezinskii, J. Michael Kosterlitz and David Thouless, who studied the behavior of so-called "exotic magnetism", which occurs in quantum magnetic systems.

SEE: Guide to Becoming a Digital Transformation Champion (TechRepublic Premium)

The Nobel Prize winners used advanced mathematical methods to describe, in the 1970s, the properties of a two-dimensional quantum magnet, which shed light on the strange – or "exotic" – states that matter can take on.

Instead of proving quantum supremacy, which happens when a quantum computer runs a calculation that is impossible to resolve with classical means, D-Wave's latest research demonstrates that the company's quantum annealing processors can lead to a computational performance advantage.

"This work is the clearest evidence yet that quantum effects provide a computational advantage in D-Wave processors," said Andrew King, director of performance research at D-Wave.

D-Wave's processors are based on quantum annealing technology, which is a quantum computing technique used to find solutions to optimization problems. While some argue that the scope of the problems that can be resolved by the technology is limited, quantum annealing processors are easier to control and operate than their gate-based equivalents, which is why D-Wave's technology has already reached much higher numbers of qubits than can be found in the devices built by big players like IBM or Google.

To simulate exotic magnetism, King and his team used the D-Wave 2,000-qubit system, which was recently revised to reduce noise , to model a programmable quantum magnetic system, just like Berezinskii, Kosterlitz and Thouless did in the 1970s to observe the unusual states of matter. The researchers also programmed a standard classical algorithm for this kind of simulation, called a "path-integral Monte Carlo" (PIMC), to compare the quantum results with CPU-run calculations. As the numbers show, the quantum simulation outperformed classical methods by a margin.

"What we see is a huge benefit in absolute terms," said King. "This simulation is a real problem that scientists have already attacked using the algorithms we compared against, marking a significant milestone and an important foundation for future development. This wouldn't have been possible today without D-Wave's lower noise processor."

To simulate exotic magnetism, King and his team programmed the D-Wave 2,000-qubit system to model a quantum magnetic system.

Equally as significant as the performance milestone, said D-Wave's team, is the fact that the quantum annealing processors were used to run a practical application, instead of a proof-of-concept or an engineered, synthetic problem with little real-world relevance. Until now, quantum methods have mostly been leveraged to prove that the technology has the potential to solve practical problems , and is yet to make tangible marks in the real world.

In contrast, D-Wave's latest experiment resolved a meaningful problem that scientists are interested in independent of quantum computing. The findings have already attracted the attention of scientists around the world.

"The search for quantum advantage in computations is becoming increasingly lively because there are special problems where genuine progress is being made. These problems may appear somewhat contrived even to physicists," said Gabriel Aeppli, professor of physics at ETH Zürich and EPF Lausanne.

"But in this paper from a collaboration between D-Wave Systems, Google, and Simon Fraser University, it appears that there is an advantage for quantum annealing using a special purpose processor over classical simulations for the more 'practical' problem of finding the equilibrium state of a particular quantum magnet."

SEE: BMW explores quantum computing to boost supply chain efficiencies

D-Wave, however, stayed clear of claiming quantum advantage, which happens when a quantum processor can demonstrate superiority over all possible classical competition; King stressed that it is still possible to design highly specialized algorithms to simulate the model once the properties of the model are already known.

The real significance of the experiment lies in the proof that a computational advantage can already be achieved using existing quantum methods to solve a valuable materials science problem.

"These experiments are an important advance in the field, providing the best look yet at the inner workings of D-Wave computers, and showing a scaling advantage over its chief classical competition," said King. "All quantum computing platforms will have to pass this kind of checkpoint on the way to widespread adoption."

Although D-Wave's 2,000-qubit system was used for the research due to the technology's lower noise rates, the company recently released a 5,000-qubit quantum processor , which is already available for programmers to build quantum applications.

From improving the logistics of retail supply chains to simulating new proteins for therapeutic drugs, through optimizing vehicles' routes through busy city streets, D-Wave is currently counting 250 early quantum annealing applications from various different customers.

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## Microsoft Research Blog

What problems will we solve with a quantum computer.

Published July 5, 2017

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## New paper suggests quantum computers will address problems that could have substantial scientific and economic impact

The MoFe protein, left, and the FeMoco, right, would be able to be analyzed by quantum computing to help reveal the complex chemical system behind nitrogen fixation by the enzyme nitorgense.

With rapid recent advances in quantum technology, we have drawn ever closer to the threshold of quantum devices whose computational powers can exceed those of classical supercomputers.

But when a useful, scalable general-purpose quantum computer arrives, what problems will it solve?

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Peter Lee, head of Microsoft Research, and Ashley Llorens, AI scientist and engineer, discuss the future of AI research and the potential for GPT-4 as a medical copilot.

Much work has already been done towards identifying areas where quantum computing provides a clear improvement over traditional classical approaches. Many suspect that quantum computers will one day revolutionize chemistry and materials science; the likely ability of quantum computers to predict specific properties of molecules and materials fits this outcome nicely.

However, a number of important questions remain. Not the least of these is the question of how exactly to use a quantum computer to solve an important problem in chemistry. The inability to point to a clear use case complete with resource and cost estimates is a major drawback. After all, even an exponential speedup may not lead to a useful algorithm if a typical, practical application requires an amount of time and memory that is beyond the reach of even a quantum computer.

Our paper (opens in new tab) published earlier this week at the Proceedings of the National Academy of Sciences confirms the feasibility of such a practical application, showing that a quantum computer can be employed to reveal reaction mechanisms in complex chemical systems, using the open problem of biological nitrogen fixation in nitrogenase as an example.

Today, we spend approximately 3 percent of the world’s total energy output on making fertilizer. This relies on a process developed in the early 1900s (opens in new tab) that is extremely energy intensive—the reaction gas required is taken from natural gas, which is in turn required in very large amounts. However, we know that a tiny anaerobic bacteria in the roots of plants performs this same process every day at very low energy cost using a specific molecule— nitrogenase (opens in new tab) .

This molecule is beyond the abilities of our largest supercomputers to analyze, but would be within the reach of a moderate scale quantum computer. Efficiently capturing carbon (to combat global warming) is in the same class of problem. The search for high-temperature superconductors is another example.

This paper shows that these kinds of necessary computations can be performed in reasonable time on realistic quantum computers—demonstrating that quantum computers will one day tackle important problems in chemistry without requiring exorbitant resources. This paper also gives us further confidence that quantum simulation will be able to provide answers to problems with a tremendous potential for scientific and economic impact.

Editor’s Note: The paper’s authors contributed to this post: Markus Reiher, Nathan Wiebe, Krysta Svore, Dave Wecker and Matthias Troyer .

- Microsoft doubles down on quantum computing bet (opens in new tab)
- Learn more about Station Q (opens in new tab)
- Read about the quest for a quantum future (opens in new tab)
- New York Times: Microsoft makes bet quantum computing is next breakthrough (opens in new tab)
- With quantum computing simulator, Microsoft offers a sneak peek into the future of computing (opens in new tab)
- Follow Krysta Svore (opens in new tab) , Dave Wecker (opens in new tab) , Matthias Troyer (opens in new tab) on Twitter

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## Problems that only Quantum Computers can solve

In this article, we will explore how quantum algorithms can solve real-world problems, and how you can get involved in this quantum revolution!

Quantum computers can solve NP-hard problems that classical computers are unable to solve.

Currently, the two most important and notable complexity classes are “P” and “NP.” P represents problems that can be solved in polynomial time by a classical computer. For instance, asking if a number is prime belongs to P. NP problems are problems that cannot be solved in polynomial time by classical computers, but the answers to the problem can be verified quickly with a classical computer. Asking what are the prime factors of a number is an NP problem, as it can be easily verified if x is the prime factor of a number y, however it is very hard for the computer to find out its prime factors. The problem of whether P=NP, whether the two complexity classes are distinct or not is an important dilemma and the one who solves it gets a million dollars!

In 1993, Ethan Bernstein and Umesh Vazirani defined a new complexity class called “bounded-error quantum polynomial time" or BQP. They defined this class to contain decision problems — problems with a yes or no answer — that quantum computers can solve efficiently. They also proved that P is a subset of BQP- that a quantum computer can solve all problems that a classical computer can solve.

They also defined another class of problems called PH or "Polynomial Hierarchy". PH is a generalization of NP. Problems in PH are NP problems that are made more complex by asking questions like is it true "for all" or "does it exist for a particular x".

But can a quantum computer solve problems that classical computers are unable to solve- the NP hard problems? Can we use quantum computing to solve practical problems that industries or companies are facing in real life? Well, you might have heard about how Shor’s algorithm might crack the encryption codes such as RSA and break into your bank account. Shor’s algorithm is able to solve the NP-hard problem of factoring large numbers- check out our implementation of Shor’s algorithm.

Recently, researchers at Chalmers University of Technology have been able to solve a small part of a logistics problem faced by the aviation industry- the Tail Assignment Problem- assigning airplanes to flights with the goal of minimizing connection times between flights and keeping in mind maintenance constraints.

This is a scheduling problem- which scales up exponentially with the number of flights and routes. The team at Chalmers was able to execute their algorithm on a processor with two qubits using the Quantum Approximate Optimization Algorithm or QAOA. The research team also simulated the optimization problem for up to 278 aircraft, however it requires a 25 qubit processor. Read this article to find out more!

So what exactly is the Quantum Approximate Optimization algorithm?

Optimization is searching for an optimal solution in a finite or countably infinite set of potential solutions of a cost function, which is set to be maximized or minimized. In the tail assignment problem, the connection times between flights should be minimized. The problem can also be defined in a way such that the total distance travelled by all airplanes over all air routes should be minimized.

Let us take a simple version of an optimization problem that is easy to visualize. Consider the Travelling Salesman Problem: A salesman wants to travel through all historic sites of the United States to sell souvenirs. The aim is to find the shortest route the salesman should travel such that he visits all the sites and returns back to his starting point.

The image above represents the shortest path to travel through all the historic sites of America. It would take the salesman 50 years to travel this path!

For a small number of cities, we can apply the “brute-force” solution: calculate all the possible routes and pick the shortest. For a large number of cities n, the complexity of this approach is O(n!), which is not efficient

How we would find this path is to use graph theory: each historic site is a vertex, and the edges are drawn between the vertices and represent the journey that the salesman takes. There will be numbers between the edges that represent the distance between the sites. How we minimize the distance is to first convert the problem into a weighted bipartite graph, and minimize the sum of the edges of the graph.

A weighted bipartite graph looks like the one above. We describe it as a hamiltonian cycle: A cycle where the start and end point is the same and uses each vertex of the graph exactly once.

In qiskit, we can map this problem to a Ising Hamiltonian, and minimize the value of the Ising Hamiltonian using the minimum variational Quantum Eigensolver optimizer. We will use the QuadraticProgram() function in Qiskit to make a model of the optimization problem. Check here to find out more.

To find out the shortest path between the vertices, we will be using the tsp module from qiskit.optimization.applications.ising class to solve our problem! Then we will find out what the shortest distance is( which is the objective or the minimum value of our cost function)

Our output will look something like this:

This means that the solution is the path from 1 to 2 to 3 to 0 to 1. The minimum distance when traversing the graph is 236.0.

If we want to make sure this is the correct solution, we can compare with the brute force method(to find the minimum sum of the edges).

We are still in the Noisy Intermediate Quantum Scale Era, and have a long way to go before running optimization algorithms will be feasible. In the paper , the authors estimate that 420 qubits will be necessary to run the QAOA in a short amount of time and scalable to complex optimization problems. Currently, IBM's supercomputers work on around 50 qubits. However, quantum optimization can be used to solve other problems such as finding the ground state energy of a molecule or optimizing portfolios in finance.

Qiskit can solve a wide range of combinatorial optimization problems. To find out more, check [this]( https://qiskit.org/textbook/ch-applications/qaoa.html.) .

In this paper , computer scientists have found out a problem that is in BQP but not in PH. This means even if classical computers were able to solve NP problems, quantum computers still have an advantage over them as a problem is found to exist in BQP and not PH- problems that only a quantum computer can solve. This further proves the fact that quantum computers have a processing capacity beyond what a classical computer can achieve.

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## Quantum Computers Are NOT Magical

What kind of problems can quantum computers solve.

Sara A. Metwalli

Level Up Coding

And What Kind of Problems Can’t They Solve?

B efore we get into the good stuff, I need to tell you that quantum computers are not magical. They may seem magical, just because the fundamental math and physics behind them are somewhat complicated. But the truth is, a quantum computer is marginally better than classical computers, nothing more.

Classical computers are better at some tasks than quantum computers, such as sending and receiving emails, creating spreadsheets and other documents, and desktop publishing. Quantum computers are not meant to replace classical computers; they are just meant to solve distinct problems than those solvable by classical computers. That being said, any problem that is impossible to solve using classical computers will be impossible to solve using quantum computers.

I know what you’re thinking…

You’re probably saying, but if they only solve problems solvable on classical computers, then what’s all the hype about?

Well, quantum computers have the potential to solve some problems better, faster, and more efficient than classical computers.

Problems like what you say? Before we get into that, let’s see what classical computers can and can’t do.

## Problems Classification in Computer Science

In computer science, they categorize problems according to how many computational steps it would take to solve them using the best-known algorithms. That is translated to how long it will take a computer to solve them. These categories are broad and often overlap. The most used three categories are P problems, NP problems, and NP-complete problems.

- P Problems:

These are problems classical computers can solve efficiently in polynomial time. An example of such problems is: Given n numbers and another number k , is there a number in n that’s larger than k ? This problem can be solved easily in linear time.

Because polynomials increase somewhat slowly as n increases, classical computers can solve even enormous P problems within a reasonable number of steps (reasonable length of time).

- NP Problems:

These are problems whose solutions are easy to verify, yet tricky to implement efficiently. For example, if we have the same set of numbers n and the constant k , but this time we want to check it, there two numbers in n that add up to k . In this case, the problem’s complexity becomes exponential, that’s because the number of sets that we need to sum and check will grow quickly even with smaller n .

Every P problem is also an NP problem, so the class P is a subclass of class NP.

- NP-complete Problems:

These are the hardest to solve problems. Theoretically speaking, if we can design an efficient solution to one NP-complete problem, then we can provide an efficient solution to all NP problems. An example of such problems is: Given a map, can you color it using only three colors so that no neighboring countries are the same color? This problem is challenging to solve, and hence why it categorizes as NP-complete. If you can find a solution to that problem — first, congratulations on being a genius — then you can extend the solution to the rest of the NP-complete problems.

So far, no known algorithm can solve a NP-complete problem efficiently.

Okay, now that we have the general idea of problem complexities, in what category do the problems solvable by quantum computers fall?

## Four Types of Problems Solvable by Quantum Computers

What we can say is that quantum algorithms can solve NP-complete problems efficiently as it provides exponential speedups. Algorithms such as Shor’s algorithm(Factoring algorithm) exploits the problems’ structure, in a way is by far beyond our present-day techniques.

Here are four types of problems we think quantum computers can solve much better than classical computers

- Encryption and Cybersecurity

These probably the most known kind of problems that quantum computers can solve. For example, the complex mathematical problem that is the core of the design of RSA encryption and other public-key encryption schemes is factoring a product of two prime numbers. Searching for the right pair using classical methods takes approximately forever. Here’s where quantum computer shine! Shor’s algorithm can be used to do the required factorization of integers in a short time.

Quantum cryptography is, without a doubt, more fascinating than classical cryptography. That is because it exploits the exciting properties of quantum mechanics. The best-known quantum encryption so far is the quantum key distribution (QKD), which uses the method of quantum communication to determine a shared key between two end-nodes. The magic of QKD comes because the mere act of listening in on this communication will cause changes in the connection, which provides absolute security to the communication channel.

- Chemistry Research

Biological systems are extraordinarily complex; that’s why it’s very challenging and time-consuming to model and simulate them. Classical computers find it difficult, if not impossible, to predict biological molecules and biochemical interactions. Because of that, early stage biomedical research has to be done by working chemically, cells, and animal labs, hoping for reproducible conditions between experiments. Because of this, the process of drug discovery and testing, which is an essential area of biomedical innovation, can take quite a long time. Using quantum computers, simulating these biological systems is not just doable, but easily doable even on current noisy quantum computers.

- Optimization Problems

Optimization problems are problems that tackle finding the best solution from all feasible solutions. Examples of optimization problems include the traveling salesman problem, speech, and image recognition. Usually, the way classical computers deal with these kinds of problems is using brute-force to check all possible answers, which is not applicable all of the time. On the quantum side, we can use superposition, or quantum “parallelism” to check all possible answers at the same time to solve these problems more efficiently.

If you want to go deeper into superposition, check this article

## The Three Pillars of Quantum Computing

The fundamentals of understanding how a quantum computer works.

- Data Analysis

We live in an era of information, the size and complexity of our datasets explode year after year. Eventually, classical computers — even supercomputers — will fail to handle the massive amount of data, and we won’t be able to process, organize, and extract actual value from the noise.

Quantum computers can complete complex calculations in only seconds — the same calculations that will take today’s computers thousands of years to resolve. They will also enable organizations to sample large amounts of information to analyze and optimize them. Moreover, quantum computers allow for quick detection, integration, and diagnosis from scattered data sets. They can search extensive, unsorted data sets to quickly and uncover patterns (an example of that is Grover’s Search Algorithm). All of that will transform the way we perceive and handle data and allow us to grow bigger and better.

## Problems Both Classical and Quantum Computers Can’t Solve

Now that we know what kind of problems quantum computers will be better at than classical computers, what problems are still impossible to solve on both types of computers?

So, quantum computers are useful in solving P problems and some NP problems, but beyond that, they still follow the laws of computability. Therefore, the Halting Problem (Can we determine from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever.)remains true whether handling classical or quantum computers.

Problems that are unsolvable by either computer are called Undecidable Problems. They include problems such as the whiteboard problem in graph theory and Hilbert’s Tenth Problem and Kruskal’s tree theorem, which are two of the toughest maths problem to solve.

In conclusion, quantum computers have a fascinating premise in solving some of the problems that classical computers failed to resolve. However, in the end, both techniques are still incapable of solving many problems out there.

[1] Aaronson, S. (2008). The limits of quantum computers. Scientific American .

[2]Monz, T., Nigg, D., Martinez, E. A., Brandl, M. F., Schindler, P., Rines, R., & Blatt, R. (2016).

[3]Mannhold, R., Kubinyi, H., & Folkers, G. (2006). Quantum medicinal chemistry (Vol. 17). John Wiley & Sons.

## Written by Sara A. Metwalli

Ph.D. candidate working on Quantum Computing. Traveler, writing lover, science enthusiast, and CS instructor. Get in touch with me bit.ly/2CvFAw6

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## Finally, a Problem That Only Quantum Computers Will Ever Be Able to Solve

June 21, 2018

Kevin Hong for Quanta Magazine

## Introduction

Early on in the study of quantum computers, computer scientists posed a question whose answer, they knew, would reveal something deep about the power of these futuristic machines. Twenty-five years later, it’s been all but solved. In a paper posted online at the end of May , computer scientists Ran Raz and Avishay Tal provide strong evidence that quantum computers possess a computing capacity beyond anything classical computers could ever achieve.

Raz, a professor at Princeton University and the Weizmann Institute of Science, and Tal, a postdoctoral fellow at Stanford University, define a specific kind of computational problem. They prove, with a certain caveat, that quantum computers could handle the problem efficiently while traditional computers would bog down forever trying to solve it. Computer scientists have been looking for such a problem since 1993, when they first defined a class of problems known as “BQP,” which encompasses all problems that quantum computers can solve.

Since then, computer scientists have hoped to contrast BQP with a class of problems known as “PH,” which encompasses all the problems workable by any possible classical computer — even unfathomably advanced ones engineered by some future civilization. Making that contrast depended on finding a problem that could be proven to be in BQP but not in PH. And now, Raz and Tal have done it.

The result does not elevate quantum computers over classical computers in any practical sense. For one, theoretical computer scientists already knew that quantum computers can solve any problems that classical computers can. And engineers are still struggling to build a useful quantum machine . But Raz and Tal’s paper demonstrates that quantum and classical computers really are a category apart — that even in a world where classical computers succeed beyond all realistic dreams, quantum computers would still stand beyond them.

## Quantum Classes

A basic task of theoretical computer science is to sort problems into complexity classes . A complexity class contains all problems that can be solved within a given resource budget, where the resource is something like time or memory.

Computer scientists have found an efficient algorithm, for example, for testing whether a number is prime. They have not, however, been able to find an efficient algorithm for identifying the prime factors of large numbers. Therefore, computer scientists believe (but have not been able to prove) that those two problems belong to different complexity classes.

The two most famous complexity classes are “P” and “NP.” P is all the problems that a classical computer can solve quickly. (“Is this number prime?” belongs to P.) NP is all the problems that classical computers can’t necessarily solve quickly, but for which they can quickly verify an answer if presented with one. (“What are its prime factors?” belongs to NP.) Computer scientists believe that P and NP are distinct classes, but actually proving that distinctness is the hardest and most important open problem in the field.

In 1993 computer scientists Ethan Bernstein and Umesh Vazirani defined a new complexity class that they called BQP, for “bounded-error quantum polynomial time.” They defined this class to contain all the decision problems — problems with a yes or no answer — that quantum computers can solve efficiently. Around the same time they also proved that quantum computers can solve all the problems that classical computers can solve. That is, BQP contains all the problems that are in P.

But they could not determine whether BQP contains problems not found in another important class of problems known as “PH,” which stands for “polynomial hierarchy.” PH is a generalization of NP. This means it contains all problems you get if you start with a problem in NP and make it more complex by layering qualifying statements like “there exists” and “for all.” 1 Classical computers today can’t solve most of the problems in PH, but you can think of PH as the class of all problems classical computers could solve if P turned out to equal NP. In other words, to compare BQP and PH is to determine whether quantum computers have an advantage over classical computers that would survive even if classical computers could (unexpectedly) solve many more problems than they can today.

“PH is one of the most basic classical complexity classes there is,” said Scott Aaronson , a computer scientist at the University of Texas at Austin. “So we sort of want to know, where does quantum computing fit into the world of classical complexity theory?”

The best way to distinguish between two complexity classes is to find a problem that is provably in one and not the other. Yet due to a combination of fundamental and technical obstacles, finding such a problem has been a challenge.

If you want a problem that is in BQP but not in PH, you have to identify something that “by definition a classical computer could not even efficiently verify the answer, let alone find it,” said Aaronson. “That rules out a lot of the problems we think about in computer science.”

## Ask the Oracle

Here’s the problem. Imagine you have two random number generators, each producing a sequence of digits. The question for your computer is this: Are the two sequences completely independent from each other, or are they related in a hidden way (where one sequence is the “Fourier transform” of the other)? Aaronson introduced this “forrelation” problem in 2009 and proved that it belongs to BQP. That left the harder, second step — to prove that forrelation is not in PH.

Which is what Raz and Tal have done, in a particular sense. Their paper achieves what is called “oracle” (or “black box”) separation between BQP and PH. This is a common kind of result in computer science and one that researchers resort to when the thing they’d really like to prove is beyond their reach.

The actual best way to distinguish between complexity classes like BQP and PH is to measure the computational time required to solve a problem in each. But computer scientists “don’t have a very sophisticated understanding of, or ability to measure, actual computation time,” said Henry Yuen , a computer scientist at the University of Toronto.

So instead, computer scientists measure something else that they hope will provide insight into the computation times they can’t measure: They work out the number of times a computer needs to consult an “oracle” in order to come back with an answer. An oracle is like a hint-giver. You don’t know how it comes up with its hints, but you do know they’re reliable.

If your problem is to figure out whether two random number generators are secretly related, you can ask the oracle questions such as “What’s the sixth number from each generator?” Then you compare computational power based on the number of hints each type of computer needs to solve the problem. The computer that needs more hints is slower.

“In some sense we understand this model much better. It talks more about information than computation,” said Tal.

The new paper by Raz and Tal proves that a quantum computer needs far fewer hints than a classical computer to solve the forrelation problem. In fact, a quantum computer needs just one hint, while even with unlimited hints, there’s no algorithm in PH that can solve the problem. “This means there is a very efficient quantum algorithm that solves that problem,” said Raz. “But if you only consider classical algorithms, even if you go to very high classes of classical algorithms, they cannot.” This establishes that with an oracle, forrelation is a problem that is in BQP but not in PH.

Raz and Tal nearly achieved this result almost four years ago, but they couldn’t complete one step in their would-be proof. Then just a month ago, Tal heard a talk on a new paper on pseudorandom number generators and realized the techniques in that paper were just what he and Raz needed to finish their own. “This was the missing piece,” said Tal.

News of the separation between BQP and PH circulated quickly. “The quantum complexity world is a-rocking,” wrote Lance Fortnow, a computer scientist at Georgia Tech, the day after Raz and Tal posted their proof.

The work provides an ironclad assurance that quantum computers exist in a different computational realm than classical computers (at least relative to an oracle). Even in a world where P equals NP — one where the traveling salesman problem is as simple as finding a best-fit line on a spreadsheet — Raz and Tal’s proof demonstrates that there would still be problems only quantum computers could solve.

“Even if P were equal to NP, even making that strong assumption,” said Fortnow, “that’s not going to be enough to capture quantum computing.”

Correction June 21, 2018: An earlier version of this article stated that the version of the traveling salesman problem that asks if a certain path is exactly the shortest distance is “likely” to be in PH. In fact, it has been proved to be in PH.

This article was reprinted on Wired.com and in Spanish at Investigacionyciencia.es .

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## Understanding how to solve problems with a quantum computer

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Microsoft Azure Quantum Team

Quantum computers can lead to breakthroughs in a wide variety of subject areas because they offer a computational strength we’ve never seen before. However, not all problems are favorable for a quantum computer. In order to identify which problems make good candidates, it’s important to have an understanding of how a quantum computer solves problems.

While quantum computers can offer an exponential boost in computational power, they can’t be programmed in the same way as a classical computer. The instruction set and algorithms change, and the resulting output is different as well. On a classical computer, the solution is found by checking possibilities one at a time. Depending upon the problem, this can take too long. A quantum computer can explore all possibilities at the same time, but there are a few challenges. Getting the right answer out of the computer isn’t easy, and because the answers are probabilistic, you may need to do extra work to uncover the desired answer.

For example, assume you wanted to page-rank the internet. To do so, the process would require loading every single page as input data. On a classical machine you would create a computation that gives you the page rank of each page, but this takes time and a significant amount of hardware. With a quantum computer, computation is exponentially faster than on classical hardware. But the caveat is that with quantum, your result will typically be the page rank of one page. And then you’d have to load the whole web again to get another, and do it again to get another, and continue until you eventually have the page rank for the entire internet. Because you have to load everything each time, the exponential speedup is lost. This example would not be favorable for quantum computing.

To solve any problem, you’ll have input, computation, and output.

- Input – The data required to run the computation
- Computation – The instructions given to the computer to process the data
- Output – The useful result received from the computation

Instead of returning the entire quantum state, a quantum computer returns one state as the result of a computation. This unique characteristic is why we write the algorithm in such a way that produces the desired answer with the highest probability. For this reason, problems that require a limited number of values are more applicable.

The amount of input data is also a consideration. As input data increases, either the number of qubits or the amount of work to ‘prepare’ the data grows quickly. Problems with highly compressed input data are more much more favorable.

What types of problems are ideal challenges for a quantum computer? Quantum computers are best-suited for solving problems with a limited volume of output, and—ideally—those with a limited amount of input. These restrictions might lead you to assume that the scope of what quantum computers can do is narrow, but the exact opposite is true. Quantum computers provide a level of computational power that allows us to tackle some of the biggest challenges we face. The nuance is in framing problems in a way that makes them solvable . Here are some great examples of how a quantum computer can be used to address some of today’s biggest challenges.

Modelling molecules is a perfect application for quantum computing. In Richard Feynman’s own words, “Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy.”

While we have an accurate understanding of organic molecules—those with S and P orbitals—molecules whose orbitals interact with each other are currently beyond our ability to model accurately. Many of the answers we need to address significant issues, such as world hunger and global warming, come by way of understanding these more difficult molecules. Current technology doesn’t allow us to analyze some of the more complex molecules, however, this is an excellent problem for a quantum computer because input and output are small. There’s a unique approach in quantum computing where, instead of loading the input data, you’re able to encode it into the quantum circuit itself. Modelling molecules are an example of this; the initial positions of the electrons would be the input—also referred to as ‘preparation’—and the final positions of the electron would be the output.

## Materials science

Modelling materials is essentially in the same problem class as modelling molecules, which means quantum computers are also helpful in identifying new possibilities in material science. The ability to develop high-temperature superconductors is a great example. We currently lose around 15% of the power in the energy grid every year due to the resistance in the wires transporting the electricity. Finding a material that can transmit energy without heating up the wires requires modelling properties of materials, a process very similar to modelling molecules. Again, this precise focus has a minimal amount of input and a highly focused output—both great candidates for quantum computing. In addition, materials have a regular structure with (mostly) local interactions making them generally easier to model than chemicals on a quantum computer.

## Cryptography

Many cryptosystems are built using math problems more difficult than a classical computer is able to solve. However, a quantum computer has the computational ability to find solutions to the cryptographic algorithms in use today. Cryptographic problems that use factoring are excellent examples of problems that can be solved with a quantum computer because both the input and output are each a single number. Note that the numbers used in the key are huge, so a significant amount of qubits are needed to calculate the result. A quantum computer’s ability to solve cryptographic algorithms is an issue we take extremely seriously at Microsoft, and we are already working on quantum-safe cryptography protocols to replace those which will be vulnerable to quantum attacks.

## Machine learning and optimization

In general, quantum computers aren’t challenged by the amount of computation needed. Instead, the challenge is getting a limited number of answers and restricting the size of the inputs. Because of this, machine learning problems often don’t make for a perfect fit because of the large amount of input data. However, optimization problems are a type of machine learning problem that can be a good fit for a quantum computer.

Imagine you have a large factory and the goal is to maximize output. To do so, each individual process would need to be optimized on its own, as well as compared against the whole. Here the possible configurations of all the processes that need to be considered are exponentially larger than the size of the input data. With a search space exponentially bigger than the input data, optimization problems are feasible for a quantum computer.

Additionally, due to the unique requirements of quantum programming, one of the unexpected benefits of developing quantum algorithms is identifying new methods to solve problems. In many cases, these new methods can be brought back to classical computing, yielding significant improvements. Implementing these new techniques in the cloud is what we refer to as quantum-inspired algorithms .

Quantum computing brings about a paradigm shift in multiple ways: Not only will quantum computing provide access to new levels of computational ability, but it will also inspire new ways of thinking. For a quantum computer to solve some of our biggest challenges, we have to understand how to frame the problem . As we look at problems in new ways, this shift can, in turn, bring new ideas to how we approach classical computation as well. With more and more individuals considering problems from different angles, more and more ideas and solutions will result. Luckily, you don’t have to wait until quantum computers are readily available to begin considering problems in new ways—you can start today by learning quantum development .

As you dive into the world of quantum development, you’ll practice your ability to think about problems in new ways, get familiar with programming a quantum computer, and even simulate your work so that you’ll be ready once quantum computers are made available.

Get started today with the Microsoft Quantum Development Kit .

## Related blog posts

Quantum networking: a roadmap to a quantum internet .

Quantum computing has the potential to tackle some of our most pressing global issues, from Read more

## Azure Quantum learning resources enable getting ready for a quantum supercomputer

As an industry, we are all collectively committed to bringing scaled quantum computing to fruition. Read more

## Announcing season 2 of the Microsoft Quantum Innovator Series

In this series, you will hear directly from the Microsoft Azure Quantum scientists and leaders Read more

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## New algorithms inspired by quantum computing for simulating polymeric materials

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## The Problem with Quantum Computers

It’s called decoherence—but while a breakthrough solution seems years away, there are ways of getting around it

- By Scott Pakin , Patrick Coles on June 10, 2019

By now, most people have heard that quantum computing is a revolutionary technology that leverages the bizarre characteristics of quantum mechanics to solve certain problems faster than regular computers can. Those problems range from the worlds of mathematics to retail business, and physics to finance. If we get quantum technology right, the benefits should lift the entire economy and enhance U.S. competitiveness.

The promise of quantum computing was first recognized in the 1980s yet remains unfulfilled. Quantum computers are exceedingly difficult to engineer, build and program. As a result, they are crippled by errors in the form of noise, faults and loss of quantum coherence, which is crucial to their operation and yet falls apart before any nontrivial program has a chance to run to completion.

This loss of coherence (called decoherence), caused by vibrations, temperature fluctuations, electromagnetic waves and other interactions with the outside environment, ultimately destroys the exotic quantum properties of the computer. Given the current pervasiveness of decoherence and other errors, contemporary quantum computers are unlikely to return correct answers for programs of even modest execution time.

While competing technologies and competing architectures are attacking these problems, no existing hardware platform can maintain coherence and provide the robust error correction required for large-scale computation. A breakthrough is probably several years away.

The billion-dollar question in the meantime is, how do we get useful results out of a computer that becomes unusably unreliable before completing a typical computation?

Answers are coming from intense investigation across a number of fronts, with researchers in industry, academia and the national laboratories pursuing a variety of methods for reducing errors. One approach is to guess what an error-free computation would look like based on the results of computations with various noise levels. A completely different approach, hybrid quantum-classical algorithms, runs only the most performance-critical sections of a program on a quantum computer, with the bulk of the program running on a more robust classical computer. These strategies and others are proving to be useful for dealing with the noisy environment of today’s quantum computers.

While classical computers are also affected by various sources of errors, these errors can be corrected with a modest amount of extra storage and logic. Quantum error correction schemes do exist but consume such a large number of qubits (quantum bits) that relatively few qubits remain for actual computation. That reduces the size of the computing task to a tiny fraction of what could run on defect-free hardware.

To put in perspective the importance of being stingy with qubit consumption, today’s state-of-the-art, gate-based quantum computers, which use logic gates analogous to those forming the digital circuits found in the computer, smartphone or tablet you’re reading this article on, boast a mere 50 qubits. That is just a tiny fraction of the number of classical bits your device has available to it, typically hundreds of billions.

TAMING DEFECTS TO GET SOMETHING DONE

The trouble is, quantum mechanics challenges our intuition. So we struggle to figure out the best algorithms for performing meaningful tasks. To help overcome these problems, our team at Los Alamos National Laboratory is developing a method to invent and optimize algorithms that perform useful tasks on noisy quantum computers.

Algorithms are the lists of operations that tell a computer to do something, analogous to a cooking recipe. Compared to classical algorithms, the quantum kind are best kept as short as possible and, we have found, best tailored to the particular defects and noise regime of a given hardware device. That enables the algorithm to execute more processing steps within the constrained time frame before decoherence reduces the likelihood of a correct result to nearly zero.

In our interdisciplinary work on quantum computing at Los Alamos, funded by the Laboratory Directed Research and Development program, we are pursuing a key step in getting algorithms to run effectively. The main idea is to reduce the number of gates in an attempt to finish execution before decoherence and other sources of errors have a chance to unacceptably reduce the likelihood of success.

We use machine learning to translate, or compile, a quantum circuit into an optimally short equivalent that is specific to a particular quantum computer. Until recently, we have employed machine-learning methods on classical computers to search for shortened versions of quantum programs. Now, in a recent breakthrough, we have devised an approach that uses currently available quantum computers to compile their own quantum algorithms. That will avoid the massive computational overhead required to simulate quantum dynamics on classical computers.

Because this approach yields shorter algorithms than the state of the art, they consequently reduce the effects of noise. This machine-learning approach can also compensate for errors in a manner specific to the algorithm and hardware platform. It might find, for instance, that one qubit is less noisy than another, so the algorithm preferentially uses better qubits. In that situation, the machine learning creates a general algorithm to compute the assigned task on that computer using the fewest computational resources and the fewest logic gates. Thus optimized, the algorithm can run longer.

This method, which has worked in a limited setting on quantum computers now available to the public on the cloud, also takes advantage of quantum computers’ superior ability to scale-up algorithms for large problems on the larger quantum computers envisioned for the future.

New work with quantum algorithms will give both experts and nonexperts the tools to perform calculations on a quantum computer. Application developers can begin to take advantage of quantum computing’s potential for accelerating execution speed beyond the limits of conventional computing. These advances may bring us all several steps closer to having robust, reliable large-scale quantum computers to solve complex real-world problems that bring even the fastest classical computers to their knees.

The views expressed are those of the author(s) and are not necessarily those of Scientific American.

## ABOUT THE AUTHOR(S)

Scott Pakin is a computer scientist in the Applied Computer Science group at Los Alamos National Laboratory. With co-principal investigator Wojciech Zurek, he leads the Taming Defects in Quantum Computers project at Los Alamos.

Patrick Coles is a quantum physicist in the Physics of Condensed Matter and Complex Systems group at Los Alamos National Laboratory and is a co-investigator on the Taming Defects in Quantum Computers project at Los Alamos.

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WGN-TV Chicago

## Quantum capital of the world: Emerging field that could solve ‘unsolvable’ problems

Posted: November 2, 2023 | Last updated: November 2, 2023

CHICAGO — Think of a corn maze as a problem. Think of the people in the maze as traditional computers trying to solve the problem. They’re limited to attempting one route at a time.

But what if they could try all of the potential routes at the same time?

## THE ‘QUANTUM LEAP’

That’s one way of thinking about the difference between our current computers and quantum computers.

The ones we use today process information using binary digits or “bits” that are either in the state of zero or one, handling one input, or one maze route, at a time.

A quantum computer processes more information faster, using quantum bits – or q-bits. The process is so fast and powerful because the data is in multiple states all at once.

“A little bit like spinning a coin on a table,” said Professor David Awschalom, a University of Chicago physicist and the director of Chicago Quantum Exchange. “Is it heads or is it tails? it’s a combination.”

Awschalom said it means instead of the single answer we might get from a classical computer, a quantum computer can try infinite answers to find the right way out of the maze, or the right way to solve any number of problems.

“So, it means you can address problems that are really unsolvable in today’s technology,” Awschalom said.

## PRACTICAL APPLICATIONS

Now imagine if we could apply quantum mechanics, quantum technology, and quantum computing power to real-world problems.

We might find solutions to traffic congestion, identity theft, risk in investing, or detecting diseases like cancer in the earliest stages and then creating the pharmaceuticals to treat them.

There are many more possible applications, according to Awschalom.

“How do you transport energy efficiently across a country?,” he said. “How does a package delivery service know the fastest way to deliver packages across the nation? So many problems that are very complex are reachable with quantum computers.”

## PRIORITY FOR PRITZKER

Quantum technology has the potential to shape the future. That’s why Illinois Gov. JB Pritzker is focused on making Chicago the quantum capital of the world.

PRITZKER: “It’s the next phase of technological development in the world, not just for the United States, this is a worldwide competition.”

WGN: “How is Chicago going to win this competition with the entire world?”

PRITZKER: “In order to make Chicago the hub of quantum development, you had to have the universities and laboratories willing to work together. The collaboration between them is vital and working with Purdue in Indiana, and University of Wisconsin in Madison, bringing all of that together and having Chicago as the center of that is vital for our future. That didn’t happen accidentally.”

WGN: “A phone call from the governor gets all the universities, gets the labs. You’re the guy who can pull it together.”

PRITZKER: “What I knew is that there are federal dollars, there are private dollars, there are foundation dollars that were available to a city, as state a locale that was making real investments and actually making progress in quantum mechanics and quantum computing. So if the state was willing to step forward with a major investment. We invested $200 million back in 2019, if the state was willing to do that, it would bring enormous attention and it would catalyze those other investments coming here.”

## PUBLIC, PRIVATE INVESTMENT CREATE ‘QUANTUM ECOSYSTEM’

Illinois receives two of every $5 the federal government spends on quantum technology. It is home to four of the nation’s ten quantum centers, the most of any state.

“We are the chosen location for the United States government to put a significant amount of its dollars toward quantum development right here in Illinois.”

Private investment is also fueling Chicago’s quantum economy, according to Robin Ficke of World Business Chicago.

FICKE: “If you look at private investment, we’re number two, so we actually really are the epicenter for interest in quantum.”

LOWE: “What are the factors that are making Chicago a quantum technology capital?”

FICKE: “There are three things. First, we have a deep bench of talent coming out of universities. then when people leave the universities, we have a robust ecosystem that they can interact with and finally when they’re ready to launch their quantum sensing products or computing products, we have a robust and diverse industry base that they can interact with.”

One company on the cutting edge of Chicago’s economic present is in a building that symbolizes the city’s economic past – the Chicago Board of Trade.

At the offices of Infleqtion, 20 employees are building the software for quantum computers. Pranav Gokhale, the company’s vice president of quantum software, says he always thought he’d start a quantum tech company in Silicon Valley.

“But a couple of years into grad school I realized that this is where we’d want to build a company, this is where the talent was, this is where technology [was], and where the business development was,” Gokhale said.

Chicago is the leader in U.S.-based quantum investments. It ranks only behind California for the number of quantum start-up companies.

“Chicago is becoming the center of that industrial revolution for what quantum technology will bring,” Gokhale said.

The Infleqtion team was celebrating the launch of its new software product this fall.

## TOP UNIVERSITIES PROVIDE THE QUANTUM WORKFORCE

Technology industry experts have said the key to any region’s success is the available workforce, and that is where Illinois shines, ranking second in the nation for producing Ph.D. graduates in quantum-related fields.

Swathi Chandrika, is working at a University of Chicago lab three stories underground. She and other doctoral students are fine-tuning experiments and building the devices that will connect to a 124-mile fiber-optic network running from the university’s campus on Chicago’s South Side to two federally funded labs in the suburbs: Argonne National Laboratory and Fermi National Accelerator Laboratory.

“We’ve built one of the first quantum links or quantum networks between this building, where you’re standing right now, downtown Hyde Park, and Argonne National Laboratories,” Awschalom said. “We’re extending it throughout the state right now, and the idea is can we use this as a testbed for companies to come, bring their technology, try it in the real-world network. There is weather in Chicago, there are big temperature changes, we use optical fibers to transmit quantum information and those change with temperature. Change with vibrations on the Eisenhower (Expressway), right? On the tollways? How does quantum information work in the real world? These are things on which we’re working together with industry to explore.”

## DUALITY, CHICAGO QUANTUM EXCHANGE, WORLD BUSINESS CHICAGO CONNECT RESEACH TO BUSINESS

Building quantum connections is also a focus of World Business Chicago CEO Michael Fassnacht.

“Quantum will be ultimately the foundation in 10, 15, 20 years of how we live our lives and how we do business, because it will be the foundation of any computing activity that’s happening,” Fassnacht said. “The great thing about quantum is, if you do it right, you solve real problems that face mankind. It’s not building another dating app, like Silicon Valley likes to do.”

Quantum has become a buzzword in popular culture from the show ‘Quantum Leap’ to Marvel’s movie ‘Quantummania.’ It seems like a concept too big to grasp, but quantum fields explore the smallest particles in the universe. “Technology on the scale of microns and sub-microns even down to the nanometer, almost down to the atomic scale,” Awschalom said.

It’s at that level where those unusual rules of Quantum mechanics exist, the ones that allow for all those possibilities to ‘solve the maze,’ because data exists in two states at once, like the spinning coin.

## LESSONS FROM THE PAST

For Pritzker, who is leading Illinois into the quantum future, it’s a story from the not-so-distant past that should guide the state’s next steps.

“I want to analogize it to something else that happened in Illinois, about 30 years ago,” Pritzker said. “That was the development of the browser for the internet.”

It was known as “Mosaic” – the first internet browser to incorporate graphics, text, and hypertext or “links” to other pages. It was the precursor to Netscape, Explorer, and Chrome. It was developed at the National Center for Supercomputing Applications at the University of Illinois, Urbana-Champaign in 1992.

But Mosaic didn’t stay in Illinois, and neither did other tech start-ups.

“Making sure we don’t lose out on this next great opportunity,” Pritzker said. “In Illinois, we lost out 30 years ago at the University of Illinois when the browser got up and left and went to Silicon Valley, when YouTube and PayPal got up and left the University of Illinois and went to California. That’s not happening now. Quantum is the next big thing, and companies are coming to Chicago to take advantage of that.”

For the latest news, weather, sports, and streaming video, head to WGN-TV.

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## IMAGES

## VIDEO

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Quantum and classical computers both try to solve problems, but the way they manipulate data to get answers is fundamentally different. This section provides an explanation of what makes quantum computers unique by introducing two principles of quantum mechanics crucial for their operation, superposition and entanglement.

As quantum computing is still a nascent field, most of the problems we know quantum computers will solve are phrased in abstract mathematics. Some of these will have "real world" applications ...

Google has set a quantum-computing road map for itself with six major milestones. ... fulfil their promise of solving problems that are beyond the reach of classical machines — such as factoring ...

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Quantum computing holds the promise of solving some of our planet's biggest challenges - in the areas of environment, agriculture, health, energy, climate, materials science, and more. For some of these problems, classical computing is increasingly challenged as the size of the system grows. When designed to scale, quantum systems will likely ...

Quantum computers are naturally good for solving complex quantum many-body problems and thus may be instrumental in applications involving quantum chemistry. Therefore, one can expect that quantum-enhanced generative models [83] including quantum GANs [84] may eventually be developed into ultimate generative chemistry algorithms.

Quantum computing (QC) is a field that has witnessed a rapid increase in interest and development over the past few decades since it was theoretically shown that quantum computers can provide an ...

R.Cumming, T.Thomas. Quantum computing is an important developing technology with the potential to revolutionise the landscape of scientific and business problems that can be practically addressed. The widespread excitement derives from the potential for a fault tolerant quantum computer to solve previously intractable problems.

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scale quantum devices (NISQ) for solving real-world problems. It is likely that quantum methods can efﬁciently solve certain (NP-)hard optimization problems where classical approaches fail. In our perspective, we examine the ﬁeld of quantum optimization where we solve optimisation problems using quantum computers.

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When classical computers solve a problem with multiple variables, they must conduct a new calculation every time a variable changes. Each calculation is a single path to a single result. Quantum computers, however, have a larger working space, which means they can explore a massive number of paths simultaneously.

The problem is, the physics underlying quantum computers is itself fundamentally linear. "It's like teaching a car to fly," said Bobak Kiani, a co-author of the MIT study. So the trick is finding a way to mathematically convert a nonlinear system into a linear one.

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Quantum computing promises transformational gains for solving some problems, but little to none for others. For anyone hoping to use quantum computers now or in the future, it is important to know which problems will benefit. In this paper, we introduce a framework for answering this question both intuitively and quantitatively. The underlying structure of the framework is a race between ...

What problems will we solve with a quantum computer? - Microsoft Research Microsoft Research Blog What problems will we solve with a quantum computer? Published July 5, 2017 New paper suggests quantum computers will address problems that could have substantial scientific and economic impact

Quantum computers can solve NP-hard problems that classical computers are unable to solve. Currently, the two most important and notable complexity classes are "P" and "NP." P represents problems that can be solved in polynomial time by a classical computer.

Simon's problem. In computational complexity theory and quantum computing, Simon's problem is a computational problem that is proven to be solved exponentially faster on a quantum computer than on a classical (that is, traditional) computer. The quantum algorithm solving Simon's problem, usually called Simon's algorithm, served as the ...

What Kind of Problems Can Quantum Computers Solve? Sara A. Metwalli · Follow Published in Level Up Coding · 6 min read · Jul 20, 2020 1 And What Kind of Problems Can't They Solve? Photo by Gabriel Crismariu on Unsplash B efore we get into the good stuff, I need to tell you that quantum computers are not magical.

Around the same time they also proved that quantum computers can solve all the problems that classical computers can solve. That is, BQP contains all the problems that are in P. 1. When thinking about complexity classes, examples help.

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What would be a sufficient amount of qubits to solve such problem and what amount do we currently have? First and foremost, complexity-theory wise, it doesn't matter how many qubits a current or future system has - from a complexity classes point of view, a quantum computer 'is' a quantum computer, and a classical computer 'is' a classical ...

Quantum computing is a promising technology that may provide breakthrough solutions to today's difficult problems such as breaking encryption and solving large-scale combinatorial optimization problems. An algorithm referred to as Quantum Approximate Optimization Algorithm (QAOA) has been recently proposed to approximately solve hard problems using a protocol know as bang-bang.

CHICAGO — Think of a corn maze as a problem. Think of the people in the maze as traditional computers trying to solve the problem. They're limited to attempting one route at a time.

Jan 15 1 There are many problems that require finding the maximum or minimum of a function, called the objective function, that depends on several (or many) variables; certain constraints may or may not need to be applied to the variables. The variables could be binary, integer, elements in sets, floating point, etc.