Can Quantum Computers Solve Linear Algebra Problems to Advance Engineering Applications?

Fundamental transformations in the basic logic of computing are few and far between. Since the invention of digital computers in the early 1940s, the logic underlying computation has remained the same, even as computing hardware evolved from vacuum tubes to silicon transistors. With the advent of quantum computation, a fundamental transformation is near. Quantum computation is based on a different type of logic: rather than being in one of the two states of a classical bit, a quantum bit or qubit can be in a superposition of two states simultaneously. Operations and measurements on these qubits obey the constraints of quantum mechanics. It is now understood that quantum computers have great power in principle to go beyond classical computers, but that not every application is well suited to implementation on quantum computers. For reasons explained in more detail in the Introduction, scientific problems in chemical and materials sciences are uniquely suited to take advantage of quantum computing in the relatively near future. Indeed, quantum computing offers the best hope to solve many of the most important and difficult problems in this field. For example, quantum materials, such as superconductors and complex magnetic materials, show novel kinds of ordered phases that are natural from the point of view of quantum mechanics but difficult to access via computation on classical computers. Quantum sensors based on solid materials are already widely used but could be greatly improved with insight from quantum computations, as could materials for information technologies. Quantum chemical dynamics is another example of a problem that is intrinsically well suited to studies on quantum computers. Applications of quantum chemical dynamics include catalysis, artificial photosynthesis, and other industrially important processes. Quantum computers exist in the laboratory and are beginning to exceed 50 qubits, which is roughly the size beyond which their behavior cannot be predicted or emulated on present-day classical supercomputers. While a quantum computer of 50 qubits is almost certainly not powerful enough to tackle the major scientific challenges in chemical and materials sciences, some of these major challenges start to become accessible with a few hundred qubits if error rates can be kept small. This roundtable was convened to ask how emerging quantum computers can be applied to major scientific problems in chemical and materials sciences, in light of Basic Energy Sciences’s leading role in these fields and the Department of Energy’s leading role in high-performance scientific computation more generally. The main outcome of the roundtable was a consensus that there are scientific problems of great importance on which emerging quantum computers have the potential for disruptive impact, and where comparable progress is unlikely to occur by other means.

Recent improvements in control of quantum systems make it seem feasible to finally build a quantum computer within a decade. While it has been shown that such a quantum computer can in principle solve certain small electronic structure problems and idealized model Hamiltonians, the highly relevant problem of directly solving a complex correlated material appears to require a prohibitive amount of resources. Here, we show that by using a hybrid quantum-classical algorithm that incorporates the power of a small quantum computer into a framework of classical embedding algorithms, the electronic structure of complex correlated materials can be efficiently tackled using a quantum computer. In our approach, the quantum computer solves a small effective quantum impurity problem that is self-consistently determined via a feedback loop between the quantum and classical computation. Use of a quantum computer enables much larger and more accurate simulations than with any known classical algorithm, and will allow many open questions in quantum materials to be resolved once a small quantum computer with around one hundred logical qubits becomes available.

As quantum computing development advances closer toward achieving fault-tolerant error-corrected realisation, debates on applications, impacts, risks, and benefits of quantum computing are timely and due. While there is awareness of the potential power and complexity of quantum computers, there has been relatively little attention on the social impacts and ethical implications of this technology. In grappling with the social impacts of quantum computing, some stakeholders have turned to applied ethics, specifically (classical) computer and Artificial Intelligence (AI) ethics, for guidance. While computer and AI ethics are useful starting points for evaluating the ethical issues posed by quantum computing, uncritically applying existing sets of ethical principles risks overlooking how quantum computing differs from these technologies. We argue that borrowing of ethical principles and guidelines from AI and computing is inappropriate for several reasons: (1) quantum computing, classical computing, and AI are different technologies with significant material differences; (2) unlike AI and classical computers which have become established, quantum computing is an emerging technology, which has implications on the levels of accessibility and immediate impact in society; and (3) there are significant differences in the way AI and quantum computing have been developed and by whom. We also briefly summarise some of the unique risks and societ al impacts posed by quantum computing. We posit that these reasons support the argument for a new sub-field of quantum computing ethics, which would allow a relevant scholarship to develop and provide guidance as quantum computing technology continues to mature.

In the world of computation, the invention of Computers has reduced human effort for accomplishing task. Depending on the type, size and nature of the work the computers are manufactured in different magnitudes. As the technology grows, it facilitated to manufacture high performing, high throughput, and high memory Classical computers. Classical computers store any information in the form of bits in the memory device, which can be either 1 or 0 (binary system) and all the processing will be done in the same logic i.e., high voltage/low voltage. Because of this logic the classical computers solve some problems by using more resources. Quantum computer are the next level computers which uses quantum theory principles for solving problems. The quantum computer uses Qubit as basic unit which uses principles of quantum computing such as superposition and entanglement, which gives qubit, the efficiency of exhibiting many logical states at a time. This paper shows a comparative study on classical and quantum computers and illustrates various problems in classical computers and respective solutions with use of quantum computers.

Quantum computing is a multidisciplinary field at the intersectionof quantum physics, computer science, and mathematics, aiming to har-ness quantum mechanical phenomena for computational advantage. Thisarticle provides a comprehensive review of both the foundational princi-ples and advanced theoretical models of quantum computing. We beginwith the basic framework of quantum computation, introducing the con-cept of qubits as two-level quantum systems, the linear algebra of Hilbertspaces, Dirac’s bra–ket notation, and the postulates of quantum mechan-ics (state superposition, unitary evolution, and projective measurement).We then discuss quantum logic gates and circuits, highlighting how clas-sical reversible computation principles extend to universal quantum gatesets. Next, we survey key quantum algorithms (from Shor’s factoring toGrover’s searching and beyond) and the complexity-theoretic implicationsof quantum computers, contrasting the class BQP with classical complex-ity classes. We cover the theoretical foundations of quantum error cor-rection and fault tolerance, explaining how quantum information can beprotected from decoherence using redundancy and how fault-tolerant pro-tocols can, in principle, allow scalable quantum computation despite noisyhardware. We then explore advanced models and paradigms of quantumcomputation, including measurement-based quantum computing (one-wayquantum computing), topological quantum computation with anyons, adi-abatic quantum computing and quantum annealing, continuous-variablequantum computing, and related approaches. We also provide an overviewof quantum information theory as it relates to computation, discussingentanglement, quantum entropy, and information-theoretic limits like theHolevo bound. Finally, we review the leading physical implementations(quantum hardware models) of quantum computers—trapped ions, super-conducting circuits, photonic systems, solid-state spin qubits, and oth-ers—outlining the challenges and achievements of each approach.

Exploring the trade-off between computational power and energy efficiency: An analysis of the evolution of quantum computing and its relation to classical computing

During recent years, quantum computers have received increasing attention, primarily due to their ability to significantly increase computational performance for specific problems. Computational performance could be improved for mathematical optimization by quantum annealers. This special type of quantum computer can solve quadratic unconstrained binary optimization problems. However, multi-energy systems optimization commonly involves integer and continuous decision variables. Due to their mixed-integer problem structure, quantum annealers cannot be directly used for multi-energy system optimization.To solve multi-energy system optimization problems, we present a hybrid Benders decomposition approach combining optimization on quantum and classical computers. In our approach, the quantum computer solves the master problem, which involves only the integer variables from the original energy system optimization problem. The subproblem includes the continuous variables and is solved by a classical computer. For better performance, we apply improvement techniques to the Benders decomposition. We test the approach on a case study to design a cost-optimal multi-energy system. While we provide a proof of concept that our Benders decomposition approach is applicable for the design of multi-energy systems, the computational time is still higher than for approaches using classical computers only. We therefore estimate the potential improvement of our approach to be expected for larger and fault-tolerant quantum computers.

Classic computing can take credit for technology ranging from mobile phones to supercomputers. But in recent years, a budding counterpart to these conventional devices has emerged: quantum computers. Whereas classic computing sometimes fails to solve complex calculations, such as factoring hundred-digit numbers, quantum computing holds the potential to easily tackle such problems. The field of quantum computing has attracted researchers, such as National Academy of Sciences member Christopher Monroe. An experimental atomic physicist at the University of Maryland and fellow of the Joint Quantum Institute and the Joint Center for Quantum Information and Computer Science in Maryland, Monroe uses lasers to exploit atomic particles to study complex computational problems. Monroe spoke to PNAS about how quantum computing might evolve. Christopher Monroe. > PNAS:Can you describe some of the basic mechanics of quantum computing? > Monroe:It all comes down to the superposition principle, in which a quantum system can exist in multiple states at the same time. Consider the fundamental unit of information: the bit. A bit is binary information that can be a 0 or a 1. A quantum bit, or qubit, can be in a superposition of both 0 and 1 as long as it’s isolated and unobserved. A single qubit is pretty trivial, but when you put many qubits together, there become exponentially many possibilities. > > Let’s consider 300 qubits. That’s an interesting number because the number of possible configurations, 2300, is more than the number of particles in the universe. You could never simulate what happens with 300 qubits on a classical machine; there’s not enough stuff in the universe to do it. The magic with quantum is the power of exponential growth: every time you add a qubit, you double the number of configurations. > > But to control these qubits and exploit this massive storage, we …

Quantum computing is an extremely promising research combining theoretical and experimental quantum physics, mathematics, quantum information theory and computer science. Classical simulation of quantum computations will cover part of the gap between the theoretical mathematical formulation of quantum mechanics and the realization of quantum computers. One of the most important problems in “quantum computer science” is the development of new symbolic languages for quantum computing and the adaptation of existing symbolic languages for classical computing to quantum algorithms. The present paper is devoted to the adaptation of the Mathematica symbolic language to known quantum algorithms and corresponding simulation on the classical computer. Concretely we shall represent in the Mathematica symbolic language Simon’s algorithm, the Deutsch-Josza algorithm, Grover’s algorithm, Shor’s algorithm and quantum error-correcting codes. We shall see that the same framework can be used for all these algorithms. This framework will contain the characteristic property of the symbolic language representation of quantum computing and it will be a straightforward matter to include this framework in future algorithms.KeywordsDeutsch-Josza algorithmGrover’s algorithmQuantum computingQuantum error-correctingShor’s algorithmSimon’s algorithmSimulation of quantum algorithms

With the explosion of big data, higher requirements for computational efficiency have emerged. Compared with classical computing, quantum computing possesses quantum parallelism due to the unique nature of quantum systems. It has been found that many classical algorithms can be accelerated using quantum computing. In addition to factorizing a large integer, quantum computers can be used for data processing and analysis. In recent years, two frontiers, i.e., big data and quantum computing have begun to merge. Though practical quantum computers have not yet been built, theoretical studies have made some important progress. In this review, we introduce the basic principles of quantum computing. As a representative example, we describe the Grover search algorithm and its important generalizations. Quantum machine learning is the entry point for the integration of big data with quantum computation. We review in detail, the applications of quantum computation in data mining, the main application of machine learning. Other aspects of quantum computing in big data are also briefly summarized.

Experimental realization of a universal set of quantum logic gates is the central requirement for the implementation of a quantum computer. In an 'all-geometric' approach to quantum computation, the quantum gates are implemented using Berry phases and their non-Abelian extensions, holonomies, from geometric transformation of quantum states in the Hilbert space. Apart from its fundamental interest and rich mathematical structure, the geometric approach has some built-in noise-resilience features. On the experimental side, geometric phases and holonomies have been observed in thermal ensembles of liquid molecules using nuclear magnetic resonance; however, such systems are known to be non-scalable for the purposes of quantum computing. There are proposals to implement geometric quantum computation in scalable experimental platforms such as trapped ions, superconducting quantum bits and quantum dots, and a recent experiment has realized geometric single-bit gates in a superconducting system. Here we report the experimental realization of a universal set of geometric quantum gates using the solid-state spins of diamond nitrogen-vacancy centres. These diamond defects provide a scalable experimental platform with the potential for room-temperature quantum computing, which has attracted strong interest in recent years. Our experiment shows that all-geometric and potentially robust quantum computation can be realized with solid-state spin quantum bits, making use of recent advances in the coherent control of this system.

The 2010 Benjamin Franklin Medal in Physics presented to J. Ignacio Cirac, David J. Wineland and Peter Zoller

Many challenging scheduling, planning, and resource allocation problems come with real-world input data and hard problem constraints, and reduce to optimizing a cost function over a combinatorially defined feasible set, such as colorings of a graph. Toward tackling such problems with quantum computers using quantum approximate optimization algorithms, we present novel efficient quantum alternating operator ansatz (QAOA) constructions for optimization problems over proper colorings of chordal graphs. As our primary application, we consider the flight-gate assignment problem , where flights are assigned to airport gates as to minimize the total transit time of all passengers, and feasible assignments correspond to proper graph colorings of a conflict graph derived instancewise from the input data. We leverage ideas from classical algorithms and graph theory to show our constructions have the desirable properties of restricting quantum state evolution to the feasible subspace, and satisfying a particular reachability condition for most problem parameter regimes. Using classical preprocessing we show that we can always find and construct a suitable initial quantum (superposition) state efficiently. We show our constructions in detail, including explicit decompositions to a universal set of basic quantum gates, which we use to bound the required resource scaling as low-degree polynomials of the input parameters. In particular, we derive novel QAOA mixing operators and show that their implementation cost is commensurate with that of the QAOA phase operator for flight-gate assignment. A number of quantum circuit diagrams are included such that our constructions may be used as a template toward development and implementation of quantum gate-model approaches for a wider variety of potentially impactful real-world applications.

Quantum computers have the potential to create important new opportunities for ongoing essential research on gauge theories. They can provide simulations that are unattainable on classical computers such as sign-problem afflicted models or time evolutions. In this work, we variationally prepare the low-lying eigenstates of a non-Abelian gauge theory with dynamically coupled matter on a quantum computer. This enables the observation of hadrons and the calculation of their associated masses. The SU(2) gauge group considered here represents an important first step towards ultimately studying quantum chromodynamics, the theory that describes the properties of protons, neutrons and other hadrons. Our calculations on an IBM superconducting platform utilize a variational quantum eigensolver to study both meson and baryon states, hadrons which have never been seen in a non-Abelian simulation on a quantum computer. We develop a hybrid resource-efficient approach by combining classical and quantum computing, that not only allows the study of an SU(2) gauge theory with dynamical matter fields on present-day quantum hardware, but further lays out the premises for future quantum simulations that will address currently unanswered questions in particle and nuclear physics.

Quantum machine learning recently gained prominence due to the computational ability of quantum computers in solving machine learning problems that are intractable on a classical computer. However, achieving a quantum advantage on present-day quantum computers remains an open challenge. In this work, we primarily focus on solving machine learning problems using a hybrid model based on both quantum and classical computers together for the classification task. We propose the quantum multi-class classifier (QMCC) as a variational circuit with a hybrid classical-quantum approach using quantum mechanical properties such as superposition and entanglement. A unitary operation on a single qubit for the state preparation is designed and also demonstrated using a real quantum computer on the IBMQX platform. The entire variational circuit for the classification task is implemented on a quantum simulator. We performed our quantum simulations on three benchmark datasets: Iris dataset, Banknote Authentication (BNA) dataset, and Wireless Indoor Localization (WIL) dataset for machine learning algorithms. Our simulation results show that the proposed QMCC model classified Iris dataset with an accuracy of 92.10%, BNA dataset with an accuracy of 89.50%, and WIL dataset with an accuracy of 91.73%. The proposed model can also be extended to multiple class classifiers.