# Quantum annealing

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Quantum annealing (QA) is a metaheuristic for finding the global minimum of a given objective function over a given set of candidate solutions (candidate states), by a process using quantum fluctuations (in other words, a meta-procedure for finding a procedure that finds an absolute minimum size/length/cost/distance from within a possibly very large, but nonetheless finite set of possible solutions using quantum fluctuation-based computation instead of classical computation). Quantum annealing is used mainly for problems where the search space is discrete (combinatorial optimization problems) with many local minima; such as finding the ground state of a spin glass [1] or the traveling salesman problem. Quantum annealing was first proposed in 1988 by B. Apolloni, N. Cesa Bianchi and D. De Falco. [2] [3] It was formulated in its present form by T. Kadowaki and H. Nishimori (ja) in "Quantum annealing in the transverse Ising model" [4] though a proposal in a different form had been made by A. B. Finnila, M. A. Gomez, C. Sebenik and J. D. Doll, in "Quantum annealing: A new method for minimizing multidimensional functions". [5]

## Contents

Quantum annealing starts from a quantum-mechanical superposition of all possible states (candidate states) with equal weights. Then the system evolves following the time-dependent Schrödinger equation, a natural quantum-mechanical evolution of physical systems. The amplitudes of all candidate states keep changing, realizing a quantum parallelism, according to the time-dependent strength of the transverse field, which causes quantum tunneling between states. If the rate of change of the transverse field is slow enough, the system stays close to the ground state of the instantaneous Hamiltonian (also see adiabatic quantum computation). [6] If the rate of change of the transverse field is accelerated, the system may leave the ground state temporarily but produce a higher likelihood of concluding in the ground state of the final problem Hamiltonian, i.e., diabatic quantum computation. [7] [8] The transverse field is finally switched off, and the system is expected to have reached the ground state of the classical Ising model that corresponds to the solution to the original optimization problem. An experimental demonstration of the success of quantum annealing for random magnets was reported immediately after the initial theoretical proposal. [9]

## Comparison to simulated annealing

Quantum annealing can be compared to simulated annealing, whose "temperature" parameter plays a similar role to QA's tunneling field strength. In simulated annealing, the temperature determines the probability of moving to a state of higher "energy" from a single current state. In quantum annealing, the strength of transverse field determines the quantum-mechanical probability to change the amplitudes of all states in parallel. Analytical [10] and numerical [11] evidence suggests that quantum annealing outperforms simulated annealing under certain conditions (see [12] for a careful analysis).

## Quantum mechanics: analogy and advantage

The tunneling field is basically a kinetic energy term that does not commute with the classical potential energy part of the original glass. The whole process can be simulated in a computer using quantum Monte Carlo (or other stochastic technique), and thus obtain a heuristic algorithm for finding the ground state of the classical glass.

In the case of annealing a purely mathematical objective function, one may consider the variables in the problem to be classical degrees of freedom, and the cost functions to be the potential energy function (classical Hamiltonian). Then a suitable term consisting of non-commuting variable(s) (i.e. variables that have non-zero commutator with the variables of the original mathematical problem) has to be introduced artificially in the Hamiltonian to play the role of the tunneling field (kinetic part). Then one may carry out the simulation with the quantum Hamiltonian thus constructed (the original function + non-commuting part) just as described above. Here, there is a choice in selecting the non-commuting term and the efficiency of annealing may depend on that.

It has been demonstrated experimentally as well as theoretically, that quantum annealing can indeed outperform thermal annealing (simulated annealing) in certain cases, especially where the potential energy (cost) landscape consists of very high but thin barriers surrounding shallow local minima. [13] Since thermal transition probabilities (proportional to ${\displaystyle e^{-{\frac {\Delta }{k_{B}T}}}}$, with ${\displaystyle T}$ the temperature and ${\displaystyle k_{B}}$ the Boltzmann constant) depend only on the height ${\displaystyle \Delta }$ of the barriers, for very high barriers, it is extremely difficult for thermal fluctuations to get the system out from such local minima. However, as argued earlier in 1989 by Ray, Chakrabarti & Chakrabarti, [1] the quantum tunneling probability through the same barrier (considered in isolation) depends not only on the height ${\displaystyle \Delta }$ of the barrier, but also on its width ${\displaystyle w}$ and is approximately given by ${\displaystyle e^{-{\frac {{\sqrt {\Delta }}w}{\Gamma }}}}$, where ${\displaystyle \Gamma }$ is the tunneling field. [14] This additional handle through the width ${\displaystyle w}$, in presence of quantum tunneling, can be of major help: If the barriers are thin enough (i.e. ${\displaystyle w\ll {\sqrt {\Delta }}}$), quantum fluctuations can surely bring the system out of the shallow local minima. For an ${\displaystyle N}$-spin glass, the barrier height ${\displaystyle \Delta }$ becomes of order ${\displaystyle N}$. For constant value of ${\displaystyle w}$ one gets ${\displaystyle \tau }$ proportional to ${\displaystyle e^{\sqrt {N}}}$ for the annealing time (instead of ${\displaystyle \tau }$ proportional to ${\displaystyle e^{N}}$ for thermal annealing), while ${\displaystyle \tau }$ can even become ${\displaystyle N}$-independent for cases where ${\displaystyle w}$ decreases as ${\displaystyle 1/{\sqrt {N}}}$. [15] [16]

It is speculated that in a quantum computer, such simulations would be much more efficient and exact than that done in a classical computer, because it can perform the tunneling directly, rather than needing to add it by hand. Moreover, it may be able to do this without the tight error controls needed to harness the quantum entanglement used in more traditional quantum algorithms. Some confirmation of this is found in exactly solvable models. [17]

Timeline for Quantum Annealing in Ising Spin Glasses: *1981- Ising spin glass model introduced and studied for its transition behavior; [18] *1989- Idea proposed that quantum fluctuations could help explore rugged energy landscapes of the Ising spin glasses by escaping from local minima (having tall but thin barriers) using tunneling; [1] *1991- Experimental realization of quantum Ising spin glass in LiHoYF; [19] *1998- First Monte Carlo simulation demonstrating quantum annealing in Ising glass systems; [4] *1999- First experimental demonstration of quantum annealing in LiHoYF Ising glass magnets; [20] *2011- Superconducting-circuit quantum annealing machine for Ising spin glass systems realized and marketed by D-Wave Systems. [21]

## D-Wave implementations

In 2011, D-Wave Systems announced the first commercial quantum annealer on the market by the name D-Wave One and published a paper in Nature on its performance. [21] The company claims this system uses a 128 qubit processor chipset. [22] On May 25, 2011, D-Wave announced that Lockheed Martin Corporation entered into an agreement to purchase a D-Wave One system. [23] On October 28, 2011 USC's Information Sciences Institute took delivery of Lockheed's D-Wave One.

In May 2013 it was announced that a consortium of Google, NASA Ames and the non-profit Universities Space Research Association purchased an adiabatic quantum computer from D-Wave Systems with 512 qubits. [24] [25] An extensive study of its performance as quantum annealer, compared to some classical annealing algorithms, is already available. [26]

In June 2014, D-Wave announced a new quantum applications ecosystem with computational finance firm 1QB Information Technologies (1QBit) and cancer research group DNA-SEQ to focus on solving real-world problems with quantum hardware. [27] As the first company dedicated to producing software applications for commercially available quantum computers, 1QBit's research and development arm has focused on D-Wave's quantum annealing processors and has successfully demonstrated that these processors are suitable for solving real-world applications. [28]

With demonstrations of entanglement published, [29] the question of whether or not the D-Wave machine can demonstrate quantum speedup over all classical computers remains unanswered. A study published in Science in June 2014, described as "likely the most thorough and precise study that has been done on the performance of the D-Wave machine" [30] and "the fairest comparison yet", attempted to define and measure quantum speedup. Several definitions were put forward as some may be unverifiable by empirical tests, while others, though falsified, would nonetheless allow for the existence of performance advantages. The study found that the D-Wave chip "produced no quantum speedup" and did not rule out the possibility in future tests. [31] The researchers, led by Matthias Troyer at the Swiss Federal Institute of Technology, found "no quantum speedup" across the entire range of their tests, and only inconclusive results when looking at subsets of the tests. Their work illustrated "the subtle nature of the quantum speedup question". Further work [32] has advanced understanding of these test metrics and their reliance on equilibrated systems, thereby missing any signatures of advantage due to quantum dynamics.

There are many open questions regarding quantum speedup. The ETH reference in the previous section is just for one class of benchmark problems. Potentially there may be other classes of problems where quantum speedup might occur. Researchers at Google, LANL, USC, Texas A&M, and D-Wave are working hard to find such problem classes. [33]

In December 2015, Google announced that the D-Wave 2X outperforms both simulated annealing and Quantum Monte Carlo by up to a factor of 100,000,000 on a set of hard optimization problems. [34]

D-Wave's architecture differs from traditional quantum computers. It is not known to be polynomially equivalent to a universal quantum computer and, in particular, cannot execute Shor's algorithm because Shor's algorithm is not a hillclimbing process. Shor's algorithm requires a universal quantum computer. D-Wave claims only to do quantum annealing.[ citation needed ]

"A cross-disciplinary introduction to quantum annealing-based algorithms" [35] presents an introduction to combinatorial optimization (NP-hard) problems, the general structure of quantum annealing-based algorithms and two examples of this kind of algorithms for solving instances of the max-SAT and Minimum Multicut problems, together with an overview of the quantum annealing systems manufactured by D-Wave Systems. Hybrid quantum-classic algorithms for large-scale discrete-continuous optimization problems were reported to illustrate the quantum advantage. [36]

## Related Research Articles

Quantum computing is the exploitation of collective properties of quantum states, such as superposition and entanglement, to perform computation. The devices that perform quantum computations are known as quantum computers. They are believed to be able to solve certain computational problems, such as integer factorization, substantially faster than classical computers. The study of quantum computing is a subfield of quantum information science. Expansion is expected in the next few years as the field shifts toward real-world use in pharmaceutical, data security and other applications.

Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. It is often used when the search space is discrete. For problems where finding an approximate global optimum is more important than finding a precise local optimum in a fixed amount of time, simulated annealing may be preferable to exact algorithms such as gradient descent or branch and bound.

This is a timeline of quantum computing.

In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a quantum computer. Although all classical algorithms can also be performed on a quantum computer, the term quantum algorithm is usually used for those algorithms which seem inherently quantum, or use some essential feature of quantum computation such as quantum superposition or quantum entanglement.

Extremal optimization (EO) is an optimization heuristic inspired by the Bak–Sneppen model of self-organized criticality from the field of statistical physics. This heuristic was designed initially to address combinatorial optimization problems such as the travelling salesman problem and spin glasses, although the technique has been demonstrated to function in optimization domains.

D-Wave Systems Inc. is a Canadian quantum computing company, based in Burnaby, British Columbia, Canada. D-Wave was the world's first company to sell computers to exploit quantum effects in their operation. D-Wave's early customers include Lockheed Martin, University of Southern California, Google/NASA and Los Alamos National Lab.

Adiabatic quantum computation (AQC) is a form of quantum computing which relies on the adiabatic theorem to do calculations and is closely related to quantum annealing.

Daniel Amihud Lidar is the holder of the Viterbi Professorship of Engineering at the University of Southern California, where he is a Professor of Electrical Engineering, Chemistry, Physics & Astronomy. He is the Director and co-founder of the USC Center for Quantum Information Science & Technology (CQIST) as well as Scientific Director of the USC-Lockheed Martin Quantum Computing Center, notable for his research on control of quantum systems and quantum information processing.

Subir Sachdev is Herchel Smith Professor of Physics at Harvard University specializing in condensed matter. He was elected to the U.S. National Academy of Sciences in 2014, and received the Lars Onsager Prize from the American Physical Society and the Dirac Medal from the ICTP in 2018.

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Quantum simulators permit the study of quantum systems that are difficult to study in the laboratory and impossible to model with a supercomputer. In this instance, simulators are special purpose devices designed to provide insight about specific physics problems. Quantum simulators may be contrasted with generally programmable "digital" quantum computers, which would be capable of solving a wider class of quantum problems.

D-Wave Two is the second commercially available quantum computer, and the successor to the first commercially available quantum computer, D-Wave One. Both computers were developed by Canadian company D-Wave Systems. The computers are not general purpose, but rather are designed for quantum annealing. Specifically, the computers are designed to use quantum annealing to solve a single type of problem known as quadratic unconstrained binary optimization. As of 2015, it was still debated whether large-scale entanglement takes place in D-Wave Two, and whether current or future generations of D-Wave computers will have any advantage over classical computers.

The quantum algorithm for linear systems of equations, also called HHL algorithm, designed by Aram Harrow, Avinatan Hassidim, and Seth Lloyd, is a quantum algorithm formulated in 2009 for solving linear systems. The algorithm estimates the result of a scalar measurement on the solution vector to a given linear system of equations.

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The USC-Lockheed Martin Quantum Computing Center (QCC) is a joint scientific research effort between Lockheed Martin Corporation and the University of Southern California (USC). The QCC is housed at the Information Sciences Institute (ISI), a computer science and engineering research unit of the USC Viterbi School of Engineering, and is jointly operated by ISI and Lockheed Martin.

Edward Farhi is a Principal Scientist at Google working on quantum computation. In 2018 he retired from his position as the Cecil and Ida Green Professor of Physics at the Massachusetts Institute of Technology. He was the Director of the Center for Theoretical Physics at MIT from 2004 until 2016. He made contributions to particle physics, general relativity and astroparticle physics before turning to his current interest, quantum computation. For a recent interview see here.

Andrew MacGregor Childs is an American computer scientist and physicist known for his work on quantum computing. He is currently a Professor in the Department of Computer Science and Institute for Advanced Computer Studies at the University of Maryland. He also co-directs the Joint Center for Quantum Information and Computer Science, a partnership between the University of Maryland and the National Institute of Standards and Technology.

Quantum optimization algorithms are quantum algorithms that are used to solve optimization problems. Mathematical optimization deals with finding the best solution to a problem from a set of possible solutions. Mostly, the optimization problem is formulated as a minimization problem, where one tries to minimize an error which depends on the solution: the optimal solution has the minimal error. Different optimization techniques are applied in various fields such as mechanics, economics and engineering, and as the complexity and amount of data involved rise, more efficient ways of solving optimization problems are needed. The power of quantum computing may allow problems which are not practically feasible on classical computers to be solved, or suggest a considerable speed up with respect to the best known classical algorithm.

In quantum computing, quantum supremacy or quantum advantage is the goal of demonstrating that a programmable quantum device can solve a problem that no classical computer can solve in any feasible amount of time. Conceptually, quantum supremacy involves both the engineering task of building a powerful quantum computer and the computational-complexity-theoretic task of finding a problem that can be solved by that quantum computer and has a superpolynomial speedup over the best known or possible classical algorithm for that task. The term was coined by John Preskill in 2012, but the concept of a quantum computational advantage, specifically for simulating quantum systems, dates back to Yuri Manin's (1980) and Richard Feynman's (1981) proposals of quantum computing. Examples of proposals to demonstrate quantum supremacy include the boson sampling proposal of Aaronson and Arkhipov, D-Wave's specialized frustrated cluster loop problems, and sampling the output of random quantum circuits.

Exact diagonalization (ED) is a numerical technique used in physics to determine the eigenstates and energy eigenvalues of a quantum Hamiltonian. In this technique, a Hamiltonian for a discrete, finite system is expressed in matrix form and diagonalized using a computer. Exact diagonalization is only feasible for systems with a few tens of particles, due to the exponential growth of the Hilbert space dimension with the size of the quantum system. It is frequently employed to study lattice models, including the Hubbard model, Ising model, Heisenberg model, t-J model, and SYK model.

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