Quantum algorithm design usually assumes access to a perfect quantum computer with ideal properties like full connectivity, noise-freedom and arbitrarily long coherence time. In Noisy Intermediate-Scale Quantum (NISQ) devices, however, the number of qubits is highly limited and quantum operation error and qubit coherence are not negligible. Besides, the connectivity of physical qubits in a quantum processing unit (QPU) is also strictly constrained. Thereby, additional operations like SWAP gates have to be inserted to satisfy this constraint while preserving the functionality of the original circuit. This process is known as quantum circuit transformation. Adding additional gates will increase both the size and depth of a quantum circuit and therefore cause further decay of the performance of a quantum circuit. Thus it is crucial to minimize the number of added gates. In this paper, we propose an efficient method to solve this problem. We first choose by using simulated annealing an initial mapping which fits well with the input circuit and then, with the help of a heuristic cost function, stepwise apply the best selected SWAP gates until all quantum gates in the circuit can be executed. Our algorithm runs in time polynomial in all parameters including the size and the qubit number of the input circuit, and the qubit number in the QPU. Its space complexity is quadratic to the number of edges in the QPU. Experimental results on extensive realistic circuits confirm that the proposed method is efficient and can reduce by 57% on average the size of the output circuits when compared with the state-of-the-art algorithm on the most recent IBM quantum device viz. IBM Q20 (Tokyo).
We analyze the performance of simulated quantum annealing (SQA) on an optimization problem for which simulated classical annealing (SA) is provably inefficient because of a high energy barrier. We present evidence that SQA can pass through this barrier to find the global minimum efficiently. This demonstrates the potential for SQA to inherit some of the advantages of quantum annealing (QA), since this problem has been previously shown to be efficiently solvable by quantum adiabatic optimization.
Simulated quantum annealing is a generic classical protocol to simulate some aspects of quantum annealing and is sometimes regarded as a classical alternative to quantum annealing in finding the ground state of a classical Ising model. We derive a generic condition for simulated quantum annealing to converge to thermal equilibrium at a given, typically low, temperature. Both closed and open systems are treated. We rewrite the classical master equation for simulated quantum annealing into an imaginary-time Schrödinger equation, to which we apply the imaginary-time variant of asymptotic adiabati…
We present a comparison between the Quantum Approximate Optimization Algorithm (QAOA) and two widely studied competing methods, Quantum Annealing (QA) and Simulated Annealing (SA). To achieve this, we define a class of optimization problems with respect to their spectral properties which are exactly solvable with QAOA. In this class, we identify instances for which QA and SA have an exponentially small probability to find the solution. Consequently, our results define a first demarcation line between QAOA, Simulated Annealing and Quantum Annealing, and highlight the fundamental differences bet…
Classical and quantum annealing are two heuristic optimization methods that search for an optimal solution by slowly decreasing thermal or quantum fluctuations. Optimizing annealing schedules is important both for performance and fair comparisons between classical annealing, quantum annealing, and other algorithms. Here we present a heuristic approach for the optimization of annealing schedules for quantum annealing and apply it to 3D Ising spin glass problems. We find that if both classical and quantum annealing schedules are similarly optimized, classical annealing outperforms quantum anneal…
We explain why quantum adiabatic evolution and simulated annealing perform similarly in certain examples of searching for the minimum of a cost function of n bits. In these examples each bit is treated symmetrically so the cost function depends only on the Hamming weight of the n bits. We also give two examples, closely related to these, where the similarity breaks down in that the quantum adiabatic algorithm succeeds in polynomial time whereas simulated annealing requires exponential time.
For many optimization algorithms the time-to-solution depends not only on the problem size but also on the specific problem instance and may vary by many orders of magnitude. It is then necessary to investigate the full distribution and especially its tail. Here we analyze the distributions of annealing times for simulated annealing and simulated quantum annealing (by path integral quantum Monte Carlo) for random Ising spin glass instances. We find power-law distributions with very heavy tails, corresponding to extremely hard instances, but far broader distributions - and thus worse performanc…
This paper summarizes a quantum algorithm of [R.D. Somma, et.al., Phys. Rev. Lett. 101, 130504 (2008)] that simulates a classical annealing process for solving discrete optimization problems. The complexity of the quantum algorithm scales with the inverse square root of the spectral gap of an associated stochastic matrix. This represents a quadratic quantum speedup, in terms of the gap, with respect to classical simulated annealing.
We develop a quantum algorithm to solve combinatorial optimization problems through quantum simulation of a classical annealing process. Our algorithm combines techniques from quantum walks, quantum phase estimation, and quantum Zeno effect. It can be viewed as a quantum analogue of the discrete-time Markov chain Monte Carlo implementation of classical simulated annealing. Our implementation scales with the inverse of the square root of the minimum spectral gap of the stochastic matrix used in the classical simulation. The quantum algorithm outperforms the classical one, which scales with the …
The observation of an unequivocal quantum speedup remains an elusive objective for quantum computing. The D-Wave quantum annealing processors have been at the forefront of experimental attempts to address this goal, given their relatively large numbers of qubits and programmability. A complete determination of the optimal time-to-solution (TTS) using these processors has not been possible to date, preventing definitive conclusions about the presence of a scaling advantage. The main technical obstacle has been the inability to verify an optimal annealing time within the available range. Here we…
This paper introduces QuSAnn v1.2 and Multiplexor Expander v1.2, two Java applications available for free. (Source code included in the distribution.) QuSAnn is a "code generator" for quantum simulated annealing: after the user inputs some parameters, it outputs a quantum circuit for performing simulated annealing on a quantum computer. The quantum circuit implements the algorithm of Wocjan et al. (arXiv:0804.4259), which improves on the original algorithm of Somma et al. (arXiv:0712.1008). The quantum circuit generated by QuSAnn includes some quantum multiplexors. The application Multiplexor …
Simulated quantum annealing based on the path-integral Monte Carlo is one of the most common tools to simulate quantum annealing on classical hardware. Nevertheless, it is in principle highly non-trivial whether or not this classical algorithm can correctly reproduce the quantum dynamics of quantum annealing, particularly in the diabatic regime. We study this problem numerically through the generalized Kibble-Zurek mechanism of defect distribution in the simplest ferromagnetic one-dimensional transverse-field Ising model with and without coupling to the environment. We find that,in the absence…
Quantum Annealing, or Quantum Stochastic Optimization, is a classical randomized algorithm which provides good heuristics for the solution of hard optimization problems. The algorithm, suggested by the behaviour of quantum systems, is an example of proficuous cross contamination between classical and quantum computer science. In this survey paper we illustrate how hard combinatorial problems are tackled by quantum computation and present some examples of the heuristics provided by Quantum Annealing. We also present preliminary results about the application of quantum dissipation (as an alterna…
The Ising model with a transverse field and an antiferromagnetic transverse interaction is represented as a matrix in the computational basis with non-zero off-diagonal elements with both positive and negative signs and thus may be regarded to be non-stoquastic. We show that the local Boltzmann factors of such a system under an appropriate Suzuki-Trotter representation can be chosen non-negative and thus may potentially be simulated classically without a sign problem if the parameter values are limited to a subspace of the whole parameter space. We then derive conditions for parameters to sati…
Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule. We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantu…
The protocol of quantum annealing is applied to an optimization problem with a one-dimensional continuous degree of freedom, a variant of the problem proposed by Shinomoto and Kabashima. The energy landscape has a number of local minima, and the classical approach of simulated annealing is predicted to have a logarithmically slow convergence to the global minimum. We show by extensive numerical analyses that quantum annealing yields a power law convergence, thus an exponential improvement over simulated annealing. The power is larger, and thus the convergence is faster, than a prediction by an…
In this work we study the convergence of a classical algorithm called Simulated Quantum Annealing (SQA) on the Spike Hamiltonian, a specific toy model Hamiltonian for quantum-mechanical tunneling introduced by [FGG02]. This toy model Hamiltonian encodes a simple bit-symmetric cost function f in the computational basis, and is used to emulate local minima in more complex optimization problems. In previous work [CH16] showed that SQA runs in polynomial time in much of the regime of spikes that QA does, pointing to evidence against an exponential speedup through tunneling. In this paper we extend…
Satisfiability filters, introduced by S. A. Weaver et al. in 2014, are a new and promising type of filters to address set membership testing. In order to construct satisfiability filters, it is necessary to find disparate solutions to hard random $k$-SAT problems. This paper compares simulated annealing, simulated quantum annealing and walkSAT, an open-source SAT solver, in terms of their ability to find such solutions. The results indicate that solutions found by simulated quantum annealing are generally less disparate than solutions found by the other solvers and therefore less useful for th…
We study various annealing dynamics, both classical and quantum, for simple mean-field models and explain how to describe their behavior in the thermodynamic limit in terms of differential equations. In particular we emphasize the differences between quantum annealing (i.e. evolution with Schrödinger equation) and simulated quantum annealing (i.e. annealing of a Quantum Monte Carlo simulation).
Quantum annealing is a heuristic algorithm for searching the ground state of an Ising model. Heuristic algorithms aim to obtain near-optimal solutions with a reasonable computation time. Accordingly, many algorithms have so far been proposed. In general, the performance of heuristic algorithms strongly depends on the instance of the combinatorial optimization problem to be solved because they escape the local minima in different ways. Therefore, combining several algorithms to exploit their complementary strength is effective for obtaining highly accurate solutions for a wide range of combinat…
The Potts model is a generalization of the Ising model with $Q>2$ components. In the fully connected ferromagnetic Potts model, a first-order phase transition is induced by varying thermal fluctuations. Therefore, the computational time required to obtain the ground states by simulated annealing exponentially increases with the system size. This study analytically confirms that the transverse magnetic-field quantum annealing induces a first-order phase transition. This result implies that quantum annealing does not exponentially accelerate the ground-state search of the ferromagnetic Potts mod…
Quantum annealing is a heuristic quantum optimization algorithm that can be used to solve combinatorial optimization problems. In recent years, advances in quantum technologies have enabled the development of small- and intermediate-scale quantum processors that implement the quantum annealing algorithm for programmable use. Specifically, quantum annealing processors produced by D-Wave Systems have been studied and tested extensively in both research and industrial settings across different disciplines. In this paper we provide a literature review of the theoretical motivations for quantum ann…
Relations of simulated annealing and quantum annealing are studied by a mapping from the transition matrix of classical Markovian dynamics of the Ising model to a quantum Hamiltonian and vice versa. It is shown that these two operators, the transition matrix and the Hamiltonian, share the eigenvalue spectrum. Thus, if simulated annealing with slow temperature change does not encounter a difficulty caused by an exponentially long relaxation time at a first-order phase transition, the same is true for the corresponding process of quantum annealing in the adiabatic limit. One of the important dif…
Solving discrete NP-hard problems is an important part of scientific discoveries and operations research as well as many commercial applications. A commonly used metric to compare meta-heuristic solvers is the time required to obtain an optimal solution, known as time to solution. However, for some applications it is desirable to have a set of high-quality and diverse solutions, instead of a single optimal one. For these applications, time to solution may not be informative of the performance of a solver, and another metric would be necessary. In particular, it is not known how well quantum so…
Combinatorial optimization problems are computationally hard in general, but they are ubiquitous in our modern life. A coherent Ising machine (CIM) based on a multiple-pulse degenerate optical parametric oscillator (DOPO) is an alternative approach to solve these problems by a specialized physical computing system. To evaluate its potential performance, computational experiments are performed on maximum cut (MAX-CUT) problems against traditional algorithms such as semidefinite programming relaxation of Goemans-Williamson and simulated annealing by Kirkpatrick, et al. The numerical results empi…
Minor embedding heuristics have become an indispensable tool for compiling problems in quadratically unconstrained binary optimization (QUBO) into the hardware graphs of quantum and CMOS annealing processors. While recent embedding heuristics have been developed for annealers of moderate size (about 2000 nodes) the size of the latest CMOS annealing processor (with 102,400 nodes) poses entirely new demands on the embedding heuristic. This raises the question, if recent embedding heuristics can maintain meaningful embedding performance on hardware graphs of increasing size. Here, we develop an i…
Both simulated quantum annealing and physical quantum annealing have shown the emergence of "heavy tails" in their performance as optimizers: The total time needed to solve a set of random input instances is dominated by a small number of very hard instances. Classical simulated annealing, in contrast, does not show such heavy tails. Here we explore the origin of these heavy tails, which appear for inputs with high local degeneracy---large isoenergetic clusters of states in Hamming space. This category includes the low-precision Chimera-structured problems studied in recent benchmarking work c…
The question of whether or not quantum computers can efficiently solve NP-complete problems is open, although indications are that BQP does not contain NP. Still, many of these problems are natural candidates for solution on quantum computers. We outline a thermodynamical formalism for the traveling salesman problem which allows for its expected polytime solution on a quantum computer with probability arbitrarily close to unity, given sufficient energy resources and subject to a weak nondegeneracy constraint on the distances. Applications to other problems are also discussed.
Conventional quantum annealing does not sample all ground states fairly. We demonstrate that fair sampling can be achieved by performing simulated annealing on a quantum annealer. We discuss the problems that occur when implementing this method and propose an alternative way to overcome them. We numerically verify the fair sampling ability of our method in a small-scale toy model.
Almost all public secure communication relies on the inability to factor large numbers. There is no known analytic or classical numeric method to rapidly factor large numbers. Shor[1] has shown that a quantum computer can factor numbers in polynomial time but there is no practical quantum computer that can yet do such computations. We show that a simulated annealing[2] approach can be adapted to find factors of large numbers.
We formulate an adiabatic approximation for the imaginary-time Schroedinger equation. The obtained adiabatic condition consists of two inequalities, one of which coincides with the conventional adiabatic condition for the real-time Schroedinger equation, but the other does not. We apply this adiabatic approximation to the analysis of Markovian dynamics of the classical Ising model, which can be formulated as the imaginary-time Schrödinger equation, to obtain an asymptotic formula for the probability that the system reaches the ground state in the limit of a long annealing time in simulated ann…
We show clear evidence of a quadratic speedup of a quantum annealing (QA) Schroedinger dynamics over a Glauber master-equation simulated annealing (SA) for a random Ising model in one dimension, via an equal-footing exact deterministic dynamics of the Jordan-Wigner fermionized problems. This is remarkable, in view of the arguments of Katzgraber et al., PRX 4, 021008 (2014), since SA does not encounter any phase transition, while QA does. We also find a second remarkable result: that a "quantum-inspired" imaginary-time Schroedinger QA provides a further exponential speedup, i.e., an asymptotic …
Quantum Annealing is a heuristic for solving optimization problems that have seen a recent surge in usage owing to the success of D-Wave Systems. This paper aims to find a good heuristic for solving the Electric Vehicle Charger Placement (EVCP) problem, a problem that stands to be very important given the costs of setting up an electric vehicle (EV) charger and the expected surge in electric vehicles across the world. The same problem statement can also be generalized to the optimal placement of any entity in a grid and can be explored for further uses. Finally, the authors introduce a novel h…
In this paper we consider the use of certain classical analogues to quantum tunneling behavior to improve the performance of simulated annealing on a discrete spin system of the general Ising form. Specifically, we consider the use of multiple simultaneous spin flips at each annealing step as an analogue to quantum spin coherence as well as modifications of the Boltzmann acceptance probability to mimic quantum tunneling. We find that the use of multiple spin flips can indeed be advantageous under certain annealing schedules, but only for long anneal times.
Simulated Quantum Annealing (SQA) is a Markov Chain Monte-Carlo algorithm that samples the equilibrium thermal state of a Quantum Annealing (QA) Hamiltonian. In addition to simulating quantum systems, SQA has also been proposed as another physics-inspired classical algorithm for combinatorial optimization, alongside classical simulated annealing. However, in many cases it remains an open challenge to determine the performance of both QA and SQA. One piece of evidence for the strength of QA over classical simulated annealing comes from an example by Farhi, Goldstone and Gutmann . There a bit-sy…
Non-equilibrium dynamics of the Ising model is a classical stochastic process whereas quantum mechanics has no stochastic elements in the classical sense. Nevertheless, it has been known that there exists a close formal relationship between these two processes. We reformulate this relationship and use it to compare the efficiency of simulated annealing that uses classical stochastic processes and quantum annealing to solve combinatorial optimization problems. It is shown that classical dynamics can be efficiently simulated by quantum-mechanical processes whereas the converse is not necessarily…
Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）；Quantumz.io Sp. z o.o.（Quantumz.io有限责任公司）；Institute of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology（弗罗茨瓦夫理工大学基础技术问题学院理论物理研究所）；Institute of Theoretical and Applied Informatics, Polish Academy of Sciences（波兰科学院理论与应用信息学研究所）
Quantum annealing and D-Wave quantum annealer attracted considerable attention for their ability to solve combinatorial optimization problems. In order to solve other type of optimization problems, it is necessary to apply certain kinds of mathematical transformations. However, linear constraints reduce the size of problems that can be represented in quantum annealers, owing to the sparseness of connections between qubits. For example, the quadratic assignment problem (QAP) with linear equality constraints can be solved only up to size 12 in the quantum annealer D-Wave Advantage, which has 564…
We analyze the performance of quantum annealing as a heuristic optimization method to find the absolute minimum of various continuous models, including landscapes with only two wells and also models with many competing minima and with disorder. The simulations performed using a projective quantum Monte Carlo (QMC) algorithm are compared with those based on the finite-temperature path-integral QMC technique and with classical annealing. We show that the projective QMC algorithm is more efficient than the finite-temperature QMC technique, and that both are inferior to classical annealing if this…
The strongest evidence for superiority of quantum annealing on spin glass problems has come from comparing simulated quantum annealing using quantum Monte Carlo (QMC) methods to simulated classical annealing [G. Santoro et al., Science 295, 2427(2002)]. Motivated by experiments on programmable quantum annealing devices we revisit the question of when quantum speedup may be expected for Ising spin glass problems. We find that even though a better scaling compared to simulated classical annealing can be achieved for QMC simulations, this advantage is due to time discretization and measurements w…
We compare the performance of quantum annealing (QA, through Schrödinger dynamics) and simulated annealing (SA, through a classical master equation) on the $p$-spin infinite range ferromagnetic Ising model, by slowly driving the system across its equilibrium, quantum or classical, phase transition. When the phase transition is second-order ($p=2$, the familiar two-spin Ising interaction) SA shows a remarkable exponential speed-up over QA. For a first-order phase transition ($p \ge 3$, i.e., with multi-spin Ising interactions), in contrast, the classical annealing dynamics appears to remain stu…
Quantum annealing method has been widely attracted attention in statistical physics and information science since it is expected to be a powerful method to obtain the best solution of optimization problem as well as simulated annealing. The quantum annealing method was incubated in quantum statistical physics. This is an alternative method of the simulated annealing which is well-adopted for many optimization problems. In the simulated annealing, we obtain a solution of optimization problem by decreasing temperature (thermal fluctuation) gradually. In the quantum annealing, in contrast, we dec…
We discuss an Ising spin glass where each $S=1/2$ spin is coupled antiferromagnetically to three other spins (3-regular graphs). Inducing quantum fluctuations by a time-dependent transverse field, we use out-of-equilibrium quantum Monte Carlo simulations to study dynamic scaling at the quantum glass transition. Comparing the dynamic exponent and other critical exponents with those of the classical (temperature-driven) transition, we conclude that quantum annealing is less efficient than classical simulated annealing in bringing the system into the glass phase. Quantum computing based on the qu…
It has been recently reported that classical systems have speed limit for state evolution, although such a concept of speed limit had been considered to be unique to quantum systems. Owing to the speed limit for classical system, the lower bound for calculation time of simulated annealing with desired calculation accuracy can be derived. However, such a lower bound does not work as a criterion for completion of calculation in a practical time. In this paper, we derive an inequality for classical system analogous to the quantum adiabatic theorem that gives calculation time for an accuracy-guara…
University of Trier（特里尔大学）；University of Bergen（卑尔根大学）
The Kemeny Rank Aggregation (KRA) problem is a well-studied problem in the field of Social Choice with a variety of applications in many different areas like databases and search engines. Intuitively, given a set of votes over a set of candidates, the problem asks to find an aggregated ranking of candidates that minimizes the overall dissatisfaction concerning the votes. Recently, a diverse version of KRA was considered which asks for a sufficiently diverse set of sufficiently good solutions. The framework of diversity of solutions is a young and thriving topic in the field of artificial intel…
Finding the global minimum in a rugged potential landscape is a computationally hard task, often equivalent to relevant optimization problems. Simulated annealing is a computational technique which explores the configuration space by mimicking thermal noise. By slow cooling, it freezes the system in a low-energy configuration, but the algorithm often gets stuck in local minima. In quantum annealing, the thermal noise is replaced by controllable quantum fluctuations, and the technique can be implemented in modern quantum simulators. However, quantum-adiabatic schemes become prohibitively slow i…
Simulated Quantum Annealing (SQA), that is emulating a Quantum Annealing (QA) dynamics on a classical computer by a Quantum Monte Carlo whose parameters are changed during the simulation, is a well established computational strategy to cope with the exponentially large Hilbert space. It has enjoyed some early successes but has also raised more recent criticisms. Here we investigate, on the paradigmatic case of a one-dimensional transverse field Ising chain, two issues related to SQA in its Path-Integral implementation: the question of Monte Carlo vs physical (Schrödinger) dynamics and the issu…
There have been multiple attempts to design synthetic benchmark problems with the goal of detecting quantum speedup in current quantum annealing machines. To date, classical heuristics have consistently outperformed quantum-annealing based approaches. Here we introduce a class of problems based on frustrated cluster loops - deceptive cluster loops - for which all currently known state-of-the-art classical heuristics are outperformed by the D-Wave 2000Q quantum annealing machine. While there is a sizable constant speedup over all known classical heuristics, a noticeable improvement in the scali…
