Quantum computers use quantum resources to carry out computational tasks and may outperform classical computers in solving certain computational problems. Special-purpose quantum computers such as quantum annealers employ quantum adiabatic theorem to solve combinatorial optimization problems. In this paper, we compare classical annealings such as simulated annealing and quantum annealings that are done by the D-Wave machines both theoretically and numerically. We show that if the classical and quantum annealing are characterized by equivalent Ising models, then solving an optimization problem, i.e., finding the minimal energy of each Ising model, by the two annealing procedures, are mathematically identical. For quantum annealing, we also derive the probability lower-bound on successfully solving an optimization problem by measuring the system at the end of the annealing procedure. Moreover, we present the Markov chain Monte Carlo (MCMC) method to realize quantum annealing by classical computers and investigate its statistical properties. In the numerical section, we discuss the discrepancies between the MCMC based annealing approaches and the quantum annealing approach in solving optimization problems.
Quantum computers use quantum resources to carry out computational tasks and may outperform classical computers in solving certain computational problems. Special-purpose quantum computers such as quantum annealers employ quantum adiabatic theorem to solve combinatorial optimization problems. In this paper, we compare classical annealings such as simulated annealing and quantum annealings that are done by the D-Wave machines both theoretically and numerically. We show that if the classical and quantum annealing are characterized by equivalent Ising models, then solving an optimization problem, i.e., finding the minimal energy of each Ising model, by the two annealing procedures, are mathematically identical. For quantum annealing, we also derive the probability lower-bound on successfully solving an optimization problem by measuring the system at the end of the annealing procedure. Moreover, we present the Markov chain Monte Carlo (MCMC) method to realize quantum annealing by classical computers and investigate its statistical properties. In the numerical section, we discuss the discrepancies between the MCMC based annealing approaches and the quantum annealing approach in solving optimization problems.
Quantum computation performs calculations by using quantum devices instead of electronic devices following classical physics and used by classical computers. Although general purpose quantum computers of practical scale may be many years away, special purpose quantum computers are being built with capabilities exceeding classical computers. One prominent case is the so-called D-Wave quantum computer, which is a computing hardware device built to implement quantum annealing for solving combinatorial optimization problems. Whether D-Wave computing hardware devices display a quantum behavior or can be described by a classical model has attracted tremendous attention, and it remains controversial to determine whether quantum or classical effects play a crucial role in exhibiting the computational input–output behaviors of the D-Wave devices. This paper consists of two parts where the first part provides a review of quantum annealing and its implementations, and the second part proposes statistical methodologies to analyze data generated from annealing experiments. Specifically, we introduce quantum annealing to solve optimization problems and describe D-Wave computing devices to implement quantum annealing. We illustrate implementations of quantum annealing using Markov chain Monte Carlo (MCMC) simulations carried out by classical computers. Computing experiments have been conducted to generate data and compare quantum annealing with classical annealing. We propose statistical methodologies to analyze computing experimental data from a D-Wave device and simulated data from the MCMC based annealing methods, and establish asymptotic theory and check finite sample performances for the proposed statistical methodologies. Our findings confirm bimodal histogram patterns displayed in input–output data from the D-Wave device and both U-shape and unimodal histogram patterns exhibited in input–output data from the MCMC based annealing methods. Further statistical explorations reveal possible sources for the U-shape patterns. On the other hand, our statistical analysis produces statistical evidence to indicate that input–output data from the D-Wave device are not consistent with the stochastic behaviors of any MCMC based annealing models under the study. We present a list of statistical research topics for the future study on quantum annealing and MCMC simulations.
Quantum annealing is a promising tool for solving optimization problems, similar in some ways to the traditional (classical) simulated annealing of Kirkpatrick et al. Simulated annealing takes advantage of thermal fluctuations in order to explore the optimization landscape of the problem at hand, whereas quantum annealing employs quantum fluctuations. Intriguingly, quantum annealing has been proved to be more effective than its classical counterpart in many applications. We illustrate the theory and the practical implementation of both classical and quantum annealing – highlighting the crucial differences between these two methods – by means of results recently obtained in experiments, in simple toy-models, and more challenging combinatorial optimization problems (namely, Random Ising model and Travelling Salesman Problem). The techniques used to implement quantum and classical annealing are either deterministic evolutions, for the simplest models, or Monte Carlo approaches, for harder optimization tasks. We discuss the pro and cons of these approaches and their possible connections to the landscape of the problem addressed.
Optimization algorithms for combinatorial problems play a critical role in a wide range of applications, from logistics and scheduling to cryptography and artificial intelligence. Traditional classical methods, such as simulated annealing, genetic algorithms, and branch-and-bound techniques, have been widely employed to solve these problems, but they often face limitations in terms of computational efficiency and scalability as problem complexity grows. In recent years, quantum computing has emerged as a promising alternative, with quantum annealing offering a novel approach to solving combinatorial optimization problems. This study provides a comparative analysis of quantum annealing and classical optimization methods, highlighting their respective strengths and limitations. We explore how quantum annealing, through leveraging quantum tunnelling and superposition, has the potential to outperform classical algorithms in specific problem domains and Noisy Mean Field Annealing (NMFA) is used to comparisons of all algorithms. Benchmarking various combinatorial problems, such as the Traveling Salesman Problem (TSP), Max-Cut, Knapsack, and Quadratic Assignment Problem (QAP), this paper discusses performance metrics, computational time, and scalability. The findings suggest that while classical methods remain robust and practical for many real-world problems, quantum annealing holds significant promise, especially as quantum hardware continues to mature. However, there remain challenges in terms of noise, coherence, and problem mapping, which currently limit the full realization of quantum annealing's potential. This study offers insights into the future directions of optimization techniques and the evolving role of quantum computing in solving complex combinatorial problems.
Ising machines have the potential to realize fast and highly accurate solvers for combinatorial optimization problems. They are classified based on their internal algorithms. Examples include simulated-annealing-based Ising machines (non-quantum-type Ising machines) and quantum-annealing-based Ising machines (quantum annealers). Herein we propose a hybrid optimization method, which utilizes the advantages of both types. In this hybrid optimization method, the preprocessing step is performed by solving the non-quantum-annealing Ising machine multiple times. Then sub-Ising models with a reduced size by spin fixing are solved using a quantum annealer. The performance of the hybrid optimization method is evaluated via simulations using Simulated Annealing (SA) as a non-quantum-type Ising machine and D-Wave Advantage as a quantum annealer. Additionally, we investigate the parameter dependence of the proposed hybrid optimization method. The hybrid optimization method outperforms the preprocessing SA and the quantum annealing machine alone in fully connected random Ising models.
Recently, several powerful machines dedicated to solving combinatorial optimization problems through the Ising-model formulation have appeared. The trigger for the paradigm shift to a specialized machine for solving optimization problems was the D-Wave machine, which implements quantum annealing. Quantum annealing employs quantum fluctuations to find an optimal solution to an optimization problem with discrete variables. In particular, we input the optimization problem in the form of the Ising Hamiltonian, which is a specialized form of the quadratic unconstrained binary optimization problem. However, when we employ quantum annealing for a practical optimization problem, there are several issues. One typical issue is absence of the detailed form of the cost function, which characterizes the optimization problem. To input problems into specialized machines for solving an optimization problem, it is necessary to determine the unknown parameters within the Ising model. We propose a method to estimate the unknown parameters in the Ising Hamiltonian using compressed sensing. Furthermore, we analyze the theoretical limitations of our proposed method by employing the replica method, which is a sophisticated tool in statistical mechanics.
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 inverse of the gap.