Quantum annealing is a new-generation tool of information technology, which helps in solving combinatorial optimization problems with high precision, based on the concepts of quantum statistical physics. Detailed discussion on quantum spin glasses and its application in solving combinatorial optimization problems is required for better understanding of quantum annealing concepts. Fulfilling this requirement, the book highlights recent development in quantum spin glasses including Nishimori line, replica method and quantum annealing methods along with the essential principles. Separate chapters on simulated annealing, quantum dynamics and classical spin models are provided for enhanced learning. Important topics including adiabatic quantum computers and quenching dynamics are discussed in detail. This text will be useful for students of quantum computation, quantum information, statistical physics and computer science.
Quantum annealing is a new-generation tool of information technology, which helps in solving combinatorial optimization problems with high precision, based on the concepts of quantum statistical physics. Detailed discussion on quantum spin glasses and its application in solving combinatorial optimization problems is required for better understanding of quantum annealing concepts. Fulfilling this requirement, the book highlights recent development in quantum spin glasses including Nishimori line, replica method and quantum annealing methods along with the essential principles. Separate chapters on simulated annealing, quantum dynamics and classical spin models are provided for enhanced learning. Important topics including adiabatic quantum computers and quenching dynamics are discussed in detail. This text will be useful for students of quantum computation, quantum information, statistical physics and computer science.
Quantum annealing is analogous to simulated annealing with a tunneling mechanism substituting for thermal activation. Its performance has been tested in numerical simulation with mixed conclusions. There is a class of optimization problems for which the efficiency can be studied analytically using techniques based on the statistical mechanics of spin glasses. Solving hard combinatorial problems by temperature annealing is a classic method in computer science.1) The problem is formulated in terms of a cost function, which one can identify as an energy. The dynamics are then a combination of a systematic descent in energy perturbed by a ’noise’, which is turned off gradually, inducing
Spin glasses are experiencing a revival due to applications in quantum information\ntheory. In particular, they are the archetypal native benchmark problem for quantum annealing\nmachines. Furthermore, they find applications in fields as diverse as satisfiability,\nneural networks, and general combinatorial optimization problems. As such, developing\nand improving algorithms and methods to study these computationally complex systems\nis of paramount importance to many disciplines. This body of work attempts to attack\nthe problem of solving combinatorial optimization problems by simulating spin glasses\nfrom three sides: classical algorithm development, suggestions for quantum annealing\ndevice design, and improving measurements in realistic physical systems with inherent\nnoise. I begin with the introduction of a cluster algorithm based on Houdayer’s cluster algorithm\nfor two-dimensional Ising spin-glasses that is applicable to any space dimension\nand speeds up thermalization by several orders of magnitude at low temperatures where\nprevious algorithms have difficulty. I show improvement for the D-Wave chimera topology\nand the three-dimensional cubic lattice that increases with the size of the problem.\nOne consequence of adding cluster moves is that for problems with degenerate solutions,\nground-state sampling is improved. I demonstrate an ergodic algorithm to sample ground\nstates through the use of simple Monte Carlo with parallel tempering and cluster moves.\nIn addition, I present a non-ergodic algorithm to generate new solutions from a bank of\nknown solutions. I compare these results against results from quantum annealing utilizing\nthe D-Wave Inc. quantum annealing device. Finally, I present an algorithm for improving\nthe recovery of ground-state solutions from problems with noise by using thermal fluctuations\nto infer the correct solution at the Nishimori temperature. While this method has been\ndemonstrated analytically and numerically for trivial ferromagnetic and Gaussian distributions, a useful metric for more complex Gaussian distributions with added Gaussian noise\nis unavailable. We show improved recovery of numerical solutions on the chimera graph\nwith a ferromagnetic distribution and added Gaussian noise. Next, I direct my focus to the\ndesign of future generations of quantum annealers. The first design is the two-dimensional\nsquare-lattice bimodal spin glass with next-nearest ferrromagnetic interactions proposed\nby Lemke and Campbell claimed to exhibit a finite-temperature spin-glass state for a particular\nrelative strength of the next-nearest to nearest neighbor interactions. Our results\nfrom finite-temperature simulations show the system is in a paramagnetic state in the thermodynamic\nlimit, thus not useful for quantum annealing device designs that would benefit\nfrom a spin-glass phase transition. The second design is the diluted next-nearest neighbor\nIsing spin-glass with Gaussian interactions in an attempt to improve the estimation\nof critical parameter with smaller system sizes by implementing averaging of observables\nover different graph dilutions. To date, this model has shown no improvement. Finally,\nI make suggestions for the choice of distributions of interactions that are robust to noise\nand present a method for using previously unaccessible continuous distributions. I begin\nwith showing the best-case performance of quantum annealing devices. I show results for\nthe resilience, the probability that the ground-state solution has changed due to inherent\nanalog noise in the device, and present strategies for developing robust instance classes.\nThe analog noise is also detrimental to interactions chosen from continuous distributions.\nUsing Gaussian quadratures, I present a method for discretizing continuous distributions\nto reduce noise effects. Simulations on the D-Wave show that the average residual of\nthe ground-state energy with the true ground-state energy is calculated and shown to be\nsmaller in the case of the discrete distribution.
The correspondence between long-range interacting quantum spin glasses and combinatorial optimization problems underpins the physical motivation for adiabatic quantum computing. On one hand, in disordered (quantum) spin systems, the focus is on exact methods such as the replica trick that allow the calculation of system quantities in the limit of infinite system and ensemble size. On the other hand, when solving a given instance of an optimization problem, disorder-averaged quantities are of no relevance, as one is solely interested in instance-specific, finite-size properties, in particular the optimal solution. Here, we apply the nonequilibrium Green's function formalism to the spin coherent-state path integral to obtain the statistical fluctuations and the collective-excitation spectrum along the annealing path. For the example of the quantum Sherrington-Kirkpatrick spin glass, by comparing to extensive numerically exact results, we show that this method provides access to the instance-specific bottlenecks of the annealing protocol.
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.
The quantum annealing technique is such a paradigm that is next generation that can be used in solving optimization problem across different scientific domains like finance and material science and is capable of breaking through past obstacles to generate solutions for problems waiting to be solved. The purpose of this paper is the detailed depiction of the application of the quantum annealing in the optimization that includes the principles, applications, methodologies and outcomes and future trends all of them. Through a multidisciplinary approach, we examine the use of quantum annealing, as the method that exploits the specific features of the quantum annealing through the quantum mechanics, to explore the multitude of solution spaces faster and better than the classical methods, leading to the superior solution quality and the faster computing times. Next, we give examples for its implementation in Portfolio optimization, Machine learning, chemical optimization, Traffic management and so on depicting its role in delivering tangible benefits such as higher financial returns, precision improvement, and time savings contrast to the classical methods. Therefore, we also emphasize the current impediments, such as the scaling down the size and the ethical aspect, through which we give the way forward for the subsequent scientific researches and studies. In this abstract, we aim to depict the helix-like nature of quantum annealing in the revolution of optimization, bringing in innovations, and thus, shaping the future in which it is the quantum computing technologies that hold the central stage in addressing the complex interconnecting real-world issues.
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.