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Retrieving the ground state of spin glasses using thermal noise: Performance of quantum annealing at finite temperatures

光嗣 西村, Kohji Nishimura, 秀稔 西森, Hidetoshi Nishimori, Andrew J. Ochoa, Helmut G. Katzgraber

Published Sep 1, 2016• 0 citations
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We study the problem to infer the ground state of a spin-glass Hamiltonian using data from another Hamiltonian with interactions disturbed by noise from the original Hamiltonian, motivated by the ground-state inference in quantum annealing on a noisy device. It is shown that the average Hamming distance between the inferred spin configuration and the true ground state is minimized when the temperature of the noisy system is kept at a finite value, and not at zero temperature. We present a spin-glass generalization of a well-established result that the ground state of a purely ferromagnetic Hamiltonian is best inferred at a finite temperature in the sense of smallest Hamming distance when the original ferromagnetic interactions are disturbed by noise. We use the numerical transfer-matrix method to establish the existence of an optimal finite temperature in one- and two-dimensional systems. Our numerical results are supported by mean-field calculations, which give an explicit expression of the optimal temperature to infer the spin-glass ground state as a function of variances of the distributions of the original interactions and the noise. The mean-field prediction is in qualitative agreement with numerical data. Implications on postprocessing of quantum annealing on a noisy device are discussed.

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A STUDY OF QUANTUM ANNEALING DEVICES FROM A CLASSICAL PERSPECTIVE

Andrew J. Ochoa

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.

•Dec 7, 2017•1 citation
We study the problem to infer the ground state of a spin-glass Hamiltonian using data from another Hamiltonian with interactions disturbed by noise from the original Hamiltonian, motivated by the ground-state inference in quantum annealing on a noisy device. It is shown that the average Hamming distance between the inferred spin configuration and the true ground state is minimized when the temperature of the noisy system is kept at a finite value, and not at zero temperature. We present a spin-glass generalization of a well-established result that the ground state of a purely ferromagnetic Hamiltonian is best inferred at a finite temperature in the sense of smallest Hamming distance when the original ferromagnetic interactions are disturbed by noise. We use the numerical transfer-matrix method to establish the existence of an optimal finite temperature in one- and two-dimensional systems. Our numerical results are supported by mean-field calculations, which give an explicit expression of the optimal temperature to infer the spin-glass ground state as a function of variances of the distributions of the original interactions and the noise. The mean-field prediction is in qualitative agreement with numerical data. Implications on postprocessing of quantum annealing on a noisy device are discussed.
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.