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arXiv:2605.25872 · Journal of Statistical Mechanics Theory and Experiment

Cluster moves with an entropic reservoir accelerate low-temperature simulations of three-dimensional spin glasses

Claudio Chilin, Enzo Marinari, Víctor Martín-Mayor, Giorgio Parisi, Juan Jesus Ruiz-Lorenzo, David Yllanes

Published Oct 1, 2026• 1 citation
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Abstract

Abstract We present an algorithm for the simulation of three-dimensional spin glasses deep in the low-temperature phase: Parallel Tempering enhanced with Houdayer moves and with an entropic reservoir (PTHR). Although differences with the standard Houdayer algorithm are small, PTHR allows us to equilibrate a large number of samples of L = 16 lattices with Gaussian couplings for temperatures T ⩾ 0.2 . We show that the computational complexity displays better size scaling than standard Parallel Tempering. For finite sizes, our method outperforms other cluster algorithms by a speedup factor of around 64. In close analogy with standard Parallel Tempering, PTHR’s computational complexity strongly relates to temperature chaos.

DOI
10.1088/1742-5468/aeafeb

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Abstract We present an algorithm for the simulation of three-dimensional spin glasses deep in the low-temperature phase: Parallel Tempering enhanced with Houdayer moves and with an entropic reservoir (PTHR). Although differences with the standard Houdayer algorithm are small, PTHR allows us to equilibrate a large number of samples of L = 16 lattices with Gaussian couplings for temperatures T ⩾ 0.2 . We show that the computational complexity displays better size scaling than standard Parallel Tempering. For finite sizes, our method outperforms other cluster algorithms by a speedup factor of around 64. In close analogy with standard Parallel Tempering, PTHR’s computational complexity strongly relates to temperature chaos.
Abstract We have introduced a variational method to improve the computation of integrated correlation times in parallel tempering dynamics, obtaining a better estimate (a lower bound, at least) of the exponential correlation time. Using this determination of the correlation times, we revisited the problem of the characterization of the chaos in temperature in finite dimensional spin glasses, by way of the study of correlations between various chaos indicators computed in the static and the correlation times of the parallel tempering dynamics. The sample-distribution of the characteristic time for the parallel tempering dynamics turns out to be fat-tailed, and to obey finite-size scaling.
We present a massive equilibrium simulation of the three-dimensional Ising spin glass at low temperatures. The Janus special-purpose computer has allowed us to equilibrate, using parallel tempering, L = 32 lattices down to T ≈ 0.64 T c . We demonstrate the relevance of equilibrium finite size simulations to understanding experimental non-equilibrium spin glasses in the thermodynamical limit by establishing a time-length dictionary. We conclude that non-equilibrium experiments performed on a timescale of 1 h can be matched with equilibrium results on L ≈ 110 lattices. A detailed investigation of the probability distribution functions of the spin and link overlap, as well as of their correlation functions, shows that Replica Symmetry Breaking is the appropriate theoretical framework for the physically relevant length scales. Besides, we improve over existing methodologies in ensuring equilibration in parallel tempering simulations.
Spin systems with frustration and disorder are notoriously difficult to study, both analytically and numerically. While the simulation of ferromagnetic statistical mechanical models benefits greatly from cluster algorithms, these accelerated dynamics methods remain elusive for generic spin-glass-like systems. Here, we present a cluster algorithm for Ising spin glasses that works in any space dimension and speeds up thermalization by at least one order of magnitude at temperatures where thermalization is typically difficult. Our isoenergetic cluster moves are based on the Houdayer cluster algorithm for two-dimensional spin glasses and lead to a speedup over conventional state-of-the-art methods that increases with the system size. We illustrate the benefits of the isoenergetic cluster moves in two and three space dimensions, as well as the nonplanar chimera topology found in the D-Wave Inc. quantum annealing machine.
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 efficiency of parallel tempering Monte Carlo is studied for a two-dimensional Ising system of length L with N=L^2 spins. An external field is used to introduce a difference in free energy between the two low temperature states.\nIt is found that the number of replicas R_opt that optimizes the parallel tempering algorithm scales as the square root of the system size N. For two symmetric low temperature states, the time needed for equilibration is observed to grow as L^2.18. If a significant difference in free energy is present between the two states, this changes to L^1.02.\nIt is therefore established that parallel tempering is sped up by a factor of roughly L if an asymmetry is introduced between the low temperature states. This confirms previously made predictions for the efficiency of parallel tempering. These findings should be especially relevant when using parallel tempering for systems like spin glasses, where no information about the degeneracy of low temperature states is available prior to the simulation.
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature range around the critical point. By combining the parallel-tempering algorithm with cluster updates and an adaptive routine to find the temperature window of interest, we introduce a flexible and powerful method for systematic investigations of critical phenomena. As a result, we gain one to two orders of magnitude in the performance for two- and three-dimensional Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.