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arXiv:2307.13065 · Advanced Quantum Technologies

Exploring Quantum Annealing Architectures: A Spin Glass Perspective

Gabriel Jaumà, Juan José García‐Ripoll, M. Pino

Published Mar 14, 2024• 13 citations
arXivHTMLPDF

Abstract

Abstract This work analyzes the spin‐glass transition across various Ising models relevant to quantum annealers. By employing the parallel tempering method, the location of the spin‐glass phase transition is extrapolated from the pseudo‐critical temperature of finite‐sized systems. The results confirm a spin‐glass phase at finite temperature in random‐regular and small‐world graphs, in agreement with previous studies. However, strong evidence is obtained that this phase only occurs at zero temperature in the quasi‐2D graphs of D‐Wave, as their pseudo‐critical temperature drifts toward zero. This implies that the asymptotic runtime to find the low‐energy configuration of those graphs is likely to be polynomial in the size of the problem. Nevertheless, this scaling may only be reached for system sizes much larger than existing annealers, as the drift in the pseudo‐critical temperature is slow. This slowness, together with an abrupt increase in thermalization times around the pseudo‐critical temperature, may render the search for low‐energy configurations with classical methods impractical.

DOI
10.1002/qute.202300245

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arXiv:1809.01944•Journal of the Physical Society of Japan•Mar 1, 2019•9 citations
Abstract This work analyzes the spin‐glass transition across various Ising models relevant to quantum annealers. By employing the parallel tempering method, the location of the spin‐glass phase transition is extrapolated from the pseudo‐critical temperature of finite‐sized systems. The results confirm a spin‐glass phase at finite temperature in random‐regular and small‐world graphs, in agreement with previous studies. However, strong evidence is obtained that this phase only occurs at zero temperature in the quasi‐2D graphs of D‐Wave, as their pseudo‐critical temperature drifts toward zero. This implies that the asymptotic runtime to find the low‐energy configuration of those graphs is likely to be polynomial in the size of the problem. Nevertheless, this scaling may only be reached for system sizes much larger than existing annealers, as the drift in the pseudo‐critical temperature is slow. This slowness, together with an abrupt increase in thermalization times around the pseudo‐critical temperature, may render the search for low‐energy configurations with classical methods impractical.
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 quantum annealing paradigm is therefore inferior to classical simulated annealing for this class of problems. We also comment on previous simulations where a parameter is changed with the simulation time, which is very different from the true Hamiltonian dynamics simulated here.
Some recent progress in Monte Carlo simulations of spin glasses will be presented. The problem of slow dynamics at low temperatures is partially alleviated by use of the parallel tempering (replica exchange) method. A useful technique to check for equilibration (applicable only for a Gaussian distribution) will be discussed. It will be argued that a finite size scaling analysis of the scaled correlation length of the system is a good approach with which to investigate phase transitions in spin glasses. This method will be used to study two questions: (i) whether there is a phase transition in zero field in the Heisenberg spin glass in three dimensions, and (ii) whether there is phase transition in a magnetic field in an Ising spin glass, also in three dimensions.
We review our earlier studies on the order parameter distribution of the quantum Sherrington-Kirkpatrick (SK) model. Through Monte Carlo technique, we investigate the behavior of the order parameter distribution at finite temperatures. The zero temperature study of the spin glass order parameter distribution is made by the exact diagonalization method. We find in low-temperature (high-transverse-field) spin glass region, the tail (extended up to zero value of order parameter) and width of the order parameter distribution become zero in thermodynamic limit. Such observations clearly suggest the existence of a low-temperature (high-transverse-field) ergodic region. We also find in high-temperature (low-transverse-field) spin glass phase the order parameter distribution has nonzero value for all values of the order parameter even in infinite system size limit, which essentially indicates the nonergodic behavior of the system. We study the annealing dynamics by the paths which pass through both ergodic and nonergodic spin glass regions. We find the average annealing time becomes system size independent for the paths which pass through the quantum-fluctuation-dominated ergodic spin glass region. In contrast to that, the annealing time becomes strongly system size dependent for annealing down through the classical-fluctuation-dominated nonergodic spin glass region. We investigate the behavior of the spin autocorrelation in the spin glass phase. We observe that the decay rate of autocorrelation towards its equilibrium value is much faster in the ergodic region with respect to the nonergodic region of the spin glass phase.
We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy ΔF(T)‾\overline{\Delta F(T)}ΔF(T)​ and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, ΔF(T)‾\overline{\Delta F(T)}ΔF(T)​ at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, ΔF(T)‾\overline{\Delta F(T)}ΔF(T)​ decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .
Although quantum annealing is usually considered as a method for locating the ground states of difficult spin-glass and optimization problems, its use in approximate optimization -- finding low- but not zero-energy states in a reasonably short amount of time -- is no less important. Here we investigate the behavior of quantum annealing at approximate optimization in the canonical mean-field spin-glass models, the spherical ppp-spin models, and find that it performs surprisingly well. Whereas it had long been assumed that infinite-range spin glasses have a unique ``threshold'' energy at which all quench and annealing dynamics become trapped until exponential timescales, recent work has shown that two-stage quenches can in fact reach states below the naive threshold in more generic situations. We demonstrate that quantum annealing is also capable of exploiting this effect to locate sub-threshold states in O(1)O(1)O(1) time. Not only can it attain energies as far below the threshold as classical annealing algorithms, but it can do so significantly faster: for an annealing schedule taking time τττ, the residual energy under quantum annealing decays as τ−ατ^{-α}τ−α with an exponent up to twice as large as that of simulated annealing in the cases considered. Importantly, by deriving and numerically solving closed integro-differential equations that hold in the thermodynamic limit, our results are free from finite-size effects and hold for annealing times that are unambiguously independent of system size.
We first review, following our earlier studies, the critical behavior of the quantum Sherrington-Kirkpatrick (SK) model at finite as well as at zero temperatures. Through the analysis of the Binder cumulant we determined the entire phase diagram of the model and from the scaling analysis of the numerical data we obtained the correlation length exponent. For both the critical Binder cumulant and the correlation length exponent, we observed a crossover from classical- to quantum-fluctuation-dominated values at a finite temperature. We studied the behavior of the order parameter distribution of the model in the glass phase (at finite and zero temperatures). Along with a classical-fluctuation-dominated nonergodic region (where the replica symmetry is broken), we also found a quantum-fluctuation-dominated low-temperature ergodic region in the spin glass phase. In this quantum-fluctuation-dominated region, the order parameter distribution has a narrow peak around its most probable value, eventually becoming a delta function in the infinite-system-size limit (indicating replica symmetry restoration or ergodicity in the system). We also found that the annealing time (to reach a very low energy level of the classical SK model) becomes practically system-size-independent when the annealing paths pass through this ergodic region. In contrast, when such paths pass through the nonergodic region, the convergence time grows rapidly with the system size. We present a new study of the autocorrelation of the spins in both ergodic and nonergodic regions. We found a significant increase in the relaxation time (and also a change in the relaxation behavior) in the classical-fluctuation-dominated (nonergodic) region compared with that in the quantum-fluctuation-dominated (ergodic) region of the spin glass phase.