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arXiv:2504.21666 · Physical Review Letters

Quantum Annealing Algorithms for Estimating Ising Partition Functions

Haowei Li, Zhiyuan Yao, Xingze Qiu

Published Feb 11, 2026• 1 citation
arXivHTMLPDF

Abstract

Estimating partition functions of Ising spin glasses is a cornerstone of statistical physics and computational science, yet it remains classically challenging due to its #P-hard complexity. While Jarzynski's equality offers a theoretical pathway, its practical application is crippled at low temperatures by rare, divergent statistical fluctuations. Here, we introduce a quantum protocol that overcomes this fundamental limitation by synergizing reverse quantum annealing with optimized nonequilibrium initial distributions. Our method dramatically suppresses the estimator variance, achieving saturation in the low-temperature regime where existing methods fail. Numerical benchmarks on the Sherrington-Kirkpatrick spin glass and the 3-SAT problem demonstrate that our protocol reduces computational scaling exponents by over an order of magnitude (e.g., from ∼8.5 to ∼0.5), despite retaining exponential system-size dependence. Crucially, our protocol circumvents stringent adiabatic constraints, making it feasible for near-term quantum devices like superconducting qubits, trapped ions, and Rydberg atom arrays. This Letter provides a methodological framework for quantum-enhanced estimation in spin glass thermodynamics and beyond by harnessing nonadiabatic quantum dynamics to address a classically difficult problem.

DOI
10.1103/8gmb-p619

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Estimating partition functions of Ising spin glasses is a cornerstone of statistical physics and computational science, yet it remains classically challenging due to its #P-hard complexity. While Jarzynski's equality offers a theoretical pathway, its practical application is crippled at low temperatures by rare, divergent statistical fluctuations. Here, we introduce a quantum protocol that overcomes this fundamental limitation by synergizing reverse quantum annealing with optimized nonequilibrium initial distributions. Our method dramatically suppresses the estimator variance, achieving saturation in the low-temperature regime where existing methods fail. Numerical benchmarks on the Sherrington-Kirkpatrick spin glass and the 3-SAT problem demonstrate that our protocol reduces computational scaling exponents by over an order of magnitude (e.g., from ∼8.5 to ∼0.5), despite retaining exponential system-size dependence. Crucially, our protocol circumvents stringent adiabatic constraints, making it feasible for near-term quantum devices like superconducting qubits, trapped ions, and Rydberg atom arrays. This Letter provides a methodological framework for quantum-enhanced estimation in spin glass thermodynamics and beyond by harnessing nonadiabatic quantum dynamics to address a classically difficult problem.
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.
We evaluate using programmable superconducting flux qubit D-Wave quantum annealers to approximate the partition function of Ising models. We propose the use of two distinct quantum annealer sampling methods: chains of Monte Carlo-like reverse quantum anneals, and standard linear-ramp quantum annealing. The control parameters used to attenuate the quality of the simulations are the effective analog energy scale of the J coupling, the total annealing time, and for the case of reverse annealing the anneal-pause. The core estimation technique is to sample across the energy spectrum of the classical Hamiltonian of interest, and therefore obtain a density of states estimate for each energy level, which in turn can be used to compute an estimate of the partition function with some sampling error. This estimation technique is powerful because once the distribution is sampled it allows thermodynamic quantity computation at arbitrary temperatures. On a 252525 spin ±J\pm J±J hardware graph native Ising model we find parameter regimes of the D-Wave processors that provide comparable result quality to two standard classical Monte Carlo methods, Multiple Histogram Reweighting and Wang-Landau. Remarkably, we find that fast quench-like anneals can quickly generate ensemble distributions that are very good estimates of the true partition function of the classical Ising model; on a Pegasus graph-structured QPU we report a logarithmic relative error of 7.6×10−67.6 \times 10^{-6}7.6×10−6, from 171,000171,000171,000 samples generated using 0.20.20.2 seconds of QPU time with an anneal time of 888 nanoseconds per sample which is interestingly within the closed system dynamics timescale of the superconducting qubits.
We point to the existence of an analytical solution to a general quantum annealing (QA) problem of finding low energy states of an arbitrary Ising spin Hamiltonian HIH_IHI​ by implementing time evolution with a Hamiltonian H(t)=HI+g(t)HtH(t)=H_I+g(t) H_tH(t)=HI​+g(t)Ht​. We will assume that the nonadiabatic annealing protocol is defined by a specific decaying coupling g(t)g(t)g(t) and a specific mixing Hamiltonian HtH_tHt​ that make the model analytically solvable arbitrarily far from the adiabatic regime. In specific cases of HIH_IHI​, the solution shows the possibility of a considerable quantum speedup of finding the Ising ground state. We then compare predictions of our solution to results of numerical simulations and argue that the solvable QA protocol produces the optimal performance in the limit of maximal complexity of the computational problem. Our solution demonstrates for the most complex spin glasses a power-law energy relaxation with the annealing time TTT and uncorrelated from the HIH_IHI​ annealing schedule. This proves the possibility for spin glasses of a faster than ∼1/log⁡βT\sim 1/\log^{\beta} T∼1/logβT energy relaxation.
Abstract Ising spin Hamiltonians are often used to encode a computational problem in their ground states. Quantum Annealing (QA) computing searches for such a state by implementing a slow time-dependent evolution from an easy-to-prepare initial state to a low energy state of a target Ising Hamiltonian of quantum spins, H I . Here, we point to the existence of an analytical solution for such a problem for an arbitrary H I beyond the adiabatic limit for QA. This solution provides insights into the accuracy of nonadiabatic computations. Our QA protocol in the pseudo-adiabatic regime leads to a monotonic power-law suppression of nonadiabatic excitations with time T of QA, without any signature of a transition to a glass phase, which is usually characterized by a logarithmic energy relaxation. This behavior suggests that the energy relaxation can differ in classical and quantum spin glasses strongly, when it is assisted by external time-dependent fields. In specific cases of H I , the solution also shows a considerable quantum speedup in computations.
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.
Advances in quantum algorithms suggest a tentative scaling advantage on certain combinatorial optimization problems. Recent work, however, has also reinforced the idea that barren plateaus render variational algorithms ineffective on large Hilbert spaces. Hence, finding annealing protocols by variation ultimately appears to be difficult. Similarly, the adiabatic theorem fails on hard problem instances with first-order quantum phase transitions. Here we show how to use the spin coherent-state path integral to shape the geometry of quantum adiabatic evolution, leading to annealing protocols at polynomial overhead that provide orders-of-magnitude improvements in the probability to measure optimal solutions, relative to linear protocols. These improvements are not obtained on a controllable toy problem but on randomly generated hard instances (Sherrington-Kirkpatrick and maximum 2-satisfiability), making them generic and robust. Our method works for large systems and may thus be used to improve the performance of state-of-the-art quantum devices.