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Wave Electronics and its Application in Information and Telecommunication Systems (WECONF)

Simulated Quantum Annealing of the Wishart Planted Ensemble

Evgenii V. Burlakov, M. A. Yarygin

Published May 11, 2026• 0 citations
Publisher

Abstract

This work investigates the performance of simulated quantum annealing on the Wishart Planted Ensemble, a tunable spin-glass model with a known global minimum and a non-monotonic hardness profile. A direct comparison between classical simulated annealing and simulated quantum annealing is carried out in the hard regime. We show that on energy landscapes where simulated annealing fails to reach the planted state, simulated quantum annealing - at optimal inverse temperatures - consistently attains significantly lower energies, approaching the global optimum. These results demonstrate the advantage of emulating quantum tunneling over purely thermal fluctuations in landscapes characterized by high but narrow energy barriers.

DOI
10.1109/weconf69969.2026.11553816

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•Jan 1, 2015•86 citations
This work investigates the performance of simulated quantum annealing on the Wishart Planted Ensemble, a tunable spin-glass model with a known global minimum and a non-monotonic hardness profile. A direct comparison between classical simulated annealing and simulated quantum annealing is carried out in the hard regime. We show that on energy landscapes where simulated annealing fails to reach the planted state, simulated quantum annealing - at optimal inverse temperatures - consistently attains significantly lower energies, approaching the global optimum. These results demonstrate the advantage of emulating quantum tunneling over purely thermal fluctuations in landscapes characterized by high but narrow energy barriers.
Quantum annealing has emerged as a promising approach for solving NP-hard optimization problems, leveraging quantum phenomena such as quantum tunneling to navigate complex energy landscapes. However, the extent to which quantum tunneling contributes to performance enhancements compared to classical methods remains unclear. In this work, we present a comprehensive investigation combining experimental analysis, theoretical modeling, and algorithmic development to characterize when and why quantum annealing provides computational advantages. We introduce a novel methodology for generating synthetic Quadratic Unconstrained Binary Optimization (QUBO) problems with controlled gradient variance, enabling systematic investigation of landscape characteristics that favor quantum approaches. Our experimental evaluation encompasses four canonical NP-hard problems: Graph Partitioning, Max Cut, Number Partitioning, and Set Cover. Using D-Wave's Advantage2 quantum annealer with 4,400+ qubits, we compare quantum annealing against classical solvers including simulated annealing, stochastic gradient descent, and commercial optimization software. Our results demonstrate that quantum annealing shows measurable advantages when energy landscapes exhibit high gradient variance (>0.3> 0.3>0.3), suggesting that quantum tunneling effects are most beneficial for rugged optimization landscapes. We provide a theoretical justification for this empirical observation through a WKB-approximation-based model connecting gradient variance to barrier width and tunneling probability. This model achieves R2=0.90R^2=0.90R2=0.90 correlation with experimental data and yields quantitative predictions for quantum advantage thresholds.
Finding the global minimum in a rugged potential landscape is a computationally hard task, often equivalent to relevant optimization problems. Annealing strategies, either classical or quantum, explore the configuration space by evolving the system under the influence of thermal or quantum fluctuations. The thermal annealing dynamics can rapidly freeze the system into a low-energy configuration, and it can be simulated well on a classical computer, but it easily gets stuck in local minima. Quantum annealing, on the other hand, can be guaranteed to find the true ground state and can be implemented in modern quantum simulators; however, quantum adiabatic schemes become prohibitively slow in the presence of quasidegeneracies. Here, we propose a strategy which combines ideas from simulated annealing and quantum annealing. In such a hybrid algorithm, the outcome of a quantum simulator is processed on a classical device. While the quantum simulator explores the configuration space by repeatedly applying quantum fluctuations and performing projective measurements, the classical computer evaluates each configuration and enforces a lowering of the energy. We have simulated this algorithm for small instances of the random energy model, showing that it potentially outperforms both simulated thermal annealing and adiabatic quantum annealing. It becomes most efficient for problems involving many quasidegenerate ground states.
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.
The path integral Monte Carlo simulated quantum annealing algorithm is applied to the optimization of a large hard instance of the random satisfiability problem (N = 10,000). The dynamical behavior of the quantum and the classical annealing are compared, showing important qualitative differences in the way of exploring the complex energy landscape of the combinatorial optimization problem. At variance with the results obtained for the Ising spin glass and for the traveling salesman problem, in the present case the linear-schedule quantum annealing performance is definitely worse than classical annealing. Nevertheless, a quantum cooling protocol based on field-cycling and able to outperform standard classical simulated annealing over short time scales is introduced.
There has been considerable progress in the design and construction of quantum annealing devices. However, a conclusive detection of quantum speedup over traditional silicon-based machines remains elusive, despite multiple careful studies. In this work we outline strategies to design hard tunable benchmark instances based on insights from the study of spin glasses - the archetypal random benchmark problem for novel algorithms and optimization devices. We propose to complement head-to-head scaling studies that compare quantum annealing machines to state-of-the-art classical codes with an approach that compares the performance of different algorithms and/or computing architectures on different classes of computationally hard tunable spin-glass instances. The advantage of such an approach lies in having to only compare the performance hit felt by a given algorithm and/or architecture when the instance complexity is increased. Furthermore, we propose a methodology that might not directly translate into the detection of quantum speedup, but might elucidate whether quantum annealing has a "`quantum advantage" over corresponding classical algorithms like simulated annealing. Our results on a 496 qubit D-Wave Two quantum annealing device are compared to recently-used state-of-the-art thermal simulated annealing codes.