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Solved: Exact Ground State of Frustrated Quantum Spin Glasses (NP-Hard) Solved

Jesse Daniel Brown

Published Oct 22, 2026• 0 citations
Publisher
DOI
10.13140/rg.2.2.29646.83521

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Limited resources motivate decomposing large-scale problems into smaller, "local" subsystems and stitching together the so-found solutions. We explore the physics underlying this approach and discuss the concept of "local hardness," i.e., the complexity of predicting local properties of the solution from local information, for the ground-state problem of both P- and NP-hard spin glasses and related frustrated spin systems. Depending on the model considered, we observe varying scaling behaviors in how errors associated with local predictions decay as a function of the size of the solved subsystem. These errors are intimately connected to global critical threshold instabilities, characterized by gapless, avalanchelike excitations that follow scale-invariant size distributions. Away from criticality, local solvers quickly achieve high accuracy, aligning closely with the results of the computationally much more expensive global minimization. We leverage these findings to introduce a heuristic contraction-based algorithm for globally studying spin-glass ground states. The local solvers further display sharp imprints of the phase transition from the spin-glass to the ferromagnetic phase as the distribution of spin-glass couplings is shifted, as well as characteristic differences for the infinite-range model, implying the existence of specific classes of local hardness. Our findings shed light on how nature may operate solely through local actions at her disposal.

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Variational neural network models have achieved remarkable success in solving ground-state problems of quantum many-body systems. However, addressing classical and quantum spin glasses remains challenging, as disorder and energy frustration give rise to an exponentially large number of local energy minima separated by high-energy barriers, hindering the efficiency of conventional Metropolis-based Monte Carlo methods. To bridge this gap, we introduce Deep Boltzmann Quantum States, a class of neural quantum states inspired by deep Boltzmann machines that inherit efficient block Gibbs sampling. We also propose two key advances in the training algorithm. Firstly, we combine natural-gradient updates with state-of-the-art stochastic optimizers. Secondly, we gradually tune the hardness of the problem Hamiltonian by interpolating from an easy to a hard regime, without the need to closely approximate the instantaneous adiabatic state at intermediate times. We match the exact solution or the best available estimate for several instances of classical and quantum Ising spin-glass models with infinite-range interactions and hundreds of spins. We also solve instances of the NP-hard Job Shop Scheduling Problem exceeding the current limitations of quantum annealing hardware. To summarize, deep neural architectures with efficient global update rules and trained within an annealing-like scheme, provide a powerful framework for solving real-world hard combinatorial optimization and for investigating disordered quantum many-body systems.
Finding an exact ground state of a three-dimensional (3D) Ising spin glass is proven to be an NP-hard problem (i.e., at least as hard as any problem in the nondeterministic polynomial-time (NP) class). Given validity of the exponential time hypothesis, its computational complexity was proven to be no less than 2 N 2 / 3 , where N is the total number of spins. Here, we report results of extensive experimentation with D-Wave 3D annealer with N ≤ 5627 . We found exact ground states (in a probabilistic sense) for typical realizations of 3D spin glasses with the efficiency, which scales as 2 N / β with β ≈ 10 3 . Based on statistical analysis of low-energy states, we argue that with an improvement of annealing protocols and device noise reduction, β can be increased even further. This suggests that, for N < β 3 , annealing devices provide most efficient way to find an exact ground state.
Limited resources motivate decomposing large-scale problems into smaller, "local" subsystems and stitching together the so-found solutions. We explore the physics underlying this approach and discuss the concept of "local hardness," i.e., the complexity of predicting local properties of the solution from local information, for the ground-state problem of both P- and NP-hard spin glasses and related frustrated spin systems. Depending on the model considered, we observe varying scaling behaviors in how errors associated with local predictions decay as a function of the size of the solved subsystem. These errors are intimately connected to global critical threshold instabilities, characterized by gapless, avalanchelike excitations that follow scale-invariant size distributions. Away from criticality, local solvers quickly achieve high accuracy, aligning closely with the results of the computationally much more expensive global minimization. We leverage these findings to introduce a heuristic contraction-based algorithm for globally studying spin-glass ground states. The local solvers further display sharp imprints of the phase transition from the spin-glass to the ferromagnetic phase as the distribution of spin-glass couplings is shifted, as well as characteristic differences for the infinite-range model, implying the existence of specific classes of local hardness. Our findings shed light on how nature may operate solely through local actions at her disposal.