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arXiv:2602.09162

Boltzmann Reinforcement Learning for Noise resilience in Analog Ising Machines

Aditya Choudhary, Saaketh Desai, Prasad Iyer

Published Feb 9, 2026• 0 citations
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Abstract

Analog Ising machines (AIMs) have emerged as a promising paradigm for combinatorial optimization, utilizing physical dynamics to solve Ising problems with high energy efficiency. However, the performance of traditional optimization and sampling algorithms on these platforms is often limited by inherent measurement noise. We introduce BRAIN (Boltzmann Reinforcement for Analog Ising Networks), a distribution learning framework that utilizes variational reinforcement learning to approximate the Boltzmann distribution. By shifting from state-by-state sampling to aggregating information across multiple noisy measurements, BRAIN is resilient to Gaussian noise characteristic of AIMs. We evaluate BRAIN across diverse combinatorial topologies, including the Curie-Weiss and 2D nearest-neighbor Ising systems. We find that under realistic 3\% Gaussian measurement noise, BRAIN maintains 98\% ground state fidelity, whereas Markov Chain Monte Carlo (MCMC) methods degrade to 51\% fidelity. Furthermore, BRAIN reaches the MCMC-equivalent solution up to 192x faster under these conditions. BRAIN exhibits $\mathcal{O}(N^{1.55})$ scaling up to 65,536 spins and maintains robustness against severe measurement uncertainty up to 40\%. Beyond ground state optimization, BRAIN accurately captures thermodynamic phase transitions and metastable states, providing a scalable and noise-resilient method for utilizing analog computing architectures in complex optimizations.

DOI
10.48550/arxiv.2602.09162

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Variational quantum eigensolvers have recently received increased attention, as they enable the use of quantum computing devices to find solutions to complex problems, such as the ground energy and ground state of strongly-correlated quantum many-body systems. In many applications, it is the optimization of both continuous and discrete parameters that poses a formidable challenge. Using reinforcement learning (RL), we present a hybrid policy gradient algorithm capable of simultaneously optimizing continuous and discrete degrees of freedom in an uncertainty-resilient way. The hybrid policy is modeled by a deep autoregressive neural network to capture causality. We employ the algorithm to prepare the ground state of the nonintegrable quantum Ising model in a unitary process, parametrized by a generalized quantum approximate optimization ansatz: the RL agent solves the discrete combinatorial problem of constructing the optimal sequences of unitaries out of a predefined set and, at the same time, it optimizes the continuous durations for which these unitaries are applied. We demonstrate the noise-robust features of the agent by considering three sources of uncertainty: classical and quantum measurement noise, and errors in the control unitary durations. Our work exhibits the beneficial synergy between reinforcement learning and quantum control.
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Several continuous dynamical systems have recently been proposed as special-purpose analog computers designed to solve combinatorial optimization problems such as kkk-SAT or the Ising problem. While combinatorial optimization problems are known to be NP-hard, and thus scale, in the worst case, exponentially with the problem size, these analog solvers promise substantial speed-up and scaling advantages in finding the solution. The underlying algorithms, which can be cast in the form of differential equations, generically involve highly chaotic dynamics and thus assume that the system variables can be processed with, in principle, arbitrary precision. However, both actual physical systems as well as finite digital machines, which are used to virtually emulate the dynamics, can process the evolution only with finite precision, be it because of intrinsic noise or because of limited precision in number representation. We investigate the impact of such noise on the solution-finding capability. To this end, we focus on two representative analog solvers, designed to address the Ising problem and the kkk-SAT problem, respectively. Our numerical analysis reveals that the ability of these algorithms to find solutions exhibits a threshold behavior under the addition of noise, where the solution-finding capability remains mostly uncompromised below a noise threshold, while it rapidly deteriorates above the threshold. As we show, these noise tolerance thresholds decrease with the problem size, following an approximate algebraic scaling. This allows us to infer principal limits on the problem sizes that can be efficiently tackled with these solvers under given noise levels.
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.