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arXiv:2506.12914

Noise resilience of deterministic analog combinatorial optimization solvers

Clemens Gneiting, Farad Khoyratee, Enrico Rinaldi, Kavita. A Jain, Rishab Khincha, Franco Nori

Published Jun 15, 2025• 0 citations
arXivHTMLPDF

Abstract

Several continuous dynamical systems have recently been proposed as special-purpose analog computers designed to solve combinatorial optimization problems such as $k$-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 $k$-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.

DOI
10.48550/arxiv.2506.12914

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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.
Many combinatorial scientific computing problems are NP-hard which in practice requires using heuristics that either decompose a large-scale problem and solve many smaller local subproblems in parallel or iteratively improve the solution by local processing for sufficiently many steps to achieve a good quality/performance trade-off. These subproblems are NP-hard by themselves and the quality of their solution that depends on the subproblem size vitally affects the global solution of the entire problem. As we approach the physical limits predicted by Moore’s law, a variety of specialized combinatorial optimization hardware (such as adiabatic quantum computers, CMOS annealers, and optical parametric oscillators) is emerging to tackle specialized tasks in different domains. Many researchers anticipate that in the near future, these devices will be hybridized with high-performance computing systems to tackle large-scale problems which will open a door for the next breakthrough in performance of the combinatorial scientific computing algorithms. Most of these devices solve the Ising combinatorial optimization model that can represent many fundamental combinatorial scientific computing models. In this paper, we tackle the main challenges of problem size and precision limitation that the Ising hardware model typically suffers from. Although, our demonstration uses one of these novel devices, the models we propose can be generally used in any local search and refinement solvers that are broadly employed in combinatorial scientific computing algorithms.
Ising machines are specialized devices designed to efficiently solve combinatorial optimization problems. Among such problems, Boolean Satisfiability (SAT) is particularly relevant in industrial applications. To solve SAT problems using Ising machines, it is crucial to incorporate higher-order interactions. However, in analog Ising machines, interactions of different orders scale unevenly with the continuous spin amplitudes, introducing imbalances that can significantly degrade performance. Here we present a numerical comparison of methods to mitigate these imbalances, evaluating time-to-solution and success rate on uniform random 3-SAT and 4-SAT instances. Our results show that the most effective approach employs spin interactions that are proportional to the signs of spins, rather than their continuous amplitudes. We further demonstrate that smooth approximations of this method make it compatible with analog hardware. These findings underscore the central role of spin-sign-based interactions in enabling robust and scalable analog Ising machine dynamics. Ising machines (IMs) offer promising solutions for combinatorial optimization problems yet face challenges with imbalances in higher-order interactions when applied to Boolean Satisfiability (SAT) tasks. Here, the authors demonstrate that employing spin-sign-based interactions effectively mitigates these imbalances, enhancing performance and compatibility with analog hardware, thus advancing robust and scalable IM dynamics.
Ising machines are hardware solvers which aim to find the absolute or approximate ground states of the Ising model. The Ising model is of fundamental computational interest because it is possible to formulate any problem in the complexity class NP as an Ising problem with only polynomial overhead. A scalable Ising machine that outperforms existing standard digital computers could have a huge impact for practical applications for a wide variety of optimization problems. In this review, we survey the current status of various approaches to constructing Ising machines and explain their underlying operational principles. The types of Ising machines considered here include classical thermal annealers based on technologies such as spintronics, optics, memristors, and digital hardware accelerators; dynamical-systems solvers implemented with optics and electronics; and superconducting-circuit quantum annealers. We compare and contrast their performance using standard metrics such as the ground-state success probability and time-to-solution, give their scaling relations with problem size, and discuss their strengths and weaknesses.
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 O(N1.55)\mathcal{O}(N^{1.55})O(N1.55) 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.
Combinatorial optimization and large-scale data analytics present NP-hard computational challenges, where quantum algorithms promise potential exponential speedups. However, the practical realization of such advantage on Noisy Intermediate-Scale Quantum (NISQ) hardware remains uncertain. This systematic review synthesizes relevant literature to critically evaluate the empirical feasibility and scaling limitations of leading quantum algorithms. The analysis reveals that Variational Quantum Algorithms (VQAs) face persistent reliability challenges such as inconsistent optimality gaps and systemic failures to produce feasible solutions for constrained problems. The principal barrier to VQA scalability is the Measurement Imprecision Wall, where stochastic noise accumulation drives the required measurement shots to scale exponentially with system size, creating a resource deadlock that undermines quantum advantage for large problem instances. For data-centric algorithms, the exponential speedup of Quantum Linear Systems Problem (QLSP) solvers, such as HHL, is largely theoretical for arbitrary classical data due to the Encoding Bottleneck, which requires O(N) runtime for input state preparation. In contrast, Quantum Kernel Methods (QKM) exhibit superior empirical performance and efficient resource scaling, representing the most promising near-term pathway. Achieving practical quantum advantage therefore depends on noise-aware algorithmic design and effective solutions to the classical data ingestion challenge.
Combinatorial optimization problems are ubiquitous in industrial applications. However, finding optimal or close-to-optimal solutions can often be extremely hard. Because some of these problems can be mapped to the ground-state search of the Ising model, tremendous effort has been devoted to developing solvers for Ising-type problems over the past decades. Recent advances in controlling and manipulating both quantum and classical systems have enabled novel computing paradigms such as quantum simulators and coherent Ising machines to tackle hard optimization problems. Here, we examine and benchmark several physics-inspired optimization algorithms, including coherent Ising machines, gain-dissipative algorithms, simulated bifurcation machines, and Hopfield neural networks, which we collectively refer to as stochastic driven nonlinear dynamical systems. Most importantly, we benchmark these algorithms against random Ising problems with planted solutions and compare them to simulated annealing as a baseline leveraging the same software stack for all solvers. We further study how different numerical integration techniques and graph connectivity affect performance. This work provides an overview of a diverse set of new optimization paradigms.