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Edge-of-chaos-enhanced quantum-inspired algorithm for combinatorial optimization

Hayato Goto1,2,*, Ryo Hidaka1, and Kosuke Tatsumura1

  • *Contact author: hayato.goto.d36@mail.toshiba
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Phys. Rev. Applied 25, 044011 – Published 6 April, 2026

DOI: https://doi.org/10.1103/2qd9-x6v8

Abstract

Nonlinear dynamical systems with continuous variables can be used for solving combinatorial optimization problems with discrete variables. Numerical simulations of them are also useful as heuristic algorithms with a desirable property, namely, parallelizability, which allows us to execute them in a massively parallel manner, leading to ultrafast performance. However, the dynamical-system approaches with continuous variables are usually less accurate than conventional approaches with discrete variables such as simulated annealing. To improve the solution accuracy of a quantum-inspired algorithm called simulated bifurcation (SB), which was found from classical simulation of a quantum nonlinear oscillator network exhibiting quantum bifurcation, here we generalize it by introducing nonlinear control of individual bifurcation parameters and show that the generalized SB (GSB) can achieve surprisingly high performance, namely, almost 100% success probabilities for some large-scale problems. As a result, the time to solution for a 2000-variable problem is shortened to 10 ms by a GSB-based machine, which is 2 orders of magnitude shorter than the best known value, 1.3 s, previously obtained by an SB-based machine. To examine the reason for the ultrahigh performance, we investigated chaos in the GSB changing the nonlinear-control strength and found that the dramatic increase of success probabilities happens near the edge of chaos. That is, the GSB can find a solution with high probability by harnessing the edge of chaos. This finding suggests that dynamical-system approaches to combinatorial optimization will be enhanced by harnessing the edge of chaos, opening a broad possibility for physics-inspired approaches to combinatorial optimization.

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Phys. Rev. Applied 25, 044011– Published 6 April, 2026
  • Received 11 October 2025
  • Revised 23 December 2025
  • Accepted 26 February 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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