Efficient Quantum Algorithms for Higher-Order Coupled Oscillators

TL;DR

This paper develops quantum algorithms to efficiently analyze synchronization and phase-locking in higher-order oscillator networks, surpassing classical computational limits.

Contribution

It introduces quantum algorithms for nonlinear dynamical analysis of higher-order networks, achieving polynomial and super-polynomial advantages over classical methods.

Findings

Quantum algorithms enable efficient synchronization estimation.

Quantum methods certify no-phase-locking regimes faster than classical approaches.

Results extend quantum analysis from network structure to nonlinear dynamics.

Abstract

Higher-order networks with multiway interactions can exhibit collective dynamical phenomena that are absent in traditional pairwise network models. However, analyzing such dynamics becomes computationally prohibitive as their state space grows combinatorially in the multiway interaction order. Here we develop quantum algorithms for two central tasks -- synchronization estimation and certification of the no-phase-locking regime -- in the simplicial Kuramoto model. This model is a higher-order generalization of the celebrated Kuramoto model for coupled oscillators on graph-based networks. Under explicit assumptions on data access and types, and simplicial structure, we derive end-to-end quantum gate complexities and identify regimes with polynomial quantum advantage for synchronization estimation and super-polynomial quantum advantage for no-phase-locking certification over classical methods. More broadly, these results extend quantum algorithms for higher-order networks from structural analysis to nonlinear dynamical diagnostics, easing a major computational bottleneck and opening a route to quantum methods for probing higher-order phenomena beyond the reach of direct classical approaches.