Complex Hyperspin Machine and Symmetry Quenching: Benchmark and scaling up to millions of degrees of freedom

Abstract

Driven-dissipative oscillator networks have been proposed to simulate spin models on physical hardware, mostly focusing on systems of binary (Ising) spins, or systems of spin vectors in low dimensions. The equalized hyperspin machine was recently introduced to provide a single platform to simulate general models of $N$ coupled spin vectors with tunable dimension $D$, realizing dynamical annealing protocols to vary the spin dimensionality in time. These schemes are limited to real spins coupled via real coupling matrices. We propose a generalization of the hyperspin machine that works with complex fields and deals with complex Hermitian coupling matrices, implementing an amplitude equalization scheme. This system, which we name complex hyperspin machine, can work on optimization problems with complex valued hyperspins that can be cast on complex $D$-vector Hamiltonians. We benchmark the machine against a differential evolution (DE) algorithm on complex $D=2$ hyperspins up to $N=80$, showing that the fraction of instances where a lower-energy solution is found grows with system size and coupling density, reaching $92.5\%$ at the largest considered sizes. When the DE finds a lower-energy solution, the machine still reaches a median relative energy deviation that lies between $10^{-15}$ and $10^{-12}$ while running at substantially lower computational cost. Using the complex nature of the field, we introduce a new dimensional reduction protocol that we name symmetry quenching, which collapses complex variables that map to XY spins onto real Ising spins by switching on the parametric pump. We show on Max-Cut instances with known ground states that symmetry quenching outperforms a fixed-pump Ising machine, reaching exact ground states in the presence of equalization. We scale the system to $N=10^6$ hyperspins, or $8\times10^6$ dynamical variables, on a Hermitian sparse two-dimensional lattice.