Variational Iterative Rotation Algorithm: Combinatorial Optimization with Classical Kicked Tops

TL;DR

VIRAL, a classical iterative rotation algorithm inspired by QAOA, outperforms quantum QAOA on spin-glass benchmarks by leveraging classical dynamics and bifurcation phenomena, with potential implementation in magnetic tunnel junctions.

Contribution

This work introduces VIRAL, a classical algorithm based on iterative rotations that surpasses quantum QAOA in optimization tasks and provides insights into its classical dynamics and physical implementation.

Findings

VIRAL outperforms QAOA on the Sherrington-Kirkpatrick benchmark at all depths.

The energy density converges linearly in 1/p to the ground state.

Classical dynamics follow a Floquet protocol with bifurcation-driven spin polarization.

Abstract

We investigate a classical formulation of the Quantum Approximate Optimization Algorithm (QAOA), realized as a Hamiltonian dynamical system of classical kicked tops, which we call the Variational Iterative Rotation Algorithm (VIRAL). The variational parameters are the transverse and longitudinal rotation angles at each of the p layers of the circuit. We find that VIRAL outperforms QAOA on the canonical Sherrington-Kirkpatrick spin-glass benchmark at all circuit depths, with the energy density converging to the ground state value linearly in 1/p. For large circuit depths, the optimized dynamics follows a Floquet protocol in which a pitchfork bifurcation destabilizes the equatorial fixed point and drives the spins toward polar Ising configurations. Our results demonstrate that the effectiveness of QAOA-like protocols derives primarily from their underlying iterative rotation structure, and that a classical implementation of it outperforms its quantum counterpart. We further elucidate its efficiency by reducing the many-body classical evolution to an effective Landau-Lifshitz dynamics for a single spin in a stochastic magnetic field. In this picture, the covariance matrix of the effective field reveals a nearly rank-one structure in which a single mode dominates the stochastic dynamics. In contrast, quantum fluctuations make the noise covariance of the effective quantum model of higher rank, hampering the control of the system. We propose nanometer-scale magnetic tunnel junctions as a natural physical platform for implementing VIRAL, where spin rotations can be realized using magnetic fields and spin torques.