Stochastic Quantum Hamiltonian Descent

TL;DR

This paper introduces Stochastic Quantum Hamiltonian Descent (SQHD), a quantum optimization method combining stochastic gradient efficiency with quantum dynamics for improved exploration in complex machine learning landscapes.

Contribution

The paper proposes a novel quantum optimization algorithm, SQHD, integrating Lindbladian dynamics with a practical gate-based implementation, and proves its convergence for convex objectives.

Findings

SQHD converges for convex and smooth functions.

Numerical experiments show advantages in non-convex optimization.

Potential for quantum-enhanced machine learning applications.

Abstract

Stochastic Gradient Descent (SGD) and its variants underpin modern machine learning by enabling efficient optimization of large-scale models. However, their local search nature limits exploration in complex landscapes. In this paper, we introduce Stochastic Quantum Hamiltonian Descent (SQHD), a quantum optimization algorithm that integrates the computational efficiency of stochastic gradient methods with the global exploration power of quantum dynamics. We propose a Lindbladian dynamics as the quantum analogue of continuous-time SGD. We further propose a discrete-time gate-based algorithm that approximates these dynamics while avoiding direct Lindbladian simulation, enabling practical implementation on near-term quantum devices. We rigorously prove the convergence of SQHD for convex and smooth objectives. Numerical experiments demonstrate that SQHD also exhibits advantages in non-convex optimization. All these results highlight its potential for quantum-enhanced machine learning.