Local Stochastic Factored Gradient Descent for Distributed Quantum State Tomography

TL;DR

This paper introduces a distributed quantum state tomography method called Local SFGD, which efficiently learns low-rank density matrices using stochastic gradient descent, with proven local convergence and validated on GHZ states.

Contribution

It proposes a novel distributed QST protocol with theoretical convergence guarantees for a class of loss functions, advancing scalable quantum state estimation.

Findings

Proves local convergence of Local SFGD for restricted strongly convex/smooth loss functions.

Demonstrates linear convergence near the optimum with constant step size.

Validates the method through numerical simulations on GHZ states.

Abstract

We propose a distributed Quantum State Tomography (QST) protocol, named Local Stochastic Factored Gradient Descent (Local SFGD), to learn the low-rank factor of a density matrix over a set of local machines. QST is the canonical procedure to characterize the state of a quantum system, which we formulate as a stochastic nonconvex smooth optimization problem. Physically, the estimation of a low-rank density matrix helps characterizing the amount of noise introduced by quantum computation. Theoretically, we prove the local convergence of Local SFGD for a general class of restricted strongly convex/smooth loss functions, i.e., Local SFGD converges locally to a small neighborhood of the global optimum at a linear rate with a constant step size, while it locally converges exactly at a sub-linear rate with diminishing step sizes. With a proper initialization, local convergence results imply global convergence. We validate our theoretical findings with numerical simulations of QST on the Greenberger-Horne-Zeilinger (GHZ) state.