Maximum-Likelihood Quantum State Tomography by Soft-Bayes

TL;DR

This paper introduces a scalable stochastic algorithm based on Soft-Bayes for maximum-likelihood quantum state tomography, providing theoretical guarantees and efficiency for high-dimensional quantum systems.

Contribution

It extends Soft-Bayes to quantum state estimation, offering a scalable stochastic method with provable convergence guarantees for maximum-likelihood quantum state tomography.

Findings

Achieves $ ext{O}((D ext{log} D)/ ext{ε}^2)$ iteration complexity

Per-iteration complexity is $ ext{O}(D^3)$

Provides sample complexity guarantees for quantum state estimation

Abstract

Quantum state tomography (QST), the task of estimating an unknown quantum state given measurement outcomes, is essential to building reliable quantum computing devices. Whereas computing the maximum-likelihood (ML) estimate corresponds to solving a finite-sum convex optimization problem, the objective function is not smooth nor Lipschitz, so most existing convex optimization methods lack sample complexity guarantees; moreover, both the sample size and dimension grow exponentially with the number of qubits in a QST experiment, so a desired algorithm should be highly scalable with respect to the dimension and sample size, just like stochastic gradient descent. In this paper, we propose a stochastic first-order algorithm that computes an $\varepsilon$-approximate ML estimate in $O( ( D \log D ) / \varepsilon ^ 2 )$ iterations with $O( D^3 )$ per-iteration time complexity, where $D$ denotes the dimension of the unknown quantum state and $\varepsilon$ denotes the optimization error. Our algorithm is an extension of Soft-Bayes to the quantum setup.