Quantum state learning advances improve tomography and confidence tools

Regret Lower Bound for Online Learning of Quantum States with Logarithmic Loss, with Applications to Quantum State Tomography and Confidence Sequences for Matrix Means

Machine Learning

Summary

This paper looks at how to better learn the state of a quantum system step by step using a mathematical loss measure called logarithmic loss. The authors establish theoretical limits on how well any learning method can perform, showing that some known algorithms are as good as possible in terms of complexity related to quantum system size. From their analysis, they develop a nearly optimal method for estimating quantum states from measurements and an algorithm to create confidence intervals when averaging matrix-valued data. They also tested the confidence interval method and found it competitive with current approaches.

What this means in practice

  • For quantum computing engineers: Improve quantum state estimation accuracy with a new nearly optimal tomography algorithm based on regret limits.
  • For statistical data scientists: Use a novel algorithm for constructing confidence sequences when estimating means of matrix-valued random variables in complex data settings.

Authors

Yen-Huan Li, Wei-Fu Tseng

Abstract

We study the problem of online learning of quantum states with logarithmic loss, which is a quantum generalization of online portfolio selection and has applications to quantum state tomography, the task of estimating an unknown quantum state from measurement outcomes. We first establish a regret lower bound, showing that the regret rates of Exponential Weights and the algorithm of Tseng et al. (2025) have optimal polynomial dependence on the dimension. Our analysis leads to two further results. First, our lower-bound proof leads to a quantum state tomography algorithm that is sample optimal up to polylogarithmic factors in the sample size. Second, we develop an algorithm for constructing confidence sequences for the mean of a matrix-valued random variable. A numerical experiment shows that the latter algorithm is competitive with existing methods.
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