A mixing time method for estimating the sample complexity of quantum state discrimination

2026-09-30

AI Summary

This study addresses the theoretical bottleneck in estimating the sample complexity of quantum state discrimination, particularly for geometrically uniform ensembles of pure and mixed states. To overcome this challenge, the work proposes a unified analytical framework based on mixing times, which reduces quantum state discrimination to the computation of classical or quantum mixing times. By introducing generalized Dobrushin coefficients alongside representation-theoretic techniques involving Hecke algebras, and by leveraging Gelfand pairs to simplify computations, the authors establish a systematic approach for deriving tight bounds. As a result, this research precisely characterizes the sample complexity in key scenarios, including phase states and hypergraph states. Ultimately, it provides tight bounds and a unified solution to several open problems in quantum state learning.

Abstract

We develop a mixing time method for estimating the sample complexity of quantum state discrimination. We start with considering the minimum-error discrimination of geometrically uniform pure state ensembles, and prove that its sample complexity has a tight estimate given by a quantum homogeneous mixing time [George et al., 2026] and a quantum version of the generalized Dobrushin coefficient [Wolfer, 2020]. This quantum mixing time further reduces to a classical one when the generating group $G$ forms a Gelfand pair with the stabilizer subgroup $H$ of the generator state. In this case the generalized Dobrushin coefficient can be fully expressed by representation-theoretic quantities of the commutative Hecke algebra $\operatorname{End}_G(\mathbb C[G/H])$. In particular, this method reduces the sample complexity estimation of learning quantum coupon collector states [Arunachalam et al., 2020] and learning phase states to classical mixing time problems. We apply this framework to answer the open problems of learning degree-$d$ phase states over $\mathbb F_q$ in [Alrabiah et al., 2026] and generalized Boolean phase states over $\mathbb Z_q$ [Arunachalam et al., 2023]. The framework also applies to hypergraph state ensembles, giving estimates expressed fully in terms of hypergraph data and recovering estimates for graph state ensembles in [Montanaro and Shao, 2022]. Finally, we extend the discussion to arbitrary mixed state ensembles with uniform priors, prove a sandwiched bound for minimum-error discrimination sample complexity by a quantum weakly mixing time, and provide a tight estimate for the minimax discrimination sample complexity from [D'Ariano et al., 2005] by a Dobrushin-type coefficient. We also discuss the method of strengthened data processing inequality [Gao and Rouz{é}, 2022] and give an upper bound in terms of a strengthened data processing inequality constant.

Problem

• quantum state discrimination
• sample complexity
• mixing time
• phase states
• minimum-error discrimination

Innovation

• Quantum state discrimination
• Mixing time
• Sample complexity
• Dobrushin coefficient
• Representation theory