一种用于估计量子态区分样本复杂度的混合时间方法

中文摘要 AI 辅助

我们发展了一种用于估计量子态区分样本复杂度的混合时间方法。首先考虑几何均匀纯态系综的最小误差区分问题，并证明其样本复杂度可由量子齐次混合时间[George等人，2026]和量子版本的广义Dobrushin系数[Wolfer，2020]给出紧致估计。当生成群G与生成态的稳定子群H构成Gelfand对时，该量子混合时间进一步退化为经典混合时间。在这种情况下，广义Dobrushin系数可以完全由交换Hecke代数End_G(C[G/H])的表示论量来表达。特别地，该方法将学习量子优惠券收集器态[Arunachalam等人，2020]和学习相位态的样本复杂度估计简化为经典混合时间问题。我们应用此框架回答了[Alrabiah等人，2026]中关于在F_q上学习次数为d的相位态以及[Arunachalam等人，2023]中关于在Z_q上学习广义布尔相位态的开问题。该框架也适用于超图态系综，给出完全由超图数据表达的估计，并恢复了[Montanaro和Shao，2022]中图态系综的估计。最后，我们将讨论扩展到具有均匀先验的任意混合态系综，证明了最小误差区分样本复杂度的一个夹逼界（由量子弱混合时间给出），并通过一个Dobrushin型系数为[D'Ariano等人，2005]中的极小极大区分样本复杂度提供了紧致估计。我们还讨论了强化数据处理不等式方法[Gao和Rouzé，2022]，并给出了用强化数据处理不等式常数表示的上界。

英文摘要

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.

发表机构

• University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
• Illinois Quantum Information Science and Technology (IQUIST) Center, University of Illinois Urbana-Champaign（伊利诺伊量子信息与科学技术（IQUIST）中心，伊利诺伊大学厄巴纳-香槟分校）

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