含噪声观测下结构化量子态层析的样本复杂度统一框架
A Unified Framework for Sample Complexity of Structured Quantum State Tomography under Noisy Observations
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中文总结 AI 辅助
本文提出含噪声观测下结构化量子态层析的统一理论框架，推导两类约束最小二乘估计器的非渐近样本复杂度保证，明确刻画样本复杂度的相关依赖关系。
中文摘要 AI 辅助
量子态层析（QST）因在量子信息处理中的基础作用而受到广泛关注。本文针对由态制备噪声、测量噪声及有限次统计射击噪声引起的含噪声观测下的结构化QST，开发了一套用于分析其样本复杂度的统一理论框架。该框架适用于广泛的结构化量子态类别，包括一般混合态、稀疏态、低秩态、矩阵乘积态（MPSs）、矩阵乘积算子（MPOs）、投影纠缠对态（PEPSs）和投影纠缠对算子（PEPOs），同时进一步引入了物理一致性结构化模型——包括低秩与稀疏态、低秩MPOs（LR-MPOs）和低秩PEPOs（LR-PEPOs）——这些模型同时利用低维结构并保留物理约束。在该框架内，我们针对含噪声观测下的两类约束最小二乘估计器推导了统一的非渐近样本复杂度保证：一类是纳入校准噪声模型的噪声感知估计器，另一类是基于理想玻恩测量模型的噪声非感知估计器。对于噪声感知估计器，我们推导了统一的迹范数恢复保证，明确刻画了样本复杂度对三个基本量的依赖关系：底层结构化态类的复杂度、测量系综的复杂度，以及态制备与测量噪声水平。对于噪声非感知估计器，我们建立了统一的非渐近恢复保证，该保证由统计误差项和额外的确定性偏差项组成，偏差项源于假设的重建模型与含噪声观测过程之间的不匹配。
英文摘要
Quantum state tomography (QST) has attracted considerable attention due to its fundamental role in quantum information processing. In this paper, we develop a unified theoretical framework for analyzing the sample complexity of structured QST under noisy observations arising from state preparation noise, measurement noise, and finite-shot statistical noise. The proposed framework applies to a broad family of structured quantum-state classes, including general mixed states, sparse states, low-rank states, matrix product states (MPSs), matrix product operators (MPOs), projected entangled-pair states (PEPSs), and projected entangled-pair operators (PEPOs), while further introducing physically consistent structured models---including low-rank and sparse states, low-rank MPOs (LR-MPOs), and low-rank PEPOs (LR-PEPOs)---that simultaneously exploit low-dimensional structures and preserve the physical constraint. Within this framework, we derive unified non-asymptotic sample complexity guarantees for two constrained least-squares estimators under noisy observations: a noise-aware estimator that incorporates the calibrated noise model and a noise-unaware estimator based on the ideal Born measurement model. For the noise-aware estimator, we derive unified trace-norm recovery guarantees that explicitly characterize the dependence of the sample complexity on three fundamental quantities: the complexity of the underlying structured state class, the complexity of the measurement ensemble, and the state preparation and measurement noise levels. For the noise-unaware estimator, we establish a unified non-asymptotic recovery guarantee consisting of a statistical error term and an additional deterministic bias term arising from the mismatch between the assumed reconstruction model and the noisy observation process.
