发表机构
School of Statistics, University of Minnesota（明尼苏达大学统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文研究量子态层析的极小极大风险，提出周期 Clifford 测量下的自适应估计器，在多项式谱衰减类上达到最优速率，并给出高效算法与形式化验证。
AI 中文摘要
量子态层析为表征态制备和预测测量结果提供了基础。我们刻画了在随机非自适应单拷贝测量下，具有多项式谱衰减的类别上量子态层析的极小极大期望迹范数风险。在两层周期 Clifford 系综的显式充分块大小条件下，我们建立了算子范数最小距离和测量加权投影最小二乘的逐态期望迹范数预言不等式，其界限依赖于每个态的实际谱尾，而估计器不需要事先知道态的 structural 特征，包括其秩、特征基或谱结构。在这些多项式谱衰减类别上，两个估计器达到常数因子内的极小极大期望迹范数速率。作为直接推论，相同的估计器在每个固定秩约束类别上达到极小极大速率，同时对态的未知结构特征保持适应性。对于可容许的对数块大小，周期 Clifford 测量以对数基本门深度达到这些保证。主要技术成分是周期 Clifford 系综投影仪第四矩的一致界。该界在维度上是尖锐的，并适用于任意测试态，包括跨块纠缠的态。协方差集中性和正性产生自适应预言界，而谱尾填充上的共享旋转信息论证建立匹配的下界。我们还提供了多项式时间重建算法、数值比较以及所有标记理论结果的 Lean 形式化。
英文摘要
Quantum state tomography provides a foundation for characterizing state preparation and predicting measurement outcomes. We characterize the minimax expected trace norm risk of quantum state tomography over classes with polynomial spectral decay under randomized nonadaptive single copy measurements. Under an explicit sufficient block size condition for the two layer periodic Clifford ensemble, we establish statewise expected trace norm oracle inequalities for operator norm minimum distance and measurement weighted projected least squares, with bounds that depend on each state's actual spectral tail, while the estimators require no prior knowledge of the state's structural characteristics, including its rank, eigenbasis, or spectral structure. Over these polynomial spectral decay classes, both estimators attain the minimax expected trace norm rates up to constants. As a direct consequence, the same estimators attain the minimax rate over every fixed rank constrained class, while remaining adaptive to the unknown structural characteristics of the state. For admissible logarithmic block sizes, the periodic Clifford measurements attain these guarantees with logarithmic elementary gate depth. The main technical ingredient is a uniform bound on the fourth moment of projectors for the periodic Clifford ensemble. The bound is sharp in dimension and holds for arbitrary test states, including those entangled across blocks. Covariance concentration and positivity yield adaptive oracle bounds, while a shared rotation information argument on spectral tail packings establishes matching lower bounds. We also provide polynomial time reconstruction algorithms, numerical comparisons, and Lean formalizations of all labeled theoretical results.
Comments39 pages, 3 figures
