发表机构
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对量子影子层析成像问题，提出一种与维度无关、样本复杂度对可观测量数量呈多对数依赖的协议，回答了Aaronson的开放问题，且相比此前成果实现了指数级改进。
AI 中文摘要
影子层析成像是量子信息理论中的基础问题。给定未知d维量子态ρ的多个副本，以及已知的可观测量集合{E₁,…,Eₘ}，目标是以至少1−δ的概率，将所有期望值{Tr(ρEᵢ)}ᵢ₌₁ᵐ估计到加性精度ε。Aaronson在其具有开创性的影子层析成像工作（STOC'18）中提出了一个悬而未决的开放问题：该任务是否存在与维度无关的样本复杂度，且仅对m呈多对数依赖，正如最知名的下界所暗示的那样。在本研究中，我们提出了一种量子影子层析成像协议，其样本复杂度为O(1/ε²·(log(m/δ))⁴/(log log(m/δ))³)，该复杂度对可观测量数量呈多对数依赖，且与未知态的维度无关，从而回答了Aaronson的原始问题，同时相比Sinha（STOC'25）之前提出的最佳维度无关影子层析成像样本复杂度实现了指数级改进。我们的方法首先通过极小极大论证将通用影子层析成像问题简化为有限系综估计问题，随后开发了一种与可观测量无关的协议，该协议反复应用优良测量，并根据测量结果更新有限系综上的先验分布，对所得估计误差进行的精细尾部分析为所有可观测量提供了同时的精度保证。
英文摘要
Shadow Tomography is a fundamental problem in quantum information theory. Given multiple copies of an unknown $d$-dimensional quantum state $ρ$ and a known collection of observables $E_1,\ldots,E_M$, the goal is to estimate all expectation values $\{\text{Tr}(ρE_i)\}_{i=1}^M$ to additive accuracy $\varepsilon$ with probability at least $1-δ$. An elusive open question from the seminal shadow tomography work of Aaronson is whether this task admits a dimension-independent sample complexity with only polylogarithmic dependence on $M$, as suggested by the best-known lower bounds. In this work, we propose two different quantum protocols for shadow tomography with the best sample complexity \[ O\left( \frac{\log(M)\log(M/δ)}{\varepsilon^2} \right), \] which is polylogarithmic in the number of observables and independent of the dimension of the unknown state, thereby answering Aaronson's original question while also providing an exponential improvement in the prior best dimension independent sample complexity of shadow tomography from Sinha (STOC 2025) and, more recently, Chen, O'Donnell, Pelecanos, and Wright. Our approach first reduces the general shadow tomography problem to a finite-ensemble estimation problem via a minimax argument. We then develop an observable-independent protocol that repeatedly applies the pretty-good measurement while updating the prior distribution over the finite ensemble according to the measurement outcomes. A tail analysis of the resulting estimation error yields simultaneous accuracy guarantees for all observables and a cubic-logarithmic upper bound. We also introduce a refined recovery-label measurement for the same finite ensemble, which yields the bound in our main theorem.
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