发表机构
Pritzker School of Molecular Engineering, The University of Chicago; Department of Physics and Astronomy, Sejong University（芝加哥大学普里茨克分子工程学院; 世宗大学物理天文学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对有限并行访问场景，推导了量子信道转移矩阵/函数元素估计的样本复杂度界，确立了信道学习资源层级，为有限多副本访问下的态学习提供了更紧下界。
AI 中文摘要
量子信道可通过其在正交算子基上的作用来表征，这些算子与量子系统的可观测性质相关。对于qudit（d级量子位）和多模玻色系统，这分别编码在由Choi态估计得到的Heisenberg-Weyl转移矩阵，以及利用双模压缩真空态探测生成的Choi态估计得到的特征函数转移函数中。我们推导了在不同资源下，以成功概率≥1−δ将这些转移矩阵/函数的元素估计至加性精度ε所需的样本复杂度界：资源包括对复共轭信道E∗的访问和/或同时访问c个信道副本。在所有设置中，学习者使用并行信道调用，搭配自适应选择的、辅助系统协助的输入态和测量。在同时访问E和E∗的情况下，转移矩阵元素的绝对值可被高效学习，其紧缩放为ε−4；若无共轭访问，对于任意c<d的副本数，学习无法高效进行，所需样本复杂度随（d级）qudits数量n（当d为素数时）呈指数增长；当c=d时，效率得以恢复，紧缩放为ε−2d。对于玻色系统，当所有c=O(1/ε)时，样本复杂度仍呈指数增长。尽管该任务是学习特定态，但这些界比标准态学习界具有更强的含义，因为学习者可控制输入和辅助协助。这确立了信道学习资源的层级：自复共轭信道需要两副本辅助访问以实现高效学习，且对每个无平方因子的d，部分信道需要d副本访问。作为推论，我们为有限多副本访问下的态学习得出了更紧的下界。
英文摘要
Quantum channels can be characterized by their action on an orthogonal operator basis, where these operators are related to observable properties of the quantum system. For qudit and multimode bosonic systems, this is encoded respectively in the Heisenberg--Weyl transfer matrix estimated from the Choi-state, and in the characteristic-function transfer map estimated from a two-mode squeezed vacuum based Choi-state. We derive sample-complexity bounds for estimating entries of the transfer matrix/map to additive accuracy $ε$ with success probability $\ge1-δ$, under different resources: access to the complex-conjugate channel $\mathcal{E}^*$ and/or parallel access to $c$ copies. In all settings, the learner uses parallel channel calls with adaptively chosen, ancilla-assisted input states and measurements. Absolute values of transfer-matrix entries can be learned efficiently with simultaneous access to $\mathcal{E}$ and $\mathcal{E}^*$, with tight scaling $ε^{-4}$. Without conjugate access, any $c<d$ copies are insufficient for efficient learning, requiring sample complexity exponential in the number of ($d$-level) qudits $n$ (for prime $d$). Efficiency is recovered at $c=d$, with tight scaling $ε^{-2d}$. For bosonic systems, exponential sample complexity holds in terms of an effective dimension induced by an energy constraint for all $c=O(1/ε)$. Although the task is learning a particular state, these bounds carry stronger implications than standard state-learning bounds since the learner controls the inputs and has ancillary assistance. This establishes a hierarchy of channel-learning resources: self-complex-conjugate channels require two-copy ancilla-assisted access for efficient learning, while for every square-free $d$, some channels require $d$-copy access. As a corollary, we derive tighter lower bounds for state learning with limited multi-copy access.
Comments18+59 pages, 5 figures, 3 tables. v2 fixes various typos and inconsistencies and has better presentation
