未知量子态块编码的样本-查询互转换
Sample-Query Interconversion of Block Encoding of Unknown Quantum States
AI总结：
该研究分析未知量子态与其块编码酉信道的双向转换限制，证明实现ε近似块编码需Ω(1/ε)个态副本、恢复态需Ω((1/λ_max(ρ))√(d/r))次查询，还确定了基态制备等任务的下界。
AI中文摘要：
块编码将矩阵嵌入酉矩阵的子块中，是基于量子奇异值变换的量子算法的基本输入模型，可实现对酉算子编码矩阵的多项式变换。未知量子态的块编码对量子学习可能有用，但人们对未知量子态与其块编码酉信道之间的转换基本限制仍知之甚少。本文研究这两个方向的可转换性：首先，证明实现未知量子态的ε近似块编码酉信道需要Ω(1/ε)个该量子态的副本，与已知上界仅差对数因子；其次，证明在给定对秩为r、维度为d的量子态ρ的块编码酉信道的查询访问权限时，恢复该量子态通常需要Ω((1/λ_max(ρ))√(d/r))次查询，其中λ_max(ρ)是ρ的最大特征值，揭示了其对量子态维度的不可避免依赖。我们的结果明确了块编码作为未知量子态表示的固有局限，揭示了量子态的学习性质与生成该态本身之间的分离。利用我们的技术，进一步确定了特定态生成任务的下界，包括基态制备和吉布斯态制备。
英文摘要：
Block encoding embeds a matrix as a sub-block of a unitary matrix and serves as a fundamental input model for quantum algorithms based on quantum singular value transformation, enabling polynomial transformations of matrices encoded in unitary operators. Block encoding of unknown quantum states can be useful for quantum learning; however, the fundamental limits on converting between unknown quantum states and their block-encoding unitary channels remain poorly understood. In this paper, we investigate this convertibility in both directions. First, we prove that implementing an $\varepsilon$-approximate block-encoding unitary channel of an unknown quantum state requires $Ω(1/\varepsilon)$ copies of the state, matching known upper bounds up to logarithmic factors. Second, we show that recovering a rank-$r$, $d$-dimensional quantum state $ρ$ given query access to its block-encoding unitary channel generally requires $Ω((1/λ_{\max}(ρ))\sqrt{d/r})$ queries, where $λ_{\max}(ρ)$ is the maximum eigenvalue of $ρ$, revealing an unavoidable dependence on the dimension of the state. Our results identify inherent limitations of block encoding as a representation of unknown quantum states and reveal a separation between learning properties of a quantum state and generating the state itself. Using our techniques, we further establish lower bounds for specific state-generation tasks, including ground-state preparation and Gibbs-state preparation.
