AI 中文总结
本文从信息论和量子物理角度建立量子压缩感知的测量复杂度下界，揭示量子并行性可降低测量次数，为量子测量科学提供理论基础。
AI 中文摘要
传统压缩感知（CS）在非自适应测量下具有测量下界 M = Ω(K log(N/K))。近期关于量子压缩感知（QCS）的实验报告了低于该经典下界的测量次数。在本工作中，我们从信息论和量子物理的角度建立了 QCS 测量复杂度的下界。QCS 利用量子并行性，使得一个酉域对齐演化能够作用于所有 N 个基态的叠加，并将稀疏基以一对一的方式映射到测量基上。因此，非零分量的位置被显式地携带在测量索引标签中。如果仅需要关于 K 个非零分量的基本统计信息，有效索引样本的总数满足粗略必要条件 Ms = Ω(K)。在独立的单索引投影测量下，当支撑集未知时，非零概率满足 pn = Θ(1/K)，并且需要以固定成功概率进行精确支撑恢复，采样覆盖要求导致 Ms = Θ(K ln K)。如果进一步要求每个非零幅度系数达到坐标方向的相对均方根误差 ε，最优有效索引样本复杂度为 Ms = Θ(K ln K + K/ε²)，对于固定的 ε，这简化为 Θ(K ln K)。多模光子数分辨探测可以减少实验轮数，但不能减少总的有效索引样本复杂度。这些结果表明，量子并行性不仅是量子计算的核心资源，也可以在测量科学中发挥变革性作用。
英文摘要
Conventional compressed sensing (CS) has a measurement lower bound of M = Ω(K log(N/K)) under non-adaptive measurements. Recent experiments on quantum compressed sensing (QCS) have reported numbers of measurements below this classical lower bound. In this work, we establish lower bounds on the measurement complexity of QCS from an information-theoretic and quantum-physical perspective. QCS exploits quantum parallelism, enabling a unitary domain-alignment evolution to act on a superposition of all N basis states and map the sparse basis onto the measurement basis in a one-to-one manner. Consequently, the locations of nonzero components are explicitly carried by measurement-index labels. If only basic statistical information about the K nonzero components is required, the total number of effective index samples satisfies the coarse necessary condition Ms = Ω(K). Under independent single-index projective measurements, when the support is unknown, the nonzero probabilities satisfy pn = Θ(1/K), and exact support recovery is required with a fixed success probability, the sampling-coverage requirement leads to Ms = Θ(K ln K). If each nonzero amplitude coefficient is further required to achieve a coordinate-wise relative root-mean-square error ε, the optimal effective index sample complexity is Ms = Θ(K ln K + K/ε2), which reduces to Θ(K ln K) for fixed ε. Multimode photon-number-resolving detection can reduce the number of experimental rounds but not the total effective index sample complexity. These results show that quantum parallelism is not only a core resource for quantum computing but can also play a transformative role in measurement science.
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