植入团与量子对称自适应测量
Planted Cliques and Quantum Symmetry-Adapted Measurements
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中文总结 AI 辅助
本文研究植入团问题的两种量子编码，证明二进制相位态检测需要大量副本，而对称自适应测量在保留标签时能保持近完美可区分性，且单个量子样本可实现高效区分，揭示量子优势潜力。
中文摘要 AI 辅助
植入团问题是一个有望实现量子优势的候选问题，它具有广泛的计算-统计差距以及经典困难性的实质性证据。我们研究了经典样本的两种量子编码，即自然的二进制相位态编码和对称自适应测量，并确定它们是否保留了足够的信息用于植入团检测，同时讨论了它们在算法效率方面的潜力。对于二进制相位态编码，我们证明即使在任意联合测量下，常数优势检测也需要 Ω(n^{1+2ε} ln² n) 个副本。在对数团阈值之上，统计上足够的测量需要 Õ(n²) 个副本。全图寄存器上的对称自适应测量自然源于 Schur 变换。我们证明弱 Schur 采样的结果分布仅通过图的边数依赖于采样图，并且无法区分这些分布；而保留表示标签和 Specht 寄存器（丢弃重数后）则保留了距离 1-o(1)。即使丢弃标签，近完美可区分性仍然存在。我们计算了保留的状态，为高效测量提供了具体目标。最后，我们证明一个提供的相干量子样本能够实现高效的量子区分器，这产生了在量子植入团困难性下与一个经典样本的条件性计算分离。我们的结果是结构性和信息论性的；在推测的困难区间内，从单个经典图进行高效检测仍然是一个开放问题。
英文摘要
We study how quantum encodings and symmetry-adapted measurements preserve information for planted-clique detection from one classical graph. For $k=\lfloor n^{1/2-\varepsilon}\rfloor$, with fixed $0<\varepsilon<1/2$, detection is statistically possible but conjectured hard for polynomial-time classical algorithms. For a compact binary phase encoding, we prove that constant-advantage detection requires $Ω(n^{1+2\varepsilon}\ln^2 n)$ copies of the phase state of the same graph, even under arbitrary joint measurements. In the large-copy limit, the optimal decision rule thresholds the total number of $k$-cliques in the graph and its complement, but this characterization provides no efficient detector. We therefore explore measurements guided by the symmetries of the input distributions, starting with the efficient Schur transform on the full graph register. We show that weak Schur sampling, which measures only the representation label, depends only on edge count and has vanishing distinguishing power in this regime. When the label and multiplicity registers are discarded, the remaining quantum states are almost perfectly distinguishable. We show that the support of the planted state occupies only a vanishing fraction of the graph Hilbert space. Any subspace containing it still permits near-perfect detection if its relative dimension also vanishes. This gives us freedom to choose a subspace that is easier to measure. We propose exploring subgroup isotypic measurements to find such subspaces. Whether they can yield an efficient detector remains open. Finally, we show that a single supplied coherent quantum sample permits efficient detection, yielding a conditional computational separation from one classical sample under quantum planted-clique hardness.
发表机构
- IBM Research（IBM研究院）
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
