固定对易约束下的部分稳定子学习与无副本率折扣
Partial Stabilizer Learning under Fixed Commuting Constraints and the Absence of a Copy-Rate Discount
- The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
- International Quantum Academy（国际量子学院）
- Nagoya University（名古屋大学）
- Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究在固定对易约束下部分稳定子态的学习，证明当副本率低于1时恢复概率趋于零，而副本数超过qudit数时整个态可被识别，表明限制学习目标无副本率折扣。
AI中文摘要：
我们考虑一个未知的纯稳定子态，涉及n个具有固定素数局域维度p的qudit，以及预先给定的对易泡利可观测量。学习器不必学习整个态，而只需恢复与给定可观测面对易的互补稳定子及其本征值。一旦执行了规定的测量，这些信息和观测结果就决定了相应的条件稳定子态。我们将学习器可用的副本数k与qudit数n进行比较，并将其比值k/n称为副本率。假设所请求的互补稳定子子空间的维度m满足m=βn+o(n)，其中0<β≤1为固定值。如果副本率收敛到小于1的值，则最优精确恢复概率和验证分数在最坏情况下以及在均匀先验下都收敛到零。相反，如果副本数k减去qudit数n趋于正无穷，则整个稳定子态可以以趋于1的概率被识别，并且所请求的互补信息随后可以被提取。因此，将学习目标限制为规定测量后所需的稳定子信息并不会带来副本率的折扣。
英文摘要：
We consider an unknown pure stabilizer state of \(n\) qudits with a fixed prime local dimension \(p\), together with prescribed commuting Pauli observables. Rather than learning the whole state, the learner must recover only complementary stabilizers that commute with the prescribed observables, together with their eigenvalues. Once the prescribed measurement is performed, this information and the observed outcome determine the corresponding conditional stabilizer state. We compare the number \(k\) of copies available to the learner with the number \(n\) of qudits and refer to their ratio \(k/n\) as the copy rate. Suppose that the dimension \(m\) of the requested complementary stabilizer subspace satisfies \(m=βn+o(n)\) for a fixed \(0<β\leq1\). If the copy rate converges to a value below one, the optimal exact recovery probabilities and verification scores converge to zero, both in the worst case and under the uniform prior. Conversely, if the number \(k\) of copies minus the number \(n\) of qudits tends to positive infinity, the entire stabilizer state can be identified with probability tending to one, and the requested complementary information can then be extracted. Thus, restricting the learning target to the stabilizer information needed after the prescribed measurement yields no copy-rate discount.
