发表机构
Los Alamos National Laboratory; Massachusetts Institute of Technology; University of Warwick; Quantum Science Center（洛斯阿拉莫斯国家实验室; 麻省理工学院; 华威大学; 量子科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了稳定子态学习所需副本数的最优常数系数，提出多项式时间量子算法达到样本复杂度下界，并推广到Clifford酉学习，证明其最优性。
AI 中文摘要
众所周知，学习一个纯 $n$ 量子比特的稳定子态 $|\psi\rangle$ 既需要访问 $|\psi\rangle$ 的副本数量，也可以通过访问数量与 $n$ 成线性关系的副本完成。然而，这一缩放关系的精确常数系数似乎尚未被确定。在此，我们证明 $L_\delta(n)$，即量子过程能够以不超过 $0<\delta<1/8$ 的失败概率识别任何稳定子态所需的最小副本数，满足 $n+\lceil\log_2(1/\delta)\rceil-3\leq L_\delta(n)\leq n+\left\lceil\log_2(1/\delta)\right\rceil+4$。我们提出了一种多项式时间的量子学习算法，该算法达到此界限，在样本复杂度上相较于先前已知方法实现了常数因子的改进。作为直接推论，我们通过 Choi-Jamiolkowski 同构获得了一种算法，用于从 $2n+\left\lceil\log_2(1/\delta)\right\rceil+4$ 次查询中学习未知的 $n$ 量子比特 Clifford 酉算子，我们证明了其 $n$ 依赖性是最优的。我们的证明技术涉及阿贝尔群 $\mathbb{Z}_4^n \times \mathbb{F}_2^{n(n-1)/2}$ 上的傅里叶分析，似乎与先前稳定子态学习方法在性质上不同，可能具有独立的研究兴趣；特别是，它自然推广到量子学习理论中的进一步问题。
英文摘要
It is well-known that learning a pure $n$-qubit stabilizer state $|ψ\rangle$ both requires, and can be accomplished with, access to a number of copies of $|ψ\rangle$ linear in $n$. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that $L_δ(n)$, the smallest number of copies from which a quantum procedure can identify any stabilizer state with failure probability at most $0<δ<1/8$, satisfies $n+\lceil\log_2(1/δ)\rceil-3\leq L_δ(n)\leq n+\left\lceil\log_2(1/δ)\right\rceil+4$. We present a polynomial-time quantum learning algorithm that saturates this bound, achieving a constant factor improvement in sample-complexity over previously known approaches. As an immediate corollary, we obtain via the Choi-Jamiolkowski isomorphism an algorithm for learning an unknown $n$-qubit Clifford unitary from $2n+\left\lceil\log_2(1/δ)\right\rceil+4$ queries, the $n$-dependence of which we show to be optimal. Our proof technique, which involves Fourier analysis on the abelian group $\mathbb{Z}_4^n \times \mathbb{F}_2^{n(n-1)/2}$, seems to be qualitatively different to previous approaches to stabilizer state learning, and may be of some independent interest; in particular, it admits natural generalisations to further problems in quantum learning theory.
