发表机构
David R. Cheriton School of Computer Science, University of Waterloo; Institute for Quantum Computing, University of Waterloo; Concordia University（滑铁卢大学大卫·R·切里顿计算机科学学院; 滑铁卢大学量子计算研究所; 康考迪亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明稳定子秩至多$k$的纯态其稳定子程度至多$2^{O(\sqrt{k\log(k+1)})}$，解决定量猜想，并给出张量幂近似稳定子秩的$\Omega(n^2/\log n)$下界及改进的层析算法。
AI 中文摘要
我们证明，稳定子秩至多为$k$的每个纯态，其稳定子程度至多为$2^{O(\sqrt{k\log(k+1)})}$，并为Clifford秩为$k$的算子的平方Clifford系数范数建立了类似的界。这意味着稳定子保真度至少为$2^{-O(\sqrt{k\log(k+1)})}$，解决了(Mehraban-Tamasbi, STOC, 2025)的定量猜想，并证明了任意非稳定子量子比特态的张量幂的近似稳定子秩的下界为$\Omega(n^2/\log n)$。后一结果将$T$态张量幂的最佳已知下界(Mehraban-Tamasbi, STOC, 2024)推广到任意非稳定子量子比特态，包括魔法态。作为Clifford秩-范数不等式的推论，我们获得了$n$比特AND函数由二次相位精确表示的$\Omega(n^2/\log n)$下界，改进了先前最佳线性界。进一步推论排除了具有近似稳定子秩和Clifford秩分别为$O((\log n)^2/\log\log n)$的伪随机态和酉系综，这比先前工作(Kalra-Sinha, Quantum, 2026)改进了$\log n$因子。我们还获得了稳定子秩至多为$k$的态层析算法，其时间和副本复杂度为$poly(n)2^{O(\sqrt{k\log(k+1)})}$，在指数上比最佳已知算法几乎平方根改进。
英文摘要
We prove that every pure state of stabilizer rank at most $k$ has stabilizer extent at most $2^{O(\sqrt{k\log(k+1)})}$, and establish the analogous bound for the squared Clifford coefficient norm of Clifford-rank-$k$ operators. This implies stabilizer fidelity at least $2^{-O(\sqrt{k\log(k+1)})}$, resolving the quantitative conjecture of (Mehraban-Tamasbi, STOC, 2025), and proves an $Ω(n^2/\log n)$ lower bound for the approximate stabilizer rank of tensor powers of any non-stabilizer qubit state. The latter result generalizes the best-known lower bound for tensor powers of $T$-states (Mehraban-Tamasbi, STOC, 2024) to arbitrary non-stabilizer qubit states, including magic states. As a consequence of the Clifford rank--norm inequality, we obtain an $Ω(n^2/\log n)$ lower bound for exact representations of $n$-bit AND function by quadratic phases, improving the previous best-known linear bound. Further consequences rule out pseudorandom state and unitary ensembles with approximate stabilizer and Clifford rank $O((\log n)^2/\log\log n)$, respectively, a $\log n$ improvement over prior work (Kalra-Sinha, Quantum, 2026). We also obtain tomography algorithms for states of stabilizer rank at most $k$, with $poly(n)2^{O(\sqrt{k\log(k+1)})}$ time and copy complexity, a nearly square-root improvement in the exponent over the best-known algorithm.
Comments31 pages
