arXiv 2106.032142021-06quant-phcs.CC

稳定子秩的下界

Lower Bounds on Stabilizer Rank

Shir Peleg, Amir Shpilka, Ben Lee Volk

发表信息：: Quantum 6, 652 (2022)
DOI：: 10.22331/q-2022-02-15-652

摘要：

量子态$ψ$的稳定子秩是最小$r$，使得对于$c_j \in \mathbb{C}$和稳定子态$φ_j$的$\left| ψ\right \rangle = \sum_{j=1}^r c_j \left|φ_j \right\rangle$。量子电路的几种经典模拟方法的运行时间取决于单量子比特幻态的$n$阶张量幂的稳定子秩。我们证明了$Ω(n)$的一个下界的稳定秩的状态，改进了以前的下限$Ω(\sqrt{n})$的Bravyi，Smith和Smolin（arXiv：1506.01396）。此外，我们证明了，对于一个足够小的常数$δ$，任何状态是$δ$ -接近这些状态的稳定秩是$Ω(\sqrt{n}/\log n)$。这是近似稳定器秩的第一个非平凡下界。我们的技术依赖于稳定状态的表示为二次函数的仿射子空间的$\mathbb{F}_2^n$，我们使用的工具，从分析布尔函数和复杂性理论。第一个结果的证明涉及二次多项式的方向导数的仔细分析，而第二个结果的证明使用Razborov-Smolensky低次多项式近似和相关界对多数函数。

英文摘要：

The stabilizer rank of a quantum state $ψ$ is the minimal $r$ such that $\left| ψ\right \rangle = \sum_{j=1}^r c_j \left|φ_j \right\rangle$ for $c_j \in \mathbb{C}$ and stabilizer states $φ_j$. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the $n$-th tensor power of single-qubit magic states. We prove a lower bound of $Ω(n)$ on the stabilizer rank of such states, improving a previous lower bound of $Ω(\sqrt{n})$ of Bravyi, Smith and Smolin (arXiv:1506.01396). Further, we prove that for a sufficiently small constant $δ$, the stabilizer rank of any state which is $δ$-close to those states is $Ω(\sqrt{n}/\log n)$. This is the first non-trivial lower bound for approximate stabilizer rank. Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of $\mathbb{F}_2^n$, and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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