The stabilizer rank of a quantum state $\psi$ is the minimal $r$ such that $\left| \psi \right \rangle = \sum_{j=1}^r c_j \left|\varphi_j \right\rangle$ for $c_j \in \mathbb{C}$ and stabilizer states $\varphi_j$. 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 $\Omega(n)$ on the stabilizer rank of such states, improving a previous lower bound of $\Omega(\sqrt{n})$ of Bravyi, Smith and Smolin (arXiv:1506.01396). Further, we prove that for a sufficiently small constant $\delta$, the stabilizer rank of any state which is $\delta$-close to those states is $\Omega(\sqrt{n}/\log n)$. 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.

中文翻译：

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稳定器等级的下限

量子态 $\psi$ 的稳定器等级是最小的 $r$，使得 $\left| \psi \right \rangle = \sum_{j=1}^r c_j \left|\varphi_j \right\rangle$ 用于 $c_j \in \mathbb{C}$ 和稳定器状态 $\varphi_j$。几种经典的量子电路模拟方法的运行时间由单量子位魔态的$n$-th张量幂的稳定器等级决定。我们证明了这些状态的稳定器等级上的 $\Omega(n)$ 下限，改进了 Bravyi、Smith 和 Smolin (arXiv:1506.01396) 之前的 $\Omega(\sqrt{n})$ 下限。此外，我们证明对于足够小的常数 $\delta$，任何 $\delta$-接近这些状态的状态的稳定器等级是 $\Omega(\sqrt{n}/\log n)$。这是近似稳定器等级的第一个非平凡下限。我们的技术依赖于将稳定器状态表示为 $\mathbb{F}_2^n$ 仿射子空间上的二次函数，并且我们使用来自布尔函数和复杂性理论的分析工具。第一个结果的证明涉及对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 低次多项式近似和针对多数函数的相关边界。