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.

翻译：量子状态 $\ psi 的稳定性值是最低值 $ 美元 = left\\\ psi \ rangle =\ sum \\ j= 1 r c_j\ left\ varphi_ j\ right\ rangle$ $ c_ j\ mathbb{ C} 美元 和 mathbb{ C} 美元 和 稳定值 $\ varphi_ j$ 。 量子电路的几种经典模拟方法运行时间由单子魔力状态 $n 的 10- sronor 的稳定性值确定。 我们看到, 美元 美元 = Omlegel2 值 = = rqrightr=n 这样的稳定值的较低值 值 。