The approximate stabilizer rank of a quantum state is the minimum number of terms in any approximate decomposition of that state into stabilizer states. Bravyi and Gosset showed that the approximate stabilizer rank of a so-called "magic" state like $|T\rangle^{\otimes n}$, up to polynomial factors, is an upper bound on the number of classical operations required to simulate an arbitrary quantum circuit with Clifford gates and $n$ number of $T$ gates. As a result, an exponential lower bound on this quantity seems inevitable. Despite this intuition, several attempts using various techniques could not lead to a better than a linear lower bound on the "exact" rank of ${|T\rangle}^{\otimes n}$, meaning the minimal size of a decomposition that exactly produces the state. For the "approximate" rank, which is more realistically related to the cost of simulating quantum circuits, no lower bound better than $\tilde \Omega(\sqrt n)$ has been known. In this paper, we improve the lower bound on the approximate rank to $\tilde \Omega (n^2)$ for a wide range of the approximation parameters. An immediate corollary of our result is the existence of polynomial time computable functions which require a super-linear number of terms in any decomposition into exponentials of quadratic forms over $\mathbb{F}_2$, resolving a question in [Wil18]. Our approach is based on a strong lower bound on the approximate rank of a quantum state sampled from the Haar measure, a step-by-step analysis of the approximate rank of a magic-state teleportation protocol to sample from the Haar measure, and a result about trading Clifford operations with $T$ gates by [LKS18].

翻译：量子态的近似稳定子秩是指将该态近似分解为稳定子态所需的最小项数。Bravyi 和 Gosset 证明，对于诸如 $|T\rangle^{\otimes n}$ 的所谓“魔法”态，其近似稳定子秩（在多项式因子范围内）是使用 Clifford 门和 $n$ 个 $T$ 门模拟任意量子电路所需经典操作数量的上界。因此，该量似乎必然存在指数级下界。然而，尽管有这种直觉，多种尝试使用不同技术未能得到优于线性下界的结果，即 ${|T\rangle}^{\otimes n}$ 的“精确”秩（产生该态的确切分解的最小规模）。对于与模拟量子电路成本更实际相关的“近似”秩，此前已知的最佳下界不超过 $\tilde \Omega(\sqrt n)$。在本文中，我们针对广泛范围的近似参数，将近似秩的下界提升至 $\tilde \Omega (n^2)$。我们的结果的一个直接推论是存在多项式时间可计算函数，其任何分解为 $\mathbb{F}_2$ 上二次型指数形式的项数都需超线性增长，从而解决了 [Wil18] 中的一个问题。我们的方法基于从 Haar 测度采样的量子态的近似秩的强下界、从 Haar 测度采样的魔法态隐形传态协议的近似秩的逐步分析，以及 [LKS18] 中关于用 $T$ 门交换 Clifford 操作的结果。