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. However, an "approximate" rank is more realistically related to the cost of simulating quantum circuits because exact rank is not robust to errors; there are quantum states with exponentially large exact ranks but constant approximate ranks even with arbitrarily small approximation parameters. No lower bound better than $\tilde \Omega(\sqrt n)$ has been known for the approximate rank. In this paper, we improve this lower bound to $\tilde \Omega (n)$ for a wide range of the approximation parameters. Our approach is based on a strong lower bound on the approximate rank of a quantum state sampled from the Haar measure and a step-by-step analysis of the approximate rank of a magic-state teleportation protocol to sample from the Haar measure.

翻译：量子态的近似稳定子秩是指该态分解为稳定子态时近似分解中项数的最小值。Bravyi和Gosset证明，所谓的“魔法”态（如 $|T\rangle^{\otimes n}$）的近似稳定子秩（最高多项式因子内）是使用Clifford门和 $n$ 个 $T$ 门模拟任意量子电路所需经典操作次数的上界。因此，该量呈指数下界似乎不可避免。然而，尽管有这种直觉，使用多种技术尝试后，仍未得到 $|T\rangle^{\otimes n}$ 的“精确”秩优于线性下界的结果（精确秩是指恰好产生该态的最小分解规模）。然而，“近似”秩更实际地关联于模拟量子电路的成本，因为精确秩对误差不鲁棒：存在一些量子态，其精确秩呈指数大，但即使采用任意小的近似参数，近似秩仍为常数。此前，对于近似秩未已知优于 $\tilde \Omega(\sqrt n)$ 的下界。在本文中，我们将此下界改进为 $\tilde \Omega (n)$，适用于广泛的近似参数。我们的方法基于对从Haar测度中采样的量子态近似秩的强下界，以及通过魔法态远程传输协议从Haar测度采样时近似秩的逐步分析。