2023/05/17 by Mehraban, Saeed, Tahmasbi, Mehrdad
#Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Physical sciences #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2305.10277
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⟩⊗ 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⟩⊗ 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 Ω(√ n) has been known. In this paper, we improve the lower bound on the approximate rank to Ω(n2) 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 \mathbbF2, 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].