Quadratic Lower bounds on the Approximate Stabilizer Rank: A Probabilistic Approach
Saeed Mehraban, Mehrdad Tahmasbi
TL;DR
The paper investigates the hardness of classically simulating Clifford+$T$ circuits through the lens of approximate stabilizer rank χ_δ. It develops a probabilistic, multi-step approach: (i) a strong concentration bound for χ_δ on Haar-random states, (ii) a reduction showing Haar-sampling can be achieved using T-state teleportation with m ∼ n 2^{n/2} T gates, and (iii) a stability argument that adaptive measurements do not inflate χ_δ for the targeted magic-state structure. Leveraging these steps, the authors prove a nearly quadratic lower bound χ_δ(|T>^{⊗ m}) = Ω(m^2 / polylog m) for a broad range of δ, with δ=0 extending to all Clifford-equivalent magic states. They also connect this bound to circuit complexity, conditional lower bounds on sampling hardness, and implications for the polynomial hierarchy under plausible average-case assumptions, while analyzing the behavior for t-designs. The results illuminate the non-Clifford resources necessary for stabilizer decompositions and offer a framework linking quantum circuit simulation costs to fundamental complexity-theoretic consequences. Overall, the work provides a substantial advance in understanding the limits of stabilizer-based classical simulation and the intrinsic cost of approximating magic states.
Abstract
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 $\tilde Ω(\sqrt n)$ has been known. In this paper, we improve the lower bound on the approximate rank to $\tilde Ω(n^2)$ 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].