Lower Bounds on Stabilizer Rank

Abstract

The stabilizer rank of a quantum state ψ is the minimal r such that |ψ⟩ = ∑_{j = 1}^r c_j |φ_j⟩ for c_j ∈ ℂ and stabilizer states φ_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 Ω(n) on the stabilizer rank of such states, improving a previous lower bound of Ω(√n) of Bravyi, Smith and Smolin [Bravyi et al., 2016]. Further, we prove that for a sufficiently small constant δ, the stabilizer rank of any state which is δ-close to those states is Ω(√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 𝔽₂ⁿ, 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.