Lower Bounds on Stabilizer Rank
share
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 (arXiv:1506.01396). 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 𝔽_2^n, 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.
READ FULL TEXT