Lower Bounds on Stabilizer Rank

The stabilizer rank of a quantum state psi is the minimal r such that vertical bar psi > = Sigma(r)(j=1) c(j) vertical bar phi(j)> for c(j) is an element of C and stabilizer states phi(j). The running time of several classical simulation methods for quantum circuits is determined by the stabi-lizer rank of the n-th tensor power of single-qubit magic states. We prove a lower bound of Omega(n) on the stabilizer rank of such states, im-proving a previous lower bound of Omega(root n) of Bravyi, Smith and Smolin [7]. Further, we prove that for a sufficiently small constant delta, the stabilizer rank of any state which is delta-close to those states is Omega(root 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 approxi-mations and correlation bounds against the majority function.