New techniques for bounding stabilizer rank

Abstract

In this work, we present number-Theoretic and algebraic-geometric techniques for bounding the stabilizer rank of quantum states. First, we refine a number-Theoretic theorem of Moulton to exhibit an explicit sequence of product states with exponential stabilizer rank but constant approximate stabilizer rank, and provide alternate (and simplified) proofs of the best-known asymptotic lower bounds on stabilizer rank and approximate stabilizer rank, up to a log factor. Second, we find the first non-Trivial examples of quantum states with multiplicative stabilizer rank under the tensor product. Third, we introduceand study the generic stabilizer rank using algebraic-geometric techniques.