New techniques for bounding stabilizer rank

TL;DR

This paper introduces new number-theoretic and algebraic-geometric methods to bound the stabilizer rank of quantum states, providing explicit sequences, lower bounds, and novel examples with multiplicative properties.

Contribution

It refines existing theorems, offers simplified proofs of bounds, and presents the first non-trivial states with multiplicative stabilizer rank, advancing understanding of quantum state complexity.

Findings

Explicit sequence of states with exponential stabilizer rank but constant approximate stabilizer rank

Simplified proofs of known asymptotic lower bounds on stabilizer rank

First examples of states with multiplicative stabilizer rank under tensor products

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 to 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 introduce and study the generic stabilizer rank using algebraic-geometric techniques.