$\mathsf{QMA}$ Lower Bounds for Approximate Counting

Abstract

We prove a query complexity lower bound for $\mathsf{QMA}$ protocols that solve approximate counting: estimating the size of a set given a membership oracle. This gives rise to an oracle $A$ such that ${\mathsf{SBP}}^A\not\subset {\mathsf{QMA}}^A$, resolving an open problem of Aaronson [2]. Our proof uses the polynomial method to derive a lower bound for the $\mathsf{SBQP}$ query complexity of the $\mathsf{AND}$ of two approximate counting instances. We use Laurent polynomials as a tool in our proof, showing that the "Laurent polynomial method" can be useful even for problems involving ordinary polynomials.

11 pages, 1 figure