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

Abstract

We prove a query complexity lower bound for 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 SBP A ⊂ QMA A , resolving an open problem of Aaronson [2]. Our proof uses the polynomial method to derive a lower bound for the SBQP query complexity of the 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.