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

## TL;DR

This paper establishes a lower bound on the query complexity of quantum Merlin-Arthur protocols for approximate counting, resolving an open problem and introducing Laurent polynomial techniques.

## Contribution

It introduces a new Laurent polynomial method to derive quantum lower bounds for approximate counting problems, connecting query complexity with complexity class separations.

## Key findings

- Proves a query complexity lower bound for QMA protocols in approximate counting.
- Constructs an oracle separating SBP and QMA, resolving an open problem.
- Demonstrates the utility of Laurent polynomials in quantum complexity lower bounds.

## 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.