Approximate Counting and Quantum Computation

TL;DR

This paper explores the relationship between approximate counting, quantum computation, and complexity classes, introducing an additive approximation scheme for functions in BQP, #P, and GapP, and discusses open problems in the area.

Contribution

It introduces a new additive approximation method for simulating BQP functions and relates it to #P and GapP classes, expanding understanding of quantum and classical complexity.

Findings

All #P and GapP functions have an additive approximation scheme under certain normalizations.

The approximation scheme can simulate functions in BQP.

Open problems remain regarding specific functions like the Jones polynomial evaluation.

Abstract

Motivated by the result that an `approximate' evaluation of the Jones polynomial of a braid at a $5^{th}$ root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP to the counting class GapP, we introduce a form of additive approximation which can be used to simulate a function in BQP. We show that all functions in the classes #P and GapP have such an approximation scheme under certain natural normalisations. However we are unable to determine whether the particular functions we are motivated by, such as the above evaluation of the Jones polynomial, can be approximated in this way. We close with some open problems motivated by this work.