Estimating Jones polynomials is a complete problem for one clean qubit

TL;DR

This paper demonstrates that approximating certain Jones polynomials is a complete problem for the one clean qubit quantum computing model, linking quantum complexity to topological invariants.

Contribution

It establishes that evaluating a specific Jones polynomial approximation is both solvable by and reducible to one clean qubit quantum computation, showing its computational completeness.

Findings

One clean qubit computers can approximate Jones polynomials efficiently.

The problem of simulating one clean qubit computers reduces to Jones polynomial approximation.

Jones polynomial evaluation is a complete problem for the one clean qubit complexity class.

Abstract

It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard quantum computers, but still capable of solving some classically intractable problems. Here we show that evaluating a certain approximation to the Jones polynomial at a fifth root of unity for the trace closure of a braid is a complete problem for the one clean qubit complexity class. That is, a one clean qubit computer can approximate these Jones polynomials in time polynomial in both the number of strands and number of crossings, and the problem of simulating a one clean qubit computer is reducible to approximating the Jones polynomial of the trace closure of a braid.