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It is known that evaluating a certain approximation to the Jones polynomialfor the plat closure of a braid is a BQP-complete problem. That is, thisproblem exactly captures the power of the quantum circuit model. The one cleanqubit model is a model of quantum computation in which all but one qubit startsin the maximally mixed state. One clean qubit computers are believed to bestrictly weaker than standard quantum computers, but still capable of solvingsome classically intractable problems. Here we show that evaluating a certainapproximation to the Jones polynomial at a fifth root of unity for the traceclosure of a braid is a complete problem for the one clean qubit complexityclass. That is, a one clean qubit computer can approximate these Jonespolynomials in time polynomial in both the number of strands and number ofcrossings, and the problem of simulating a one clean qubit computer isreducible to approximating the Jones polynomial of the trace closure of abraid.

Estimating Jones polynomials is a complete problem for one clean qubit