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.

1 aShor, Peter, W.1 aJordan, Stephen, P. uhttp://dl.acm.org/citation.cfm?id=2017011.201701201177nas a2200145 4500008004100000245006100041210006100102260001500163490000700178520074000185100002400925700001800949700002000967856004400987 2006 eng d00aError correcting codes for adiabatic quantum computation0 aError correcting codes for adiabatic quantum computation c2006/11/140 v743 a Recently, there has been growing interest in using adiabatic quantum computation as an architecture for experimentally realizable quantum computers. One of the reasons for this is the idea that the energy gap should provide some inherent resistance to noise. It is now known that universal quantum computation can be achieved adiabatically using 2-local Hamiltonians. The energy gap in these Hamiltonians scales as an inverse polynomial in the problem size. Here we present stabilizer codes which can be used to produce a constant energy gap against 1-local and 2-local noise. The corresponding fault-tolerant universal Hamiltonians are 4-local and 6-local respectively, which is the optimal result achievable within this framework. 1 aJordan, Stephen, P.1 aFarhi, Edward1 aShor, Peter, W. uhttp://arxiv.org/abs/quant-ph/0512170v3