Estimating Jones polynomials is a complete problem for one clean qubit
Peter W. Shor, Stephen P. Jordan · 2008
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[12, 3, 1]. 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 [20]. 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 One Clean Qubit The one clean qubit model of quantum computation originated as an idealized model of quantum computation on highly mixed initial states, such as appear in NMR implementations[20, 4]. In this model, one is given an initial quantum state consisting of a single qubit in the pure state |0〉, and n qubits in the maximally mixed