  The Jones and HOMFLY polynomials are link invariants with close connections
to quantum computing. It was recently shown that finding a certain
approximation to the Jones polynomial of the trace closure of a braid at the
fifth root of unity is a complete problem for the one clean qubit complexity
class. This is the class of problems solvable in polynomial time on a quantum
computer acting on an initial state in which one qubit is pure and the rest are
maximally mixed. Here we generalize this result by showing that one clean qubit
computers can efficiently approximate the Jones and single-variable HOMFLY
polynomials of the trace closure of a braid at any root of unity.
