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The Jones and HOMFLY polynomials are link invariants with close connections to quan-tum computing. It was recently shown that finding a certain approximation to the Jonespolynomial of the trace closure of a braid at the fifth root of unity is a complete problemfor the one clean qubit complexity class[18]. This is the class of problems solvable inpolynomial time on a quantum computer acting on an initial state in which one qubitis pure and the rest are maximally mixed. Here we generalize this result by showingthat one clean qubit computers can efficiently approximate the Jones and single-variableHOMFLY polynomials of the trace closure of a braid at any root of unity.

Estimating Jones and HOMFLY polynomials with one clean qubit