Efficient quantum circuits for approximating the Jones polynomial

Authors:
Yumi Nakajima;Yasuhito Kawano;Hiroshi Sekigawa
Affiliations:
NTT Communication Science Laboratories, Nippon Telegraph and Telephone Corporation, Atsugi, Kanagawa, Japan;NTT Communication Science Laboratories, Nippon Telegraph and Telephone Corporation, Atsugi, Kanagawa, Japan;NTT Communication Science Laboratories, Nippon Telegraph and Telephone Corporation, Atsugi, Kanagawa, Japan
Venue:
Quantum Information & Computation
Year:
2008

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Abstract

Freedman, Kitaev, and Wang proved the equivalence between quantum field theory and quantum computation, and consequently showed that the problem of approximating the Jones polynomial (a knot invariant) at the fifth root of unity is BQP-complete. Recently, Aharonov, Jones, and Landau proposed a concrete quantum algorithm, called the AJL algorithm, that approximates the Jones polynomial at the kth root of unity in polynomial time. In this paper, we propose a new method for implementing the AJL algorithm, which improves the performance from O(mnlog2 k) to O(mn). Here, n is the number of strands, m is the number of the crossings in a braid. Since, in the AJL algorithm, m and k are assumed to be given as polynomials in n, the difference in the performance between the original implementation and our design is significant if k is a large-degree polynomial.