The Jones polynomial: Quantum algorithms and applications in quantum complexity theory

    Authors

    P. Wocjan;J. Yard

    Comments

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    Abbreviated Journal Title

    Quantum Inform. Comput.

    Keywords

    quantum algorithms; quantum complexity theory; topological quantum; computation; BRAID-GROUPS; COMPUTATION; SUBFACTORS; Computer Science, Theory & Methods; Physics, Particles & Fields; Physics, Mathematical

    Abstract

    We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-cornplete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

    Journal Title

    Quantum Information & Computation

    Volume

    8

    Issue/Number

    1-2

    Publication Date

    1-1-2008

    Document Type

    Article

    Language

    English

    First Page

    147

    Last Page

    180

    WOS Identifier

    WOS:000251978500010

    ISSN

    1533-7146

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