dorsal/arxiv
View SchemaThe Jones polynomial: quantum algorithms and applications in quantum complexity theory
| Authors | Pawel Wocjan, Jon Yard |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0603069 |
| URL | https://arxiv.org/abs/quant-ph/0603069 |
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-complete 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.
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"abstract": "We analyze relationships between quantum computation and a family of\ngeneralizations of the Jones polynomial. Extending recent work by Aharonov et\nal., we give efficient quantum circuits for implementing the unitary\nJones-Wenzl representations of the braid group. We use these to provide new\nquantum algorithms for approximately evaluating a family of specializations of\nthe HOMFLYPT two-variable polynomial of trace closures of braids. We also give\nalgorithms for approximating the Jones polynomial of a general class of\nclosures of braids at roots of unity. Next we provide a self-contained proof of\na result of Freedman et al. that any quantum computation can be replaced by an\nadditive approximation of the Jones polynomial, evaluated at almost any\nprimitive root of unity. Our proof encodes two-qubit unitaries into the\nrectangular representation of the eight-strand braid group. We then give\nQCMA-complete and PSPACE-complete problems which are based on braids. We\nconclude with direct proofs that evaluating the Jones polynomial of the plat\nclosure at most primitive roots of unity is a #P-hard problem, while learning\nits most significant bit is PP-hard, circumventing the usual route through the\nTutte polynomial and graph coloring.",
"arxiv_id": "quant-ph/0603069",
"authors": [
"Pawel Wocjan",
"Jon Yard"
],
"categories": [
"quant-ph",
"math.GT"
],
"title": "The Jones polynomial: quantum algorithms and applications in quantum complexity theory",
"url": "https://arxiv.org/abs/quant-ph/0603069"
},
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