A polynomial quantum algorithm for approximating the Jones polynomial

Dorit Aharonov*, Vaughan Jones, Zeph Landau

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

76 Scopus citations

Abstract

The Jones polynomial, discovered in 1984 [18], is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten [32]) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang [13, 14] provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, e2πi/5, and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results in [13, 14] are heavily based on TQFT, which makes the algorithm essentially inaccessible to computer scientists. We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an n strands braid with m crossings at any primitive root of unity e2πi/k, where the running time of the algorithm is polynomial in m, n and k. Our algorithm is based, rather than on TQFT, on well known mathematical results (specifically, the path model representation of the braid group and the uniqueness of the Markov trace for the Temperly Lieb algebra). By the results of [14], our algorithm solves a BQP complete problem. The algorithm we provide exhibits a structure which we hope is generalizable to other quantum algorithmic problems. Candidates of particular interest are the approximations of other downwards self-reducible #P-hard problems, most notably, the Potts model.

Original languageAmerican English
Title of host publicationSTOC'06
Subtitle of host publicationProceedings of the 38th Annual ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery
Pages427-436
Number of pages10
ISBN (Print)1595931341, 9781595931344
DOIs
StatePublished - 2006
Event38th Annual ACM Symposium on Theory of Computing, STOC'06 - Seattle, WA, United States
Duration: 21 May 200623 May 2006

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
Volume2006
ISSN (Print)0737-8017

Conference

Conference38th Annual ACM Symposium on Theory of Computing, STOC'06
Country/TerritoryUnited States
CitySeattle, WA
Period21/05/0623/05/06

Keywords

  • Algorithm
  • Approximation
  • Braids
  • Jones Polynomial
  • Knots
  • Quantum
  • Temperley-Lieb
  • Unitary Representation

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