How hard is it to approximate the Jones polynomial?
Abstract
Freedman, Kitaev, and Wang [arXiv:quant-ph/0001071], and later Aharonov, Jones, and Landau [arXiv:quant-ph/0511096], established a quantum algorithm to "additively" approximate the Jones polynomial V(L,t) at any principal root of unity t. The strength of this additive approximation depends exponentially on the bridge number of the link presentation. Freedman, Larsen, and Wang [arXiv:math/0103200] established that the approximation is universal for quantum computation at a non-lattice, principal root of unity; and Aharonov and Arad [arXiv:quant-ph/0605181] established a uniform version of this result. In this article, we show that any value-dependent approximation of the Jones polynomial at these non-lattice roots of unity is #P-hard. If given the power to decide whether |V(L,t)| > a or |V(L,t)| < b for fixed constants a > b > 0, there is a polynomial-time algorithm to exactly count the solutions to arbitrary combinatorial equations. In our argument, the result follows fairly directly from the universality result and Aaronson's theorem that PostBQP = PP [arXiv:quant-ph/0412187].
Cite
@article{arxiv.0908.0512,
title = {How hard is it to approximate the Jones polynomial?},
author = {Greg Kuperberg},
journal= {arXiv preprint arXiv:0908.0512},
year = {2019}
}
Comments
19 pages. Don't miss this major revision! Includes more complete explanations of all results, and refinements of both Aaronson's theorem and the Solovay-Kitaev theorem. To appear in ToC