Physics – Quantum Physics
Scientific paper
2006-05-21
New J. Phys. 13 (2011) 035019
Physics
Quantum Physics
36 pages, 8 figures. A much improved and revised version, updated to Feb 2011, which includes discussions and implications of
Scientific paper
10.1088/1367-2630/13/3/035019
A celebrated important result due to Freedman, Larsen and Wang states that providing additive approximations of the Jones polynomial at the k'th root of unity, for constant k=5 and k>6, is BQP-hard. Together with the algorithmic results of Freedman et al and Aharonov et al, this gives perhaps the most natural BQP-complete problem known today and motivates further study of the topic. In this paper we focus on the universality proof; we extend the universality result of Freedman et al to k's that grow polynomially with the number of strands and crossings in the link, thus extending the BQP-hardness of Jones polynomial approximations to all values for which the AJL algorithm applies, proving that for all those values, the problems are BQP-complete. As a side benefit, we derive a fairly elementary proof of the Freedman et al density result, without referring to advanced results from Lie algebra representation theory, making this important result accessible to computer science audience. We make use of two general lemmas we prove, the Bridge lemma and the Decoupling lemma, which provide tools for establishing density of subgroups in SU(n). Those tools seem to be of independent interest in more general contexts of proving quantum universality. Our result also implies a completely classical statement, that the_multiplicative_ approximations of the Jones polynomial, at exactly the same values, are #P-hard, via a recent result due to Kuperberg. Since the first publication of those results in their preliminary form (arXiv:quant-ph/0605181v2), the methods we present here were used in several other contexts. This paper is an improved and extended version of the original results, and also includes discussions of the developments since then.
Aharonov Dorit
Arad Itai
No associations
LandOfFree
The BQP-hardness of approximating the Jones Polynomial does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The BQP-hardness of approximating the Jones Polynomial, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The BQP-hardness of approximating the Jones Polynomial will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-16534