Circuit partitions and #P-complete products of inner products
Computational Complexity
2010-01-15 v1 Discrete Mathematics
Combinatorics
Quantum Physics
Abstract
We present a simple, natural #P-complete problem. Let G be a directed graph, and let k be a positive integer. We define q(G;k) as follows. At each vertex v, we place a k-dimensional complex vector x_v. We take the product, over all edges (u,v), of the inner product <x_u,x_v>. Finally, q(G;k) is the expectation of this product, where the x_v are chosen uniformly and independently from all vectors of norm 1 (or, alternately, from the Gaussian distribution). We show that q(G;k) is proportional to G's cycle partition polynomial, and therefore that it is #P-complete for any k>1.
Keywords
Cite
@article{arxiv.1001.2314,
title = {Circuit partitions and #P-complete products of inner products},
author = {Cristopher Moore and Alexander Russell},
journal= {arXiv preprint arXiv:1001.2314},
year = {2010}
}