On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers
Abstract
We present an efficient quantum algorithm for the exact evaluation of either the fully ferromagnetic or antiferromagnetic qstate Potts partition function Z for a family of graphs related to irreducible cyclic codes. This problem is related to the evaluation of the Jones and Tutte polynomials. We consider the connection between the weight enumerator polynomial from coding theory and Z and exploit the fact that there exists a quantum algorithm for efficiently estimating Gauss sums in order to obtain the weight enumerator for a certain class of linear codes. In this way we demonstrate that for a certain class of sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCC_{∊}) graphs, quantum computers provide a polynomial speed up in the difference between the number of edges and vertices of the graph, and an exponential speed up in q, over the best classical algorithms known to date.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 May 2008
 DOI:
 10.1007/s0022000804380
 arXiv:
 arXiv:quantph/0703023
 Bibcode:
 2008CMaPh.279..735G
 Keywords:

 Quantum Physics
 EPrint:
 Comm. Math. Phys. 279, 735 (2008)