Quantum algorithm for an additive approximation of Ising partition functions
Abstract
We investigate quantumcomputational complexity of calculating partition functions of Ising models. We construct a quantum algorithm for an additive approximation of Ising partition functions on square lattices. To this end, we utilize the overlap mapping developed by M. Van den Nest, W. Dür, and H. J. Briegel [Phys. Rev. Lett. 98, 117207 (2007), 10.1103/PhysRevLett.98.117207] and its interpretation through measurementbased quantum computation (MBQC). We specify an algorithmic domain, on which the proposed algorithm works, and an approximation scale, which determines the accuracy of the approximation. We show that the proposed algorithm performs a nontrivial task, which would be intractable on any classical computer, by showing that the problem that is solvable by the proposed quantum algorithm is BQPcomplete. In the construction of the BQPcomplete problem coupling strengths and magnetic fields take complex values. However, the Ising models that are of central interest in statistical physics and computer science consist of real coupling strengths and magnetic fields. Thus we extend the algorithmic domain of the proposed algorithm to such a real physical parameter region and calculate the approximation scale explicitly. We found that the overlap mapping and its MBQC interpretation improve the approximation scale exponentially compared to a straightforward constantdepth quantum algorithm. On the other hand, the proposed quantum algorithm also provides partial evidence that there exist no efficient classical algorithm for a multiplicative approximation of the Ising partition functions even on the square lattice. This result supports the observation that the proposed quantum algorithm also performs a nontrivial task in the physical parameter region.
 Publication:

Physical Review A
 Pub Date:
 August 2014
 DOI:
 10.1103/PhysRevA.90.022304
 arXiv:
 arXiv:1405.2749
 Bibcode:
 2014PhRvA..90b2304M
 Keywords:

 03.67.Ac;
 03.67.Lx;
 75.10.Hk;
 Quantum algorithms protocols and simulations;
 Quantum computation;
 Classical spin models;
 Quantum Physics;
 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics;
 Computer Science  Computational Complexity
 EPrint:
 18 pages, 12 figures