Classical Ising model test for quantum circuits

We exploit a recently constructed mapping between quantum circuits and graphs in order to prove that circuits corresponding to certain planar graphs can be efficiently simulated classically. The proof uses an expression for the Ising model partition function in terms of quadratically signed weight enumerators (QWGTs), which are polynomials that arise naturally in an expansion of quantum circuits in terms of rotations involving Pauli matrices. We combine this expression with a known efficient classical algorithm for the Ising partition function of any planar graph in the absence of an external magnetic field, and the Robertson–Seymour theorem from graph theory. We give as an example a set of quantum circuits with a small number of non-nearest-neighbor gates which admit an efficient classical simulation. 4 Current address: The Ontario Cancer Biomarker Network, MaRS-TMDT, Toronto, Ontario M5G 1L7, Canada. 5 Author to whom any correspondence should be addressed. New Journal of Physics 12 (2010) 075026 1367-2630/10/075026+29$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft