Completeness of classical spin models and universal quantum computation

Abstract

We study mappings between different classical spin systems that leave the partitionfunction invariant. As recently shown in Van den Nest et al (2008 Phys. Rev. Lett. 100110501), the partition function of the 2D square lattice Ising model in the presence of aninhomogeneous magnetic field can specialize to the partition function of any Ising systemon an arbitrary graph. In this sense the 2D Ising model is said to be ‘complete’. However,in order to obtain the above result, the coupling strengths on the 2D lattice must assumecomplex values, and thus do not allow for a physical interpretation. Here we show how acomplete model with real—and, hence, ‘physical’—couplings can be obtainedif the 3D Ising model is considered. We furthermore show how to map generalq-state systems with possibly many-body interactions to the 2D Ising model with complexparameters, and give completeness results for these models with real parameters. We alsodemonstrate that the computational overhead in these constructions is in all relevant casespolynomial. These results are proved by invoking a recently found cross-connection betweenstatistical mechanics and quantum information theory, where partition functions areexpressed as quantum mechanical amplitudes. Within this framework, there exists anatural correspondence between many-body quantum states that allow for universalquantum computation via local measurements only, and complete classical spinsystems.