Completeness of classical spin models and universal quantum computation

TL;DR

This paper demonstrates how certain classical spin models, including the 3D Ising model, can be used to represent any other spin system's partition function with real, physical couplings, linking statistical mechanics to quantum computation.

Contribution

It shows that the 3D Ising model with real couplings is complete for representing all classical spin systems, extending previous complex-valued models and connecting to universal quantum computation.

Findings

3D Ising model can simulate any classical spin system with real couplings.

Mappings between spin models and quantum states enable universal quantum computation.

Polynomial overhead in constructing these mappings.

Abstract

We study mappings between distinct classical spin systems that leave the partition function invariant. As recently shown in [Phys. Rev. Lett. 100, 110501 (2008)], the partition function of the 2D square lattice Ising model in the presence of an inhomogeneous magnetic field, can specialize to the partition function of any Ising system on 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 assume complex values, and thus do not allow for a physical interpretation. Here we show how a complete model with real -and, hence, "physical"- couplings can be obtained if the 3D Ising model is considered. We furthermore show how to map general q-state systems with possibly many-body interactions to the 2D Ising model with complex parameters, and give completeness results for these models with real parameters. We also demonstrate that the computational overhead in these constructions is in all relevant cases polynomial. These results are proved by invoking a recently found cross-connection between statistical mechanics and quantum information theory, where partition functions are expressed as quantum mechanical amplitudes. Within this framework, there exists a natural correspondence between many-body quantum states that allow universal quantum computation via local measurements only, and complete classical spin systems.