Classical Ising model test for quantum circuits

TL;DR

This paper demonstrates that certain quantum circuits, specifically those associated with planar graphs, can be efficiently simulated classically by leveraging a novel mapping to the Ising model and existing algorithms.

Contribution

It introduces a new method connecting quantum circuits to the Ising model, enabling classical simulation of specific non-trivial quantum circuits.

Findings

Efficient classical simulation for quantum circuits on planar graphs.

Use of Ising model partition function and quadratically signed weight enumerators.

Identification of quantum circuits with limited non-nearest neighbor gates that are classically simulatable.

Abstract

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.