Statistical Mechanical Models and Topological Color Codes

TL;DR

This paper establishes a connection between topological quantum color codes and classical statistical models, enabling analysis of their computational capabilities and classical simulatability through mappings to Ising models.

Contribution

It introduces a novel mapping of topological color codes to classical Ising models, broadening the understanding of their computational properties and simulatability.

Findings

Color code states map to 3-body Ising models on specific lattices.

Different computational capabilities relate to different universality classes.

The classical simulatability of measurement-based quantum computation remains unresolved.

Abstract

We find that the overlapping of a topological quantum color code state, representing a quantum memory, with a factorized state of qubits can be written as the partition function of a 3-body classical Ising model on triangular or Union Jack lattices. This mapping allows us to test that different computational capabilities of color codes correspond to qualitatively different universality classes of their associated classical spin models. By generalizing these statistical mechanical models for arbitrary inhomogeneous and complex couplings, it is possible to study a measurement-based quantum computation with a color code state and we find that their classical simulatability remains an open problem. We complement the meaurement-based computation with the construction of a cluster state that yields the topological color code and this also gives the possibility to represent statistical models with external magnetic fields.