Noisy Toric code and random bond Ising model: The error threshold in a dual picture

TL;DR

This paper explores the duality between classical random bond Ising models and quantum topological codes, revealing how classical phase transitions relate to quantum error thresholds in topological CSS codes, especially the Toric code.

Contribution

It establishes a duality mapping between classical phase transitions and quantum coherence loss, providing a new framework to analyze error thresholds in topological quantum codes.

Findings

Classical order parameter maps to quantum coherence in noisy Toric code

Quantum phase transition corresponds to error threshold in the Toric code

Duality offers a new approach to study error thresholds in topological CSS codes

Abstract

It is known that noisy topological quantum codes are related to random bond Ising models where the order-disorder phase transition in the classical model is mapped to the error threshold of the corresponding topological code. On the other hand, there is a dual mapping between classical spin models and quantum Calderbank-Shor-Stean (CSS) states where the partition function of a classical model defined on a hypergraph $H$ is written as an inner product of a product state and a CSS state on dual hypergraph $\tilde{H}$. It is then interesting to see what is the interpretation of the classical phase transition in the random bond Ising model within the framework of the above duality and whether such an interpretation has any connection to the error threshold of the corresponding topological CSS code. In this paper, we consider the above duality relation specifically for a two-dimensional random bond Ising model. We show that the order parameter of this classical system is mapped to a coherence order parameter in a noisy Toric code model. In particular, a quantum phase transition from a coherent phase to a non-coherent phase occurs when the initial coherent state is affected by two sequences of bit-flip quantum channels where a quenched disorder is induced by measurement of the errors after the first channel. On the other hand, the above transition is directly related to error threshold of the Toric code model. Accordingly, and since the noisy process can be applied to other topological CSS states, we conclude that the dual correspondence can also provide a useful tool for the study of error thresholds in different topological CSS codes.