Quantum Decoherence of the Surface Code: A Generalized Caldeira-Leggett Approach

TL;DR

This paper explores the fundamental limits of quantum error correction using the surface code in continuous environments, revealing conditions under which topological protection is compromised due to long-range environmental effects.

Contribution

It introduces a generalized Caldeira-Leggett framework to analyze the surface code's decoherence, mapping it to a boundary conformal field theory and establishing a link to the anisotropic Kondo model.

Findings

A true thermodynamic threshold exists only for short-range environments.

Long-range environments hinder topological protection by the continuous bath.

The analysis maps the logical qubit evolution to a boundary conformal field theory.

Abstract

Standard quantum error correction (QEC) models typically assume discrete, Markovian noise, obscuring the continuous quantum nature of physical environments. In this manuscript, we investigate the fundamental limits of an actively corrected surface code coupled to a continuous, un-reset quantum environment at zero and finite temperature. Using the generalized Caldeira-Leggett framework, we map the long-time evolution of the logical qubit to a boundary conformal field theory, establishing an exact equivalence to the anisotropic Kondo model. We evaluate computational times for a finite code distance $L$ for all spatial and temporal correlations. Our analysis reveals that a true thermodynamic threshold exists strictly for short-range environments ($z>1/(s+1)$). In critical or long-range regimes, the macroscopic footprint of the code weaponizes the continuous bath, hindering the topological protection.