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

Standard quantum error correction (QEC) models typically assume discrete, Markovian noise. This assumption obscures the continuous quantum nature of physical environments. 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 ( $$z>1/(s+1)$$ ), where z is the dynamical exponent of the environment and s is the exponent of the spectral density. In critical or long-range regimes, the macroscopic footprint of the code amplifies the continuous bath, hindering the topological protection.