Fundamental Thresholds for Computational and Erasure Errors via the Coherent Information

Quantum error correcting (QEC) codes protect quantum information against environmental noise. Computational errors caused by the environment change the quantum state within the qubit subspace, whereas quantum erasures correspond to the loss of qubits at known positions. Correcting either type of error involves different correction mechanisms, which makes studying the interplay between erasure and computational errors particularly challenging. In this work, we propose a framework based on the coherent information (CI) of the mixed-state density operator associated to noisy QEC codes, for treating both types of errors together. We show how to rigorously derive different families of statistical mechanics mappings for generic stabilizer QEC codes in the presence of both types of errors. We observe that the erasure errors enter as a classical average over fully depolarizing channels. Further, we show that computing the CI for erasure errors only can be done efficiently upon sampling over erasure configurations. We then test our approach on the two-dimensional toric and color codes and compute optimal thresholds for erasure errors only, finding a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mn>50</a:mn> <a:mi mathvariant="normal">%</a:mi> </a:math> threshold for both codes. This strengthens the notion that both codes share the same optimal thresholds. When considering both computational and erasure errors, the CI of small-size codes yields thresholds in very accurate agreement with established results that have been obtained in the thermodynamic limit. Next, we perform a similar analysis for a low-density parity-check (LDPC) code—the lift-connected surface code. We find a <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mn>50</d:mn> <d:mi mathvariant="normal">%</d:mi> </d:math> threshold under erasure errors alone and, for the first time, derive the exact statistical mechanics mappings in the presence of both computational and erasure errors. We thereby further establish the CI as a practical tool for studying optimal thresholds for code classes beyond topological codes under realistic noise, and as a means for uncovering new relations between QEC codes and statistical physics models.