Extracting Error Thresholds through the Framework of Approximate Quantum Error Correction Condition

Understand

The robustness of quantum memory against physical noises is measured by two methods: the exact and approximate quantum error correction (QEC) conditions for error recoverability, and the decoder-dependent error threshold which assesses if the logical error rate diminishes with system size.

• Here we unravel their relations and propose a unified framework to extract an intrinsic error threshold from the approximate QEC condition, which could upper bound other decoder-dependent error thresholds.
• Our proof establishes that relative entropy, effectively measuring deviations from exact QEC conditions, serves as the order parameter delineating the transition from asymptotic recoverability to unrecoverability.
• Consequently, we establish a unified framework for determining the error threshold across both exact and approximate QEC codes, addressing errors originating from noise channels as well as those from code space imperfections.