Information-Theoretic Framework for Quantum State Purification and Error Correction via Entropy Compression Mechanisms

The extreme sensitivity of qubits to environmental noise constitutes the central bottleneck impeding the practical deployment of near-term quantum computing. Quantum error correction suppresses physical errors by encoding logical qubits into redundant physical qubits, yet its corrective gain deteriorates sharply when physical error rates approach the fault-tolerant threshold. This paper proposes a purification-assisted quantum error correction framework that systematically embeds a purification preprocessing module between the encoding layer and the physical layer, founded on the permutation symmetry of multiple noisy state copies. From an information-theoretic perspective, this framework treats the purification process as an entropy reduction mechanism that filters noise entropy from quantum states, concentrating quantum information into lower-entropy subspaces and achieving entropy suppression before downstream error correction processing. By projecting the joint state of identical copies onto the symmetric subspace, the module achieves noise entropy filtering, thereby concentrating quantum information and exponentially compressing the effective physical error rate before it enters the error-correcting code, establishing a compound exponential logical error rate suppression mechanism under ideal assumptions. Analytical derivations under depolarizing noise yield closed-form expressions for the purification fidelity and the enhanced equivalent fault-tolerant threshold, demonstrating that purification with three copies elevates the surface code threshold from approximately 1.1% to approximately 2.0%. Building on this framework, an Iterative Purification-assisted Error Correction (IPEC) algorithm is designed, which dynamically adjusts the purification depth via real-time syndrome feedback to achieve an adaptive balance between fidelity gain and resource consumption. Monte Carlo simulations under both independent depolarizing noise and circuit-level noise models validate the theoretical predictions: the IPEC algorithm reduces the logical error rate by approximately 46-fold at code distance 7 with a physical error rate of 1.0%, achieves an approximately 8-fold improvement on quantum LDPC codes, and maintains robust performance in non-stationary noise environments through its adaptive mechanism. From an entropy-theoretic perspective, the proposed framework constitutes an entropy management mechanism for quantum computation, in which the purification process compresses state entropy at the input stage and error correction maintains the low-entropy condition during computation, together forming a closed-loop entropy flow control system.