Universal Optimization and Tighter Fidelity Bounds for Approximate Quantum Error Correction

Identifying the optimal recovery map to maximize entanglement fidelity in approximate quantum error correction (AQEC) is typically formulated as a semidefinite program (SDP). However, the number of Kraus operators grows exponentially with system size, creating a computational bottleneck that makes high-precision optimization intractable for large-scale discrete-variable (DV) and continuous-variable (CV) systems. This paper seeks to overcome this curse of dimensionality.

The authors leverage the duality between recovery and environment decoupling to reformulate the entanglement fidelity maximization. By mapping the problem to a quotient manifold and utilizing the spectral properties of the environment-reference state, they derive a new 'in-phase' analytical lower bound that is tighter than the conventional transpose channel limit. Furthermore, they introduce a PCA-based framework that compresses the optimization space by focusing on the dominant components of the density matrix, effectively reducing the number of parameters required for high-precision optimization.

The proposed framework provides a robust toolset for AQEC. The in-phase lower bound consistently outperforms the near-optimal transpose channel bound, remaining particularly tight for the Shor nine-qubit code. For the GKP code in a thermal loss channel, the PCA-based approach achieves a 33-fold computational speedup while maintaining rigorous accuracy. The authors demonstrate that this efficiency gain is physically rooted in the exponential decay of Kraus operator weights, which allows for significant dimensionality reduction without sacrificing the fidelity of the recovery map.