Achieving pure loss channel capacity using GKP codes with near optimal recovery formalism

Most codes are approximate quantum error correcting codes against physical error sources. While the Knill-Laflamme condition determines whether certain errors are fully correctable by a certain encoding, the effect of the uncorrectable errors could only be quantized through numerical optimizations but not through analytical methods. Here, we develop a concise analytical expression for the near-optimal performance of quantum error correction codes that is optimization-free, solely depends on the quantum error correction matrix, and applies to arbitrary encodings and noise channels.



Our formula is promising from both numerical and analytical perspectives. Numerically, we demonstrate the broad scope of applicability by examples of stabilizer codes, non-stabilizer codes, bosonic codes, and others. Depending on the parameter regime of interest, our method shows polynomial to exponential speedups compared to the state-of-the-art optimization-based approach. Analytically, we focus on the Gottesman-Kitaev-Preskill (GKP) code under pure loss channel. We prove that there exist good multi-mode GKP codes that achieves the quantum capacity of pure loss channel. Our work demonstrates the potential of our method in future explorations of quantum information science.