Approximate quantum error correction theory of non-isometric codes

Abstract: Non-isometric encoding arises in various important contexts in quantum error correction, most notably in the finite-energy, non-ideal codewords inevitable in experimental realizations of continuous-variable codes, and holographic quantum gravity. In this work, we present a general and systematic theory of non-isometric quantum error-correcting codes. In particular, we employ the approximate quantum error correction framework to quantitatively study the fundamental limitations imposed by non-isometric encodings on the accuracy of quantum error correction and implementation of logical operations. We apply our theory to analyze GKP and tiger codes under energy constraints, and discuss the implications to holography.