Partial Self-Correction in Layer Codes

The storage of large-scale quantum information at finite temperature requires an autonomous and reliable quantum hard drive, also known as a self-correcting quantum memory. It is a long-standing open problem to find a self-correcting quantum memory in three dimensions. The recently introduced layer codes achieve the best possible scaling of code parameters and logical energy barrier in three dimensions; these are tantalizing features for the purposes of self-correction. In this Letter, we show that a family of layer codes based on good quantum Tanner codes exhibit partial self-correction. Their memory time grows exponentially with linear system size, up to a length scale that is exponential in the inverse temperature. At this length scale, the memory time scales as a double exponential of inverse temperature. To establish this result, we introduce a concatenated matching decoder that combines three rounds of parallelized minimum-weight perfect matching with a decoder for good quantum Tanner codes. We show that our decoder corrects errors up to a constant fraction of the energy barrier, and a constant fraction of the code distance, for a family of layer codes. Our results position layer codes as the leading candidate for a partially self-correcting memory in three dimensions in the asymptotically large system size regime. While they fall short of achieving strict self-correction in the thermodynamic limit, our Letter highlights the potential of these local codes in three dimensions, with fast distance and logical qubit growth, fast decoders, and a long memory time over a wide range of parameters. In particular, recent work by Gu et al. reports promising numerical results for the memory time of small explicit instances of layer codes based on random input codes.