Benedikt Placke, Max Planck Institute for the Physics of Complex Systems

Statistical mechanics mapping for finite rate qLDPC codes

It is well known that the decoding of zero rate stabilizer codes can be mapped onto statistical mechanics models with quenched disorder. Statistical mechanics mappings were first developed for estimating the maximum-likelihood decoding threshold of the toric code under phenomenological noise models (Dennis et al, 2002), They have since been developed for other stabilizer codes and more general noise models (Chubb and Flammia, 2021). Recently, quantum LDPC codes with finite rate have generated much interest because they could significantly reduce the overhead required for quantum error correction. This makes a generalization of the statistical mechanics mappings to this class of codes desirable. Such a generalization is non-trivial because the mapping has to account for the fact that there is an exponential (in the number of physical qubits) number of possible logical errors. In the language of statistical mechanics this implies that they have a non-vanishing contribution to the entropy of the system. This is in contrast to the case of decoding finite-rate codes, where the number of possible logical errors does not scale with the number of physical qubits. We solve this problem for the case of hyperbolic surface codes, which are close cousins of the (euclidean) surface code but have finite rate. In particular, we show that decoding them under independent bit- and phase-flip noise maps onto what we call the “dual random bond Ising model” in hyperbolic space. As an application of the mapping, we compute the maximum likelihood decoding threshold of a range of hyperbolic surface codes under independent bit- and phase-flip noise.