A partition function framework for estimating logical error curves in stabilizer codes

Based on the mapping between stabilizer quantum error correcting codes and disordered statistical mechanics models, we define a ratio of partition functions that measures the success probability for maximum partition function decoding, which at the Nishimori temperature corresponds to maximum likelihood (ML) decoding. We show that this ratio differs from the similarly defined order probability and describe the decoding strategy whose success rate is described by the order probability. We refer to the latter as a probabilistic partition function decoding and show that it is the strategy that at zero temperature corresponds to maximum probability (MP) decoding. Based on the difference between the two decoders, we discuss the possibility of a maximum partition function decodability boundary outside the order-disorder phase boundary. At zero temperature, the difference between the two ratios measures to what degree MP decoding can be improved by accounting for degeneracy among maximum probability errors, through methods such as ensembling. We consider in detail the example of the toric code under bitflip noise, which maps to the Random Bond Ising Model. We demonstrate that estimation of logical performance through decoding probability and order probability is more sample efficient than estimation by counting failures of the corresponding decoders, especially in the regime of low noise. We consider both uniform noise and noise where qubits are given individual error rates. The latter noise model lifts the degeneracy among maximum probability errors, but we show that ensembling remains useful as long as it also samples less probable errors. We also consider, in less detail, the color code under bitflip and depolarizing noise.