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

TL;DR

This paper introduces a partition function-based framework for estimating logical error rates in stabilizer codes, connecting statistical mechanics and quantum error correction, and demonstrates improved efficiency in performance estimation.

Contribution

It develops a novel partition function approach for decoding analysis in stabilizer codes, linking statistical mechanics models with quantum error correction strategies.

Findings

Partition function ratio measures success probability at Nishimori temperature.

Probabilistic partition function decoding aligns with maximum likelihood decoding at zero temperature.

Estimation of logical error rates is more sample-efficient than failure counting.

Abstract

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. 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.