Error-rate-agnostic decoding of topological stabilizer codes

Abstract

Efficient high-performance decoding of topological stabilizer codes has the potential to crucially improve the balance between logical failure rates and the number and individual error rates of the constituent qubits. High-threshold maximum-likelihood decoders require an explicit error model for Pauli errors to decode a specific syndrome, whereas lower-threshold heuristic approaches such as minimum weight matching are "error agnostic". Here we consider an intermediate approach, formulating a decoder that depends on the bias, i.e., the relative probability of phase-flip to bit-flip errors, but is agnostic to error rate. Our decoder is based on counting the number and effective weight of the most likely error chains in each equivalence class of a given syndrome. We use Metropolisbased Monte Carlo sampling to explore the space of error chains and find unique chains, which can be run efficiently by exploiting the error-rate invariance and without the need for "thermalization" between chains in different equivalence classes. Applied to the surface code and the XZZX code, the decoder matches maximum-likelihood decoders for moderate code sizes or low error rates. We anticipate that, because of the compressed information content per syndrome, it can be taken full advantage of in combination with machine-learning methods to extrapolate Monte Carlo-generated data.

The XYZ$^2$ hexagonal stabilizer code

“…[63]. As shown in [62], the two methods give almost identical results for the XZZX code for moderate code distances.…”

“…These four chains are then deformed by acting with random stabilizers to preserve only the syndrome of the initial chain. A maximum-likelihood decoder [17,[59][60][61][62] (MLD) then calculates which of the four equivalence classes of chains is most likely to correspond to the syndrome. The event is counted as a successful error correction if the initial chain is in this most likely class, and a failed error correction otherwise.…”

“…In this work, we use the decoder recently presented in Ref. [62], the "effective weight and degeneracy" (EWD) decoder, which is based on Monte Carlo sampling of error configurations using the Metropolis algorithm. This decoder is essentially a more efficient, simplified version of the Markov-chain Monte Carlo decoder formulated in Refs.…”

“…As discussed in Ref. [62], the EWD requires the tuning of a hyperparameter, corresponding to the error rate p sample (which acts as an effective temperature) of the Metropolis sampling, to minimize the logical failure rate for a given physical error rate p. There is thus some systematic uncertainty in the results, in the sense that the results provide upper bounds on the failure rate. For the XZZX code, we use the matrix-product-state (MPS) decoder as implemented in Ref.…”

“…Error bars indicate the statistical error (one standard deviation) based on the number of sampled syndromes and the mean logical failure rate. We attribute larger-than-error-bar variations of the EWD decoder for large physical error rates to difficulties with optimizing the decoder's sampling error rate [62].…”