Error mitigation for logical circuits using decoder confidence

TL;DR

This work demonstrates that decoder confidence scores, particularly the swim distance, reliably signal logical-error risk in surface-code decoding. By linking DCS values to the log success odds via calibration curves and tensor-network benchmarks, the authors show how per-window risk information can be used for low-overhead quantum error mitigation, including abort protocols and DCS-based maximum-likelihood estimation. They provide both single-window and multi-window analyses, revealing that high-risk windows drive circuit-level error rates and that discarding or reweighting such events can dramatically reduce the overall logical error rate with modest overhead. The results are applied to resource estimation for a Hubbard-model calculation, suggesting practical pathways to reduce code distance and resource requirements in the early fault-tolerant era. Overall, DCS-based strategies offer a scalable, near-term approach to improve the accuracy of quantum computations without substantial quantum-resource costs.

Abstract

Fault-tolerant quantum computers use decoders to monitor for errors and find a plausible correction. A decoder may provide a decoder confidence score (DCS) to gauge its success. We adopt a swim distance DCS, computed from the shortest path between syndrome clusters. By contracting tensor networks, we compare its performance to the well-known complementary gap and find that both reliably estimate the logical error probability (LEP) in a decoding window. We explore ways to use this to mitigate the LEP in entire circuits. For shallow circuits, we just abort if any decoding window produces an exceptionally low DCS: for a distance-13 surface code, rejecting a mere 0.1% of possible DCS values improves the entire circuit's LEP by more than 5 orders of magnitude. For larger algorithms comprising up to trillions of windows, DCS-based rejection remains effective for enhancing observable estimation. Moreover, one can use DCS to assign each circuit's output a unique LEP, and use it as a basis for maximum likelihood inference. This can reduce the effects of noise by an order of magnitude at no quantum cost; methods can be combined for further improvements.

Figures (13)

• Figure 1: (a) The $X$ decoding graph for the $\llbracket n, k, d_X, d_Z \rrbracket =\llbracket 23, 1, 5, 3 \rrbracket$ unrotated surface code under code capacity noise. Detectors are green circles, boundary nodes are blue circles. In this noise model (but not necessarily in others), each detector corresponds to a $Z$ stabiliser, and each edge corresponds to a data qubit (emphasised by a grey circle on each edge). (b) It later helps if there is only one boundary node for each unique boundary of the surface code, so we merge equivalent boundary nodes as shown. Here, we no longer show detectors nor data qubit circles.
• Figure 2: Illustrating the complementary gap on the decoding graph $G$ in \ref{['fig:to_decoding_graph']}b. The modified graph $G'$, shown in grey, has only one remaining boundary node, shown as a blue dot. The syndrome $\mathbb S$ ($\mathbb S'$) is the set of red (yellow) dots. The correction $\mathbb C$ ($\mathbb C'$) is the set of red (yellow) edges. The complementary gap is the weight difference between these two corrections; here it is $5 -3 =2$ if all edges have weight 1.
• Figure 3: Illustrating the swim distance, which is here the sum of the lengths of the blue lines: $2 \times 0.5 =1$ if all edges have weight 1. Clusters are shaded in red.
• Figure 4: The two sources of variation for the complementary gap (blue data) and the swim distance (orange data). The first source $\sigma_{\alpha(\lambda)}$ is due to the variation of the log success odds. The second source $\sigma_r$ comes from the approximate nature of the DCS. Results are obtained using the MWPM decoder on the unrotated surface code under a perturbed phenomenological noise model (described in \ref{['sec:perturbed_phenomenological_noise_model']}).
• Figure 5: Single-decoding-window statistics. (a) Histograms of DCS (swim distance) values, normalized by the number of shots. Under circuit-level noise, $10^8$ memory experiments were run for all code distances. The measured DCS values are grouped in 50 bins of equal linear width. (b) The log success odds conditioned on the DCS value, plotted against the DCS value. The upper and lower bounds on the Bernoulli variable $p_L$ are given by the Wilson score interval. Only datapoints with $p_L >0$ are shown.

Theorems & Definitions (5)

• Definition 1
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• Definition 5