Thresholds and decoders for quantum error-correcting codes under circuit-level noise
Improve reproducible, circuit-level-noise thresholds and logical error rates for surface codes and quantum LDPC codes through better codes, syndrome circuits and decoders, benchmarked with open simulators such as Stim.
Cite
@misc{cairn-quantum-error-correction-thresholds,
title = {Thresholds and decoders for quantum error-correcting codes under circuit-level noise},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/quantum-error-correction-thresholds}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
} Also: CITATION.cff · Atom feed of results
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Current state
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The problem
The question. For a given code family, syndrome-extraction circuit and decoder, what is the threshold error rate under a standard circuit-level noise model, and how many physical qubits are needed to reach a target logical error rate? Which combinations are best?
Known status. Bravyi et al. (Nature 2024) introduced bivariate-bicycle qLDPC codes with a threshold of about 0.8% under the standard circuit noise model, on par with the surface code, and a [[144,12,12]] code storing 12 logical qubits in 288 physical qubits. Google Quantum AI (Nature 2024) operated a distance-7 surface-code memory below threshold, with an error suppression factor Λ = 2.14 ± 0.02 and 0.143% logical error per cycle. Open tools include Stim (Gidney 2021) for stabilizer circuit sampling and PyMatching 2 with sparse blossom (Higgott and Gidney 2023).
What counts as progress
- A new code / circuit / decoder combination with lower logical error rate or higher threshold on a stated noise model, submitted as Stim circuit files plus decoder code and sampling statistics.
- Faster decoders that match an existing decoder's accuracy, with timing on stated hardware.
- Reproductions of published thresholds and documented negative results (e.g. "decoder X does not reach threshold on code Y under noise Z").
How it is checked. Reviewers re-run the provided circuits and decoders (e.g. with Stim and sinter) with the stated shot counts and verify logical error rates within the reported confidence intervals; thresholds are checked from the crossing of curves for several distances.