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Level B · Reproducible Quantum information P-quantum-error-correction-thresholds

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.

Status (as of 2026-10-03): Open frontier. Surface codes now run below threshold in hardware (Google 2024/25), and qLDPC codes reach about 0.8% circuit-level thresholds.
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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 2026-11-12}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

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.