Here is a list of codes related to single-shot codes.
| Code | Relation |
|---|---|
| 3D subsystem color code | The 3D subsystem color code defined on the cube-truncated rhombic dodecahedral honeycomb, i.e., a tessellation of cubes and chamfered cubes (a.k.a. tetratruncated rhombic dodecahedra) [1; Fig. 1], is a single-shot code [1,2]. |
| 3D subsystem surface code | The 3D subsystem surface code is a single-shot code [3,4]; see Ref. [5] for an alternative formulation. |
| Campbell double homological product code | For a minimal input chain complex associated with a classical \([n,k,d]\) code, the Campbell double homological product code is a single-shot code with \(d_{\text{ss}}=\infty\), \((d,f)\)-soundness for \(f(x)=x^3/4\), and check redundancy bounded by \(<2\) [6]. |
| Generalized bicycle (GB) code | A qubit GB code \([[n,k,d]]_2\) has \(k\) non-trivial relations between the syndrome bits, which is expected to help with operation in a fault-tolerant regime (in the presence of syndrome measurement errors). See Ref. [7] for many examples of such codes. There is numerical evidence that a particular family is single shot [8]. |
| Homological product code | It is conjectured that a particular class of codes called three-dimensional product codes is single shot [9]. |
| Hyperbolic surface code | A 4D hyperbolic surface code can be decoded with the Hastings decoder [10] in time \(O(n\log n)\) and with a logical error scaling inverse polynomially with \(n\). |
| Hypergraph product (HGP) code | Two-fold application of the hypergraph product to a pair of binary linear codes yields single-shot QLDPC codes that exploit redundancy in their stabilizer generators [6]. |
| Quantum Tanner code | Certain quantum Tanner codes facilitate single-shot decoding [11]. |
| Quantum data-syndrome (QDS) code | QDS codes are closely related to single-shot codes because both use redundant syndrome information to suppress measurement errors in a single round of syndrome extraction [6]. |
| Quantum expander code | Quantum expander codes are single-shot [12]. |
| Qubit QLDPC code | Qubit QLDPC codes satisfying linear confinement are single shot [9]. Any code that admits a local greedy decoder also satisfies linear confinement, and so is single shot [13]. |
| Qubit stabilizer code | Any stabilizer code can be single shot if sufficiently non-local high-weight stabilizer generators are used for syndrome measurements. These can be obtained with a Gaussian elimination procedure [6]. |
| Self-correcting quantum code | The presence of an energy barrier (i.e., confinement) is sufficient for a code to be single shot, and is also conjectured to be necessary for a code to be a self-correcting memory. Linear confinement of QLDPC (LDPC) codes implies (classical) self-correction [14]. |
| Single-shot code | |
| Subsystem Hypergraph Product Simplex (SHYPS) code | SHYPS codes exhibit practical single-shot signatures, including logical error-rate stability under small-window sliding-window decoding and constant single-shot distance \(d_{\mathrm{ss}}=3\), which supports using one syndrome-extraction round between logical generators [15]. |
| \((2,2)\) Loop toric code | Single-shot QEC has been realized using the \([[33,1,4]]\) loop toric code on the Quantinuum H2 device [16]. |
References
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- B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
- [2]
- H. Bombín, “Single-Shot Fault-Tolerant Quantum Error Correction”, Physical Review X 5, (2015) arXiv:1404.5504 DOI
- [3]
- A. Kubica and M. Vasmer, “Single-shot quantum error correction with the three-dimensional subsystem toric code”, Nature Communications 13, (2022) arXiv:2106.02621 DOI
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- J. C. Bridgeman, A. Kubica, and M. Vasmer, “Lifting Topological Codes: Three-Dimensional Subsystem Codes from Two-Dimensional Anyon Models”, PRX Quantum 5, (2024) arXiv:2305.06365 DOI
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- C. Stahl, “Single-shot quantum error correction in intertwined toric codes”, Physical Review B 110, (2024) arXiv:2307.08118 DOI
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- E. T. Campbell, “A theory of single-shot error correction for adversarial noise”, Quantum Science and Technology 4, 025006 (2019) arXiv:1805.09271 DOI
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- H.-K. Lin and L. P. Pryadko, “Quantum two-block group algebra codes”, (2023) arXiv:2306.16400
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- T. R. Scruby, T. Hillmann, and J. Roffe, “High-Threshold, Low-Overhead and Single-Shot Decodable Fault-Tolerant Quantum Memory”, PRX Quantum 7, (2026) arXiv:2406.14445 DOI
- [9]
- A. O. Quintavalle, M. Vasmer, J. Roffe, and E. T. Campbell, “Single-Shot Error Correction of Three-Dimensional Homological Product Codes”, PRX Quantum 2, (2021) arXiv:2009.11790 DOI
- [10]
- M. B. Hastings, “Decoding in Hyperbolic Spaces: LDPC Codes With Linear Rate and Efficient Error Correction”, (2013) arXiv:1312.2546
- [11]
- S. Gu, E. Tang, L. Caha, S. H. Choe, Z. He, and A. Kubica, “Single-Shot Decoding of Good Quantum LDPC Codes”, Communications in Mathematical Physics 405, (2024) arXiv:2306.12470 DOI
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- O. Fawzi, A. Grospellier, and A. Leverrier, “Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes”, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) 743 (2018) arXiv:1808.03821 DOI
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- Q. Xu, J. P. B. Ataides, C. A. Pattison, N. Raveendran, D. Bluvstein, J. Wurtz, B. Vasic, M. D. Lukin, L. Jiang, and H. Zhou, “Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays”, (2023) arXiv:2308.08648
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- Y. Hong, J. Guo, and A. Lucas, “Quantum memory at nonzero temperature in a thermodynamically trivial system”, Nature Communications 16, (2025) arXiv:2403.10599 DOI
- [15]
- A. J. Malcolm et al., “Computing Efficiently in QLDPC Codes”, (2025) arXiv:2502.07150
- [16]
- N. Berthusen et al., “Experiments with the four-dimensional surface code on a quantum charge-coupled device quantum computer”, Physical Review A 110, (2024) arXiv:2408.08865 DOI