Simple Quantum Error Correcting Codes

Andrew Steane (Clarendon Laboratory; Oxford University)

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Simple Quantum Error Correcting Codes

Andrew Steane (Clarendon Laboratory, Oxford University)

Quantum Physics · Submitted 15 May 1996

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This systematic approach leads to single-error correcting codes for 3, 4 and 5 information qubits with block lengths of 8, 10 and 11 qubits respectively.

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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,225 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.26 · mentions code or data availability 0.04.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Methods of finding good quantum error correcting codes are discussed, and many example codes are presented. The recipe C_2^{\perp} \subseteq C_1, where C_1 and C_2 are classical codes, is used to obtain codes for up to 16 information qubits with correction of small numbers of errors. The results are tabulated. More efficient codes are obtained by allowing C_1 to have reduced distance, and introducing sign changes among the code words in a systematic manner. This systematic approach leads to single-error correcting codes for 3, 4 and 5 information qubits with block lengths of 8, 10 and 11 qubits respectively.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Methods of finding good quantum error correcting codes are discussed, and many example codes are presented.	contribution 0.99	0.78	0.17	asserted 0.99
s1	The recipe C_2^{\perp} \subseteq C_1, where C_1 and C_2 are classical codes, is used to obtain codes for up to 16 information qubits with correction of small numbers of errors.	method 0.95	0.78	0.21	asserted 1.00
s2	The results are tabulated.	result 0.42	0.75	0.04	asserted 0.94
s3	More efficient codes are obtained by allowing C_1 to have reduced distance, and introducing sign changes among the code words in a systematic manner.	method 0.82	0.85	0.22	asserted 1.00
s4	This systematic approach leads to single-error correcting codes for 3, 4 and 5 information qubits with block lengths of 8, 10 and 11 qubits respectively.	result 1.00	0.88	0.27	asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from proof or derivation 0.71.

What was examined, and what was not

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arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

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P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

Methods of finding good quantum error correcting codes are discussed, and many example codes are presented. The recipe C_2^{\perp} \subseteq C_1, where C_1 and C_2 are classical codes, is used to obtain codes for up to 16 information qubits with correction of small numbers of errors. The results are tabulated. More efficient codes are obtained by allowing C_1 to have reduced distance, and introducing sign changes among the code words in a systematic manner. This systematic approach leads to single-error correcting codes for 3, 4 and 5 information qubits with block lengths of 8, 10 and 11 qubits respectively.

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References

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound
quant-ph 1996-04 accept novelty 8.0 of 10

An infinite family of quantum codes with n = 2^j physical qubits encoding n - j - 2 qubits against one error saturates the quantum Hamming bound exactly, constructed via a new stabilizer-group method.
Hyper-optimized Quantum Lego Contraction Schedules
quant-ph 2025-10 conditional novelty 7.0 of 10

A new Sparse Stabilizer Tensor cost function enables hyper-optimized contraction schedules for Quantum LEGO WEP calculations, delivering orders-of-magnitude improvements over dense tensor baselines for stabilizer codes.
The Utility of Sparse Error Detection in Quantum Simulations
quant-ph 2026-08 conditional novelty 6.0 of 10

Sparse error detection in small Iceberg codes reduces systematic errors in simulated Schwinger-model observables under depolarizing noise, with diminishing returns after a few detection layers.
Realizing Error Suppression in Partially Fault-Tolerant Quantum Simulations with IBM Quantum Computers
quant-ph 2026-07 conditional novelty 6.0 of 10

Partially fault-tolerant [[4,2,2]] Iceberg-code simulations on ibm_boston improve local Ising observables over unencoded baselines by a few percent in 1D and over 200% in 2D at late times via Observable-Ranked Postselection.
Complex instruction set computing architecture for performing accurate quantum $Z$ rotations with less magic
quant-ph 2013-02 conditional novelty 5.0 of 10

Direct distillation of Z(π/2^k) magic states with shortened quantum Reed-Muller codes reduces resource-state counts for accurate Z rotations compared with a T-gate RISC architecture, for target errors above roughly 10^-10.

Show 1 more citing papers

Fundamental Physics at the Frontier of Noisy Quantum Computation
quant-ph 2026-09

How later papers used it

90 later papers cite it

Evidence: some. 90 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 7 of the 119 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool.

Uses its method or tool 2 of 4 papers with this label

Used the [[7,1,3]] Steane code from this paper to prepare a high-fidelity Bell state.

Demonstration of logical qubits and repeated error correction with better-than-physical error rates (2024) · 0.88 · paraphrased: the licence does not allow quoting
Encoded ancilla qubits in a 7-qubit Steane code from this paper so that only Clifford operations are applied to the lattice gauge theory qubits.

Gauss law codes and vacuum codes from lattice gauge theories (2026) · 0.87 · paraphrased: the licence does not allow quoting

How the literature uses it

15 arXiv papers cite this in 18 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 9 sentences without a confident label

Uses its method or tool 1 paper

The Trotterization applies exponentiations of Pauli operators in the Hamiltonian, which are typically approximated using many $T$ gates. ] can be performed on ancilla qubits encoded in a 7-qubit Steane code [ref] , so that one must apply only Clifford operations to the qubits of the lattice gauge theory.

Gauss law codes and vacuum codes from lattice gauge theories · Connections with Quantum Simulations · 0.87

Cites it as background 7 papers

A general framework for doing this is quantum error correction (QEC) [ref] , which encode a logical qubit containing the quantum information on multiple physical qubits, and then usually by performing syndrome measurements on these physical qubits, we can determine a certain set of errors, and apply the corresponding correction [ref] .

5-qubit quantum error correction in a charge qubit quantum computer · Quantum error correction with a perfect 5 qubit code · 0.83
The introduction of magic state distillation by Bravyi and Kitaev [ref] led to the construction of a sequence of CSS codes [ref] , where the code space is preserved by a transversal $Z$-rotation of the underlying physical space [ref] .

Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers · Discussion of Main Results · 0.94
Performing quantum simulations with sufficient precision to compute many scientifically relevant observables requires levels of error suppression attainable only through quantum error correction (QEC) [ref] .

Realizing Error Suppression in Partially Fault-Tolerant Quantum Simulations with IBM Quantum Computers · Introduction · 0.84

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