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The Clifford group, stabilizer states, and linear and quadratic operations over GF(2)

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Jeroen Dehaene, Bart De Moor

Quantum Physics · Submitted 18 Apr 2003

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

The authors' main claim We show how the n-qubit Clifford group is isomorphic to a group with an operation that is defined in terms of a (2n+1)x(2n+1) binary matrix product and binary quadratic forms.

how it is stated · asserted 1.00 read the record →

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arXiv IDquant-ph/0304125 (version 1)
Pith IDS7ODI2VE, Pith's permanent record of this paper and its review
CitationBart De Moor, Jeroen Dehaene. "The Clifford group, stabilizer states, and linear and quadratic operations over GF(2)." arXiv:quant-ph/0304125 (2003). https://pith.science/paper/S7ODI2VE
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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 0.20 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.27 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We describe stabilizer states and Clifford group operations using linear operations and quadratic forms over binary vector spaces. We show how the n-qubit Clifford group is isomorphic to a group with an operation that is defined in terms of a (2n+1)x(2n+1) binary matrix product and binary quadratic forms. As an application we give two schemes to efficiently decompose Clifford group operations into one and two-qubit operations. We also show how the coefficients of stabilizer states and Clifford group operations in a standard basis expansion can be described by binary quadratic forms. Our results are useful for quantum error correction, entanglement distillation and possibly quantum computing.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We describe stabilizer states and Clifford group operations using linear operations and quadratic forms over binary vector spaces. contribution 0.87 0.79 0.31 asserted 1.00
s1 We show how the n-qubit Clifford group is isomorphic to a group with an operation that is defined in terms of a (2n+1)x(2n+1) binary matrix product and binary quadratic forms. result 0.81 0.93 0.44 asserted 1.00
s2 As an application we give two schemes to efficiently decompose Clifford group operations into one and two-qubit operations. contribution 0.87 0.95 0.16 asserted 1.00
s3 We also show how the coefficients of stabilizer states and Clifford group operations in a standard basis expansion can be described by binary quadratic forms. result 0.75 0.93 0.16 asserted 1.00
s4 Our results are useful for quantum error correction, entanglement distillation and possibly quantum computing. interpretation 0.98 0.40 0.13 hedged possibility 0.74

What kind of paper the classifier reports this is: theoretical result 0.77 · evidence from proof or derivation 1.00.

What was examined, and what was not

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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • 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

We describe stabilizer states and Clifford group operations using linear operations and quadratic forms over binary vector spaces. We show how the n-qubit Clifford group is isomorphic to a group with an operation that is defined in terms of a (2n+1)x(2n+1) binary matrix product and binary quadratic forms. As an application we give two schemes to efficiently decompose Clifford group operations into one and two-qubit operations. We also show how the coefficients of stabilizer states and Clifford group operations in a standard basis expansion can be described by binary quadratic forms. Our results are useful for quantum error correction, entanglement distillation and possibly quantum computing.

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References
How later papers used it

78 later papers cite it

Evidence: substantial. 78 of the citing papers share no author with it, and 15 of those carry a confident label from Pith's citation classifier.

Pith read 14 of the 142 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

Builds on it 1 of 2 papers with this label

  • Building on its results, showed how to classically compute a minimal pattern in time O(n^3 / log n) by solving the MPI for a quadratic form expansion for U.

    Quadratic Form Expansions for Unitaries (2008) · 0.87 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 of 12 papers with this label
How the literature uses it

18 arXiv papers cite this in 45 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 · 28 sentences without a confident label

Builds on it 2 papers
  • To achieve an efficient classical simulation of normalizer circuits over $^a×^b ×F$ (theorem [ref]), we develop new stabilizer formalism techniques which extend the stabilizer formalism for finite Abelian groups developed in [ref] (which was in turn a generalization of the well-known stabilizer formalism for qubit/qudit systems [ref] ) .

    Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems · Introduction · 0.98

  • Building on the results of [ref] , we show how to classically compute such a minimal pattern in time $O(n^3 / logn)$ by solving the MPI for a quadratic form expansion for $U$.

    Quadratic Form Expansions for Unitaries · Synthesizing measurement patterns for the Clifford group · 0.87

Uses its method or tool 4 papers
  • Provided a Leuven tableau $(C,h⃗)$ for a Clifford group operation $U$, [ref] provides a matrix formula for $U$ which we may obtain for $U$, as follows.

    Quadratic Form Expansions for Unitaries · Obtaining a quadratic form expansion. · 0.96

  • In particular, Appendix [ref] shows how the problem size can be reduced for channels diagonal in the computational basis, using a representation of stabiliser states in terms of affine spaces over binary vectors due to Dehaene and De Moor [ref] .

    Quantifying magic for multi-qubit operations · Multiqubit phase gates · 0.96

  • We represent operators as binary symplectic vectors (BSV) [ref] for $X$ and $Z$ components defined for $n$-qubit operator $Â$ as

    Decoding Holographic Codes with an Integer Optimisation Decoder · Introduction · 0.90

Compares with it 1 paper
  • This difference from Refs. [ref] is significant when using Weyl operators to represent Pauli operators, as illustrated in Eq. [ref]; and varying coefficients of the tableau by $±d$ may not preserve the property of the tableau being proper.

    A linearized stabilizer formalism for systems of finite dimension · Stabilizer tableaus for qudits · 0.80

Cites it as background 6 papers
  • For the sake of completeness, we outline the relevant results of [ref] .

    Quadratic Form Expansions for Unitaries · Obtaining a quadratic form expansion. · 0.93

  • Aside from generalizations of Clifford circuits [ref] (which includes normalizer circuits), many other classes of restricted quantum circuits have been studied in the literature.

    The computational power of normalizer circuits over black-box groups · Relationship to previous work · 0.90

  • We use standard surface-code facts without reproving them, and the stabilizer formalism describes the code and its logical operators [ref] , while the toric-code phase and its gapped boundaries describe the anyons, the boundaries, and the edge condensates [ref] .

    The resource cost of magic in a code block · Setup and definitions · 0.97

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