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Clifford group, stabilizer states, and linear and quadratic operations over GF(2)
Open accessAbstract
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 $(2n+1)\ifmmode\times\else\texttimes\fi{}(2n+1)$ 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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“To prove Eq. (5.4) using the semidefinite program in [49] first note that Φ = E 1 -E 2 has action,”
From Characterizing quantum gates via randomized benchmarking (2012) · cites this paper as [49]
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Cites
Quantum Computation and Quantum Information
Nielsen, M. A., Chuang, I., & Grover, L. K. (2002) · American Journal of Physics · doi:10.1119/1.1463744 · Cited by 21,895
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Reconstruction of Multi-user Binary Subspace Chirps
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“Based on the underlying binary symplectic geometry, we provide a systematic way of looking at the reconstruction algorithm by making use of stabilizer states [14] and related notions in quantum computation.”
MethodImproved quantum state tomography for systems with XX+YY couplings and Z readouts
Xin, T. (2020) · Physical Review A · doi:10.1103/physreva.102.052410 · Cited by 12
A Full Dichotomy for $\hol^{c}$, Inspired by Quantum Computation
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“1 Specifically, we demonstrate that these rewrites reduce diagrams to a normal form that we call the affine with phases (AP) form, which originally appeared in [30].”
BackgroundMapping quantum circuits to shallow-depth measurement patterns based on graph states
Kaldenbach, T. N., & Heller, M. (2024) · Quantum Science and Technology · doi:10.1088/2058-9565/ad802b · Cited by 13
“Since we sample from a stabilizer state, all bit strings (with non-zero amplitudes) appear with equal probability [52].”
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BackgroundFast simulation of planar Clifford circuits
Gosset, D., Grier, D., Kerzner, A., & Schaeffer, L. (2024) · Quantum · doi:10.22331/q-2024-02-12-1251 · Cited by 11
“Most of the results in this section follow from the work of Dehaene and De Moor [43] which describes the composition of Clifford operations as the product of binary matrices.11 We begin with a short summary of their techniques.”
MethodResultInfluentialImprovement of Stabilizer-Based Entanglement Distillation Protocols by Encoding OperatorsPreprint
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New techniques for bounding stabilizer rank
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BackgroundInfluentialSymbolic Synthesis of Clifford Circuits and Beyond
Amy, M., Bennett-Gibbs, O., & Ross, N. J. (2023) · Electronic Proceedings in Theoretical Computer Science · doi:10.4204/eptcs.394.17 · Cited by 7
“In the context of stabilizer states — states |ψ〉 = C |~0〉 for some C ∈ Cn — this representation is well-known by various names, including the quadratic form expansion [14] and the affine representation [16, 26].”
BackgroundCovariant mutually unbiased bases
Carmeli, C., Schultz, J., & Toigo, A. (2016) · Reviews in Mathematical Physics · doi:10.1142/s0129055x16500094 · Cited by 6
Edge local complementation for logical cluster states
Joo, J., & Feder, D. L. (2011) · New Journal of Physics · doi:10.1088/1367-2630/13/6/063025 · Cited by 6
Clifford gates in the Holant framework
Cai, J., Guo, H., & Williams, T. (2018) · Theoretical Computer Science · doi:10.1016/j.tcs.2018.06.010 · Cited by 7
Authenticity, Integrity, and Replay Protection in Quantum Data Communications and Networking
Barbeau, M., Kranakis, E., & Perez, N. (2022) · ACM Transactions on Quantum Computing · doi:10.1145/3517341 · Cited by 8
“That is, there is an equivalence between the quotient group C (n)/Gn and the group Sp (2n) [19, 21, 29, 30, 47].”
BackgroundOpening Krylov Space to Access All-Time Dynamics via Dynamical Symmetries
Loizeau, N., Buča, B., & Sels, D. (2025) · Physical Review Letters · doi:10.1103/qpyp-1mfj · Cited by 8
Kerdock Codes Determine Unitary 2-Designs
Can, T., Rengaswamy, N., Calderbank, R., & Pfister, H. D. (2019) · IEEE International Symposium on Information Theory (ISIT) · doi:10.1109/isit.2019.8849504 · Cited by 4
“These eigenvectors are called stabilizer states in the quantum information literature [27] (also see end of Section III).”
Fully-Parallel Quantum Turbo Decoder
Babar, Z., Nguyen, H. V., Botsinis, P., Alanis, D., Chandra, D., Ng, S. X., Maunder, R. G., & Hanzo, L. (2016) · IEEE Access · doi:10.1109/access.2016.2581978 · Cited by 6
“Both the encoders V1 and V2 are Clifford unitary operators [38], acting on n1 and n2 qubits, respectively.”
BackgroundClassical codes in quantum state space
Howard, M. (2015) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/48/49/495303 · Cited by 5
“(46) we can identify them as stabilizer states [20], just as we had in the odd prime power case.”
BackgroundDetermination of stabilizer states
Wu, X., Yang, Y., Wang, Y., Wen, Q., Qin, S., & Gao, F. (2015) · Physical Review A · doi:10.1103/physreva.92.012305 · Cited by 5
Improvement of stabilizer-based entanglement distillation protocols by encoding operators
Watanabe, S., Matsumoto, R., & Uyematsu, T. (2006) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/39/16/013 · Cited by 5
“Although the method for enumerating all elements of the Clifford group is known [8, 17], the method for enumerating all elements of Mcl(S) for a given stabilizer S is not known.”
BackgroundMethodFrom Holant to Quantum Entanglement and BackPreprint
Cai, J., Fu, Z., & Shao, S. (2020) · arXiv (Cornell University) · doi:10.4230/lipics.icalp.2020.22 · Cited by 12
Single-Copy Stabilizer Testing
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Tan, S., Lu, L., Xiang, D., Chu, T., Lang, C., Chen, J., Hu, X., & Yin, J. (2025) · Proceedings of the ACM on software engineering. · doi:10.1145/3715751 · Cited by 9
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Hamaguchi, H., Hamada, K., Marumo, N., & Yoshioka, N. (2025) · Physical Review Applied · doi:10.1103/physrevapplied.23.014069 · Cited by 5
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Song, H.-J., & Zhou, D. L. (2021) · Physics Letters A · doi:10.1016/j.physleta.2021.127298 · Cited by 4
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Guan, C., & Regan, K. W. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1904.00101 · Cited by 4
Sun, S., Goodenough, K., Bhatti, D., Azuma, K., & Elkouss, D. (2025) · Physical Review A · doi:10.1103/95k8-3zxd · Cited by 3
“[34] and Ref.”
Tailoring Fault-Tolerance to Trotter Circuits
Chen, Z., & Rengaswamy, N. (2024) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce60285.2024.00025 · Cited by 3
“For this work we set θ = π 2 throughout, i.e., consider Clifford [14] QSK (C-QSK) circuits, to take the first steps towards the above goal.”
Tailoring Fault-Tolerance to Quantum Algorithms
Chen, Z., & Rengaswamy, N. (2025) · IEEE Journal on Selected Areas in Information Theory · doi:10.1109/jsait.2025.3602446 · Cited by 4
“For this work, we set θ = π 2 throughout, i.e., consider Clifford [15] QSK (C-QSK) circuits, to take the first steps towards the above goal.”
Abelian Hypergroups and Quantum ComputationPreprint
Bermejo-Vega, J., & Zatloukal, K. C. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1509.05806 · Cited by 4
“The techniques we develop to cope with the issues are unique in the stabilizer formalism literature since both the original PSF and all of its previously known extensions [37–42, 36, 61–65, 59, 66] were tailored to handle unitary monomial stabilizer matrices.”
MethodNormalizer Circuits and Quantum ComputationPreprint
Bermejo-Vega, J. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1611.09274 · Cited by 8
“Indeed for any n-qubit stabilizer state |ψ〉 there exists a (poly-size) unitary Clifford circuit C such that |ψ〉 = γC|0〉⊗n, for some global phase γ [135].”
BackgroundMethodInfluentialImproved Strong Simulation of Universal Quantum CircuitsPreprint
Kocia, L. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2012.11739 · Cited by 3
“m-qubit stabilizer states can be expressed as a quadratic form defined on an affine space in the computational basis [24, 25]: |Ψ〉 = 2−m2 ∑”
BackgroundDistilling GHZ States using Stabilizer CodesPreprint
Rengaswamy, N., Raina, A., Raveendran, N., & Vasic, B. V. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2109.06248 · Cited by 8
“For example, the local Clifford on C must be determined by solving a set of linear equations and finding a binary symmetric matrix that specifies the diagonal Clifford, via the connection to binary symplectic matrices [32], [33].”
Clifford-dressed variational principles for precise Loschmidt echoes
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Measurement-based quantum compiling via gauge invariance
Corli, S., & Prati, E. (2026) · Physical Review Applied · doi:10.1103/lpkt-zgfn · Cited by 4
Polynomially restricted operator growth in dynamically integrable models
Ermakov, I., Byrnes, T., & Lychkovskiy, O. V. (2025) · Physical review. B./Physical review. B · doi:10.1103/physrevb.111.094314 · Cited by 2
“The efficient implementation of Algorithm 1 can be achieved by using specialized methods that enable the effective representation of Pauli strings in computer memory and allow for the fast computation of commutation relations between them 11,53 .”
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Castryck, W., Demeyer, J., Vos, A. D., Keszöcze, O., & Soeken, M. (2018) · IEEE International Symposium on Multiple-Valued Logic (ISMVL) · doi:10.1109/ismvl.2018.00051 · Cited by 2
“Together with operations described by translation matrices, of which the Hadamard matrix forms a special case, many gate libraries such as the Clifford group [1], [2], the Clifford+T group [3], [4], or recently the Clifford+Tn group [5] appear in the literature.”
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Luo, M.-X., Fei, S.-M., & Chen, J. (2022) · iScience · doi:10.1016/j.isci.2022.103972 · Cited by 2
Lai, C., Tseng, P.-C., & Yu, W.-H. (2024) · IEEE Transactions on Information Theory · doi:10.1109/tit.2024.3433550 · Cited by 2
“Using the canonical representation of stabilizer states [48], [49], we know that the conjugate state of a stabilizer state | S ⟩ is of the form : where ζ is an n -fold Pauli operator that is the tensor product of I ’s and σ z ’s. Theorem 7: Let Q be an (( n, M, d ; 0)) CWS code with CWS group S and…”
BackgroundCharacterization of non-adaptive Clifford channels
Yashin, V. I., & Elovenkova, M. A. (2025) · Quantum Information Processing · doi:10.1007/s11128-025-04682-0 · Cited by 2
“…efficiently solvable on classical computers using various approaches: stabilizer tableau methods [3, 10, 11, 34], graph state representations [35, 36], quadratic form expansions [12, 21, 37, 38], Wigner functions and other quasiprobability representations [7, 39–44], hidden variable models [45, 46].”
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Magesan, E. (2012) · UWSpace (University of Waterloo) · Cited by 4
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Rengaswamy, N. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2004.06834 · Cited by 2
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de Jong, J., Hahn, F., Tcholtchev, N., Hauswirth, M., & Pappa, A. (2022) · arXiv (Cornell University) · doi:10.17169/refubium-43258 · Cited by 2
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Gerhard, C., & Brun, T. A. (2026) · Physical Review A · doi:10.1103/7pv5-46nd · Cited by 8
“Although some readers may already be familiar with the binary symplectic representation, we briefly introduce it for completeness (see [58], [59] and [60] for a more complete treatment).”
Quantum Advantage via Efficient Postprocessing on Qudit Classical Shadow Tomography
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Group-theoretical approach to the construction of bases in 2 n -dimensional Hilbert space
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Global Synthesis of CNOT Circuits with Holes
Murphy, E., & Kissinger, A. (2023) · Electronic Proceedings in Theoretical Computer Science · doi:10.4204/eptcs.384.5 · Cited by 1
“Similar ideas involving the decomposition of symplectic matrices into basic generators have also been applied for synthesising Clifford circuits from stabiliser tableaux [1, 13, 6, 17].”
MethodExtended Binary Chirps Codebooks for Non-Coherent Communications
Bayanifar, M., Heikkilä, E., Calderbank, R., & Tirkkonen, O. (2023) · Globecom · doi:10.1109/globecom54140.2023.10437925 · Cited by 1
“The Clifford group which lives in the 2nd Clifford hierarchy level, is a group of unitary operations that maps the Pauli to Pauli under conjugation and play central role in the theory of the stabilizer codes [7], [8].”
BackgroundTargeted Clifford logical gates for hypergraph product codes
Patra, A., & Barg, A. (2025) · Quantum · doi:10.22331/q-2025-08-29-1842 · Cited by 5
“Thereafter, one can convert any of these symplectic matrices into circuits using elementary Clifford gates [13, 33].”
Probabilistic Representation of Commutative Quantum Circuit Models
Yu, F. R., Osorio, J. R., & Wong, E. (2025) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce65121.2025.10348 · Cited by 1
“Expressiveness is related to computational complexity [5] as well as trainability [3], and has been quantified in various ways, including the rank of the Jacobian matrix [6] or quantum Fisher information matrix [7], distance to a quantum t -design [2], [3], or closeness of a pairwise fidelity…”
Parity flow formalism: Tracking quantum information throughout computation
Klaver, B., Ludwig, K., Messinger, A., Rombouts, S. M. A., Fellner, M., Ender, K., & Lechner, W. (2025) · Physical Review Research · doi:10.1103/6xlb-l92j · Cited by 3
“[see Thm. 1 in [34], Sec.”
InfluentialTheory of measurement-based quantum computingPreprint
de Beaudrap, N., & Niel, J. R. (2008) · arXiv (Cornell University) · Cited by 1
Quantum Theory from Principles, Quantum Software from DiagramsPreprint
van de Wetering, J. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2101.03608 · Cited by 4
“The Eilenberg-Moore category for the Kalmbach monad is isomorphic to the category of effect algebras, and hence effect algebras form algebras in the category of bounded posets.”
MethodQUANTUM ENTANGLEMENT DISTILLATION IN THE STABILIZER FORMALISM
Ingenieurswetenschappen, F., & Elektrotechniek, D. (2007)
Maximally symmetric stabilizer MUBs in even prime-power dimensions
Carmeli, C., Schultz, J., & Toigo, A. (2017) · Journal of Mathematical Physics · doi:10.1063/1.4977830
Quadratic Form Expansions for UnitariesPreprint
de Beaudrap, N., Vincent Danos, A., Kashefi, E., & Roetteler, M. (2008) · arXiv (Cornell University) · doi:10.48550/arxiv.0801.2461
“Then Theorem 6 of [ 17 ] states that the unitary operation U for the Clifford operation characterized by (C, h) is given by the matrix formula”
BackgroundMethodResultInfluentialSampling Strategy Optimization for Randomized Benchmarking
Itoko, T., & Raymond, R. (2021) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce52317.2021.00036
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Butt, F., Heußen, S., Rispler, M., & Müller, M. (2023) · arXiv (Cornell University) · doi:10.48550/arxiv.2306.17686
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Bocharov, A. (2023) · Contemporary mathematics - American Mathematical Society · doi:10.1090/conm/789/15840
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Novak, O., & Rengaswamy, N. (2024) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce60285.2024.00022
“We can describe a stabilizer code as a 2 n × 2 n matrix U which has the following form [11], [12], [10]: where its rows ⟨ u 1 , . . . , u 2 n ⟩ span F 2 n 2 .”
BackgroundMethodResultInfluentialApproximate Optimisation of Quantum Circuits Using the ZX Calculus with Phase Squashing
Kelly, T., & Kissinger, A. (2025) · Lecture notes in computer science · doi:10.1007/978-3-031-97063-4_8
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“A DRB circuit begins with stabilizer state preparation [34, 35] from | 0 ⟩ via the noisy initialization operation ˜ G I , followed by m randomly selected layers of noisy quantum processes ˜ G 1 , . . . , ˜ G m , and ends with the application of ˜ G M prior to measurement (see Fig.”
A Streamlined Demonstration That Stabilizer Circuits Simulation Reduces to Boolean Linear Algebra
Yashin, V. I. (2025) · Lobachevskii Journal of Mathematics · doi:10.1134/s1995080225607842
Nonstabilizerness as a diagnostic of criticality and exceptional points in non-Hermitian spin chains
Moca, C. P., Sticlet, D., & Dóra, B. (2026) · Physical review. B./Physical review. B · doi:10.1103/6vv5-tsql
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“Then, the diagonal matrix D = 2 s/ 2 ( d + κ ) is by construction Clifford because its diagonal is proportional to a (full-support) stabilizer state [78, 79].”
Páez-Velasco, P., Schilling, N., Scalet, S. O., Verstraete, F., & Capel, Á. (2026) · Physical Review Research · doi:10.1103/sbxq-sqvp · Cited by 8
“In-deed, as any Hamiltonian composed of commuting Pauli strings can be diagonalized using a poly-depth Clifford circuit [17, 18], it is possible to perform efficient Gibbs sampling for any such Hamiltonian by sampling its classical poly-depth dual counterparts, provided these can be efficiently…”
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Kožić, S. B., Torre, G., & Giampaolo, S. M. (2026) · Journal of Statistical Mechanics Theory and Experiment · doi:10.1088/1742-5468/ae1ff9
Exact stabilizer scars in two-dimensional U ( 1 ) lattice gauge theory
Gupta, S., Sierant, P., Santos, L., & Stornati, P. (2026) · Physical review. D/Physical review. D. · doi:10.1103/hvdp-4m6f
Quantum simulation in imaginary time for gauge-invariant models
Cuomo, D., & van Leeuwen, R. (2026) · Physica Scripta · doi:10.1088/1402-4896/ae4dce
Efficient classical computation of the neural tangent kernel of quantum neural networks
Hernandez, A. M., Pastorello, D., & Palma, G. D. (2026) · Quantum · doi:10.22331/q-2026-05-29-2118
Limits of Clifford disentangling in tensor network states
Masot-Llima, S., Sierant, P., Stornati, P., & García-Sáez, A. (2026) · Physical review. B./Physical review. B · doi:10.1103/5g5b-5rrz
“It can also be shown with the reduced matrix of qubit n ; if this was a stabilizer state, the reduced matrix should be either a pure state or the maximally mixed state (this can be proven using the closed form of stabilizer states [105, 106]).”
Predicting Magic from Very Few Measurements
Varela, J. M., Keller, L. L., de Oliveira, A., Moreira, D. A., Chaves, R., & Macêdo, R. A. (2026) · PRX Quantum · doi:10.1103/2xnf-jcyn
Ünal, Z., & Vurdu, C. D. (2026) · Mathematics · doi:10.3390/math14203693
Jardine, J. (2026) · Zenodo (CERN European Organization for Nuclear Research) · doi:10.5281/zenodo.21658097
Jardine, J. (2026) · Zenodo (CERN European Organization for Nuclear Research) · doi:10.5281/zenodo.21680387
Metadata
- Authors
- Dehaene, J., & De Moor, B.
- Affiliations
- KU Leuven (BE)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2003-10-20
- Volume
- 68
- Issue
- 4
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2009169733
- Cited by
- 183
- References
- 3
- Citations by year
- 2026: 16 · 2025: 21 · 2024: 14 · 2023: 14 · 2022: 16 · 2021: 10 · 2020: 13 · 2019: 9 · 2018: 6 · 2017: 6 · 2016: 7 · 2015: 6 · 2014: 7 · 2013: 2 · 2012: 5
- FWCI
- 5.32
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Lirias (submittedVersion, other-oa)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- Clifford group, stabilizer states, binary quadratic forms, binary vector spaces, GF(2), group isomorphism, quantum error correction, entanglement distillation
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Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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