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.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
For generalized Pauli channels E1 and E2, the diamond norm distance is given by Eq. (5.4).
“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]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
<i>Quantum Computation and Quantum Information</i>
Nielsen, M. A., Chuang, I., & Grover, L. K. (2002) · American Journal of Physics · doi:10.1119/1.1463744 · Cited by 21,895 · see citing works
The Heisenberg Representation of Quantum Computers
Gottesman, D. (1998) · University of North Texas Digital Library (University of North Texas) · Cited by 385 · see citing works
Local permutations of products of Bell states and entanglement distillation
Dehaene, J., den Nest, M. V., Moor, B. D., & Verstraete, F. (2003) · Physical Review A · doi:10.1103/physreva.67.022310 · Cited by 84 · see citing works
Self-citation
Flagged first, then by citations
Research works printed but not linked: 1
- [1]D. Gottesman, Ph.D. thesis, Caltech (1997), quantph/9705052.
Cited by
Sorted by citations
Improved simulation of stabilizer circuits
Aaronson, S., & Gottesman, D. (2004) · Physical Review A · doi:10.1103/physreva.70.052328 · Cited by 1,515 · see citing works
Characterizing quantum gates via randomized benchmarking
Magesan, E., Gambetta, J. M., & Emerson, J. (2012) · Physical Review A · doi:10.1103/physreva.85.042311 · Cited by 464 · see citing works
A Universal Operator Growth Hypothesis
Parker, D. E., Cao, X., Avdoshkin, A., Scaffidi, T., & Altman, E. (2019) · Physical Review X · doi:10.1103/physrevx.9.041017 · Cited by 482 · see citing works
“[105] Jeroen Dehaene and Bart De Moor, “Clifford group, stabilizer states, and linear and quadratic operations over gf(2),” Phys.”
BackgroundHudson’s theorem for finite-dimensional quantum systems
Gross, D. (2006) · Journal of Mathematical Physics · doi:10.1063/1.2393152 · Cited by 366 · see citing works
Trading Classical and Quantum Computational Resources
Bravyi, S., Smith, G., & Smolin, J. (2015) · Physical Review X · doi:10.1103/physrevx.6.021043 · Cited by 458 · see citing works
Graphical description of the action of local Clifford transformations on graph states
Van den Nest, M., Dehaene, J., & De Moor, B. (2004) · Physical Review A · doi:10.1103/physreva.69.022316 · Cited by 335 · see citing works
Self-citationSymmetric informationally complete–positive operator valued measures and the extended Clifford group
Appleby, D. M. (2005) · Journal of Mathematical Physics · doi:10.1063/1.1896384 · Cited by 304 · see citing works
“ar X iv :q ua nt -p h/ 04 12 00 1v 2 1 7 Ja n 20 05 SIC-POVMs AND THE EXTENDED CLIFFORD GROUP”
Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus
Duncan, R., Kissinger, A., Perdrix, S., & van de Wetering, J. (2020) · Quantum · doi:10.22331/q-2020-06-04-279 · Cited by 201 · see citing works
Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic
Hostens, E., Dehaene, J., & Moor, B. D. (2005) · Physical Review A · doi:10.1103/physreva.71.042315 · Cited by 137 · see citing works
Self-citationReducing Qubit Requirements for Quantum Simulations Using Molecular Point Group Symmetries
Setia, K., Chen, R., Rice, J. E., Mezzacapo, A., Pistoia, M., & Whitfield, J. D. (2020) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.0c00113 · Cited by 121 · see citing works
Quantifying magic for multi-qubit operations
Seddon, J. R., & Campbell, E. T. (2019) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2019.0251 · Cited by 117 · see citing works
“In particular, electronic supplementary material, section V shows how the problem size can be reduced for channels diagonal in the computational basis, using a representation of stabilizer states in terms of affine spaces over binary vectors due to Dehaene and De Moor [55,56].”
BackgroundDirect Randomized Benchmarking for Multiqubit Devices
Proctor, T. J., Carignan-Dugas, A., Rudinger, K., Nielsen, E., Blume-Kohout, R., & Young, K. (2019) · Physical Review Letters · doi:10.1103/physrevlett.123.030503 · Cited by 118 · see citing works
“S and v pairs define a valid Clifford – given any S, half of the possible vectors define a valid Clifford [74, 75]).”
BackgroundMethodInfluentialScalable Measures of Magic Resource for Quantum Computers
Haug, T., & Kim, M. S. (2023) · PRX Quantum · doi:10.1103/prxquantum.4.010301 · Cited by 104 · see citing works
Classical simulation versus universality in measurement-based quantum computation
Van den Nest, M., Dür, W., Vidal, G., & Briegel, H. J. (2007) · Physical Review A · doi:10.1103/physreva.75.012337 · Cited by 95 · see citing works
Experimental Estimation of Quantum State Properties from Classical Shadows
Struchalin, G. I., Zagorovskii, Y. A., Kovlakov, E. V., Straupe, S. S., & Kulik, S. P. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.010307 · Cited by 94 · see citing works
Campbell, E. T., & Howard, M. (2017) · Physical Review A · doi:10.1103/physreva.95.022316 · Cited by 91 · see citing works
Optimized Compiler for Distributed Quantum Computing
Cuomo, D., Caleffi, M., Krsulich, K., Tramonto, F., Agliardi, G., Prati, E., & Cacciapuoti, A. S. (2023) · ACM Transactions on Quantum Computing · doi:10.1145/3579367 · Cited by 94 · see citing works
How to efficiently select an arbitrary Clifford group element
König, R., & Smolin, J. A. (2014) · Journal of Mathematical Physics · doi:10.1063/1.4903507 · Cited by 84 · see citing works
Catalysis and activation of magic states in fault-tolerant architectures
Campbell, E. T. (2011) · Physical Review A · doi:10.1103/physreva.83.032317 · Cited by 78 · see citing works
Typical entanglement of stabilizer states
Smith, G., & Leung, D. (2006) · Physical Review A · doi:10.1103/physreva.74.062314 · Cited by 73 · see citing works
Constructing Smaller Pauli Twirling Sets for Arbitrary Error Channels
Cai, Z., & Benjamin, S. C. (2019) · Scientific Reports · doi:10.1038/s41598-019-46722-7 · Cited by 63 · see citing works
Efficient representation of topologically ordered states with restricted Boltzmann machines
Lu, S., Gao, X., & Duan, L.-M. (2019) · Physical review. B./Physical review. B · doi:10.1103/physrevb.99.155136 · Cited by 66 · see citing works
“[55] Jeroen Dehaene and Bart De Moor, “Clifford group, stabilizer states, and linear and quadratic operations over gf(2),” Phys.”
BackgroundResultQuantum computation via translation-invariant operations on a chain of qubits
Raussendorf, R. (2005) · Physical Review A · doi:10.1103/physreva.72.052301 · Cited by 55 · see citing works
EXIT-Chart-Aided Near-Capacity Quantum Turbo Code Design
Babar, Z., Ng, S. X., & Hanzo, L. (2014) · IEEE Transactions on Vehicular Technology · doi:10.1109/tvt.2014.2328638 · Cited by 53 · see citing works
Unifying Gate Synthesis and Magic State Distillation
Campbell, E. T., & Howard, M. (2017) · Physical Review Letters · doi:10.1103/physrevlett.118.060501 · Cited by 52 · see citing works
Hashagen, A. K., Flammia, S. T., Gross, D., & Wallman, J. J. (2018) · Quantum · doi:10.22331/q-2018-08-22-85 · Cited by 56 · see citing works
“[13, 32]) for generating a gate sequence that implements a given symplectic matrix as a Clifford operation, and 3) multiply with a randomly chosen element of E(n).”
MethodNumerically probing the universal operator growth hypothesis
Heveling, R., Wang, J., & Jochen Gemmer, B. (2022) · Physical review. E · doi:10.1103/physreve.106.014152 · Cited by 50 · see citing works
Noh, J. D. (2021) · Physical review. E · doi:10.1103/physreve.104.034112 · Cited by 49 · see citing works
Ji, Z., Chen, J.-X., Wei, Z., & Ying, M. (2010) · Quantum Information and Computation · doi:10.26421/qic10.1-2-8 · Cited by 46 · see citing works
Improved upper bounds on the stabilizer rank of magic states
Qassim, H., Pashayan, H., & Gosset, D. (2021) · Quantum · doi:10.22331/q-2021-12-20-606 · Cited by 43 · see citing works
“2A state is a stabilizer state if and only if its expansion in the computational basis is of the form |A|−1/2 ∑ x∈A i l(x)(−1)q(x)|x〉, where A is an affine subspace of F2 and l : F2 → F2 and q : F2 → F2 are linear and quadratic functions, respectively [12, 13].”
BackgroundThe Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure
Babar, Z., Botsinis, P., Alanis, D., Ng, S. X., & Hanzo, L. (2015) · IEEE Access · doi:10.1109/access.2015.2405533 · Cited by 40 · see citing works
“(69) carries out an n -qubit Clifford transformation, which maps an n -qubit Pauli group G n onto itself under conjugation [105], i.e. we have: In other word, a Clifford operation preserves the elements of the Pauli group under conjugation such that for P ∈ G n , VPV † ∈ G n . matrix Furthermore,…”
BackgroundUnveiling the Stabilizer Group of a Matrix Product State
Lami, G., & Collura, M. (2024) · Physical Review Letters · doi:10.1103/physrevlett.133.010602 · Cited by 45 · see citing works
Fast and Effective Techniques for T-Count Reduction via Spider Nest IdentitiesPreprint
de Beaudrap, N., Bian, X., & Wang, Q. (2020) · arXiv (Cornell University) · doi:10.4230/lipics.tqc.2020.11 · Cited by 42 · see citing works
Hashagen, A.-L. K., Flammia, S. T., Gross, D., & Wallman, J. J. (2018) · Kölner Universitäts PublikationsServer (Universität zu Köln) · Cited by 49 · see citing works
Learning Quantum Circuits of Some T Gates
Lai, C., & Cheng, H. (2022) · IEEE Transactions on Information Theory · doi:10.1109/tit.2022.3151760 · Cited by 36 · see citing works
“Remark: The above corollary is slightly weaken than [15, Lemma 2], where the Pauli operator σr0 is shown to be Z s for some s ∈ {0, 1} by exploiting the mathematical structure of a stabilizer state [37], [38].”
BackgroundLearning efficient decoders for quasichaotic quantum scramblers
Leone, L., Oliviero, S. F. E., Lloyd, S., & Hamma, A. (2024) · Physical Review A · doi:10.1103/physreva.109.022429 · Cited by 41 · see citing works
Classical simulation complexity of extended clifford circuits
Jozsa, R., & den Nest, M. V. (2014) · Cambridge University Engineering Department Publications Database · doi:10.5555/2638682.2638689 · Cited by 35 · see citing works
Stabilizer Notation for Spekkens’ Toy Theory
Pusey, M. F. (2012) · Foundations of Physics · doi:10.1007/s10701-012-9639-7 · Cited by 35 · see citing works
One T Gate Makes Distribution Learning Hard
Hinsche, M., Ioannou, M., Nietner, A., Haferkamp, J., Quek, Y., Hangleiter, D., Seifert, J., Eisert, J., & Sweke, R. (2023) · Physical Review Letters · doi:10.1103/physrevlett.130.240602 · Cited by 42 · see citing works
Classical simulation complexity of extended Clifford circuits
Jozsa, R., & den Nest, M. V. (2014) · Quantum Information and Computation · doi:10.26421/qic14.7-8-7 · Cited by 33 · see citing works
Quadratic Form Expansions for Unitaries
de Beaudrap, N., Vincent Danos, A., Kashefi, E., & Roetteler, M. (2008) · Lecture notes in computer science · doi:10.1007/978-3-540-89304-2_4 · Cited by 32 · see citing works
“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”
BackgroundMethodResultInfluentialA linearized stabilizer formalism for systems of finite dimension
de Beaudrap, N. (2013) · Quantum Information and Computation · doi:10.26421/qic13.1-2-6 · Cited by 31 · see citing works
“However, in existing algorithms [11, 13], the exact state of the system is determined by scalar factors which depend quadratically on the exponents of Pauli operators σ xσ b z ; similar remarks apply to the formalism of Ref.”
BackgroundMethodInfluentialAsymptotic adaptive bipartite entanglement-distillation protocol
Hostens, E., Dehaene, J., & Moor, B. D. (2006) · Physical Review A · doi:10.1103/physreva.73.062337 · Cited by 30 · see citing works
Self-citationClassical simulations of Abelian-group normalizer circuits with intermediate measurements
Bermejo-Vega, J., & den Nest, M. V. (2014) · Quantum Information and Computation · doi:10.26421/qic14.3-4-1 · Cited by 30 · see citing works
“(3) Here g ∈ G and χg is a homomorphism3 from G into the multiplicative group of nonzero complex numbers C∗, defined as χg(h) = exp ( 2πi m∑ i=1 g(i)h(i) di ) ; (4) the functions χg are known as the character functions of G.”
BackgroundSzymusiak, A., & Słomczyński, W. (2016) · Physical Review A · doi:10.1103/physreva.94.012122 · Cited by 29 · see citing works
Fault-Tolerant Code-Switching Protocols for Near-Term Quantum Processors
Butt, F., Heußen, S., Rispler, M., & Müller, M. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.020345 · Cited by 30 · see citing works
A non-commuting stabilizer formalism
Ni, X., Buerschaper, O., & den Nest, M. V. (2015) · Journal of Mathematical Physics · doi:10.1063/1.4920923 · Cited by 28 · see citing works
Quantum Entanglement: Separability, Measure, Fidelity of Teleportation, and Distillation
Li, M., Fei, S.-M., & Li‐Jost, X. (2010) · Advances in Mathematical Physics · doi:10.1155/2010/301072 · Cited by 26 · see citing works
Fast Stabiliser Simulation with Quadratic Form Expansions
de Beaudrap, N., & Herbert, S. (2022) · Quantum · doi:10.22331/q-2022-09-15-803 · Cited by 26 · see citing works
Logical Clifford Synthesis for Stabilizer Codes
Rengaswamy, N., Calderbank, R., Kadhe, S., & Pfister, H. D. (2020) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2020.3023419 · Cited by 25 · see citing works
“A closely related algorithm was given earlier by Dehaene and De Moor [56].”
Hashing protocol for distilling multipartite Calderbank-Shor-Steane states
Hostens, E., Dehaene, J., & Moor, B. D. (2006) · Physical Review A · doi:10.1103/physreva.73.042316 · Cited by 24 · see citing works
Self-citationMutually unbiased bases as minimal Clifford covariant 2-designs
Zhu, H. (2015) · Physical Review A · doi:10.1103/physreva.91.060301 · Cited by 24 · see citing works
Unifying the Clifford hierarchy via symmetric matrices over rings
Rengaswamy, N., Calderbank, R., & Pfister, H. D. (2019) · Physical Review A · doi:10.1103/physreva.100.022304 · Cited by 23 · see citing works
Enumerating all bilocal Clifford distillation protocols through symmetry reduction
Jansen, S., Goodenough, K., de Bone, S., Gijswijt, D., & Elkouss, D. (2022) · Quantum · doi:10.22331/q-2022-05-19-715 · Cited by 23 · see citing works
“This subgroup consists of all Pauli strings, and quotienting out the Cliffords by this subgroup leads to the symplectic group (over F2), Sp(2n,F2) [36, 39].”
Efficient rate-adaptive reconciliation for continuous-variable quantum key distribution
Wang, X., Zhang, Y.-C., Li, Z., Xu, B., Yu, S., & Guo, H. (2017) · Quantum Information and Computation · doi:10.26421/qic17.13-14 · Cited by 22 · see citing works
“Besides the stabilizer formalism, other techniques have been used to compute the outcome probabilities of Clifford circuits efficiently (for some examples, see [7, 8, 9, 10]).”
MethodHybrid Stabilizer Matrix Product Operator
Mello, A. F., Santini, A., & Collura, M. (2024) · Physical Review Letters · doi:10.1103/physrevlett.133.150604 · Cited by 22 · see citing works
Graph concatenation for quantum codes
Beigi, S., Chuang, I. L., Grassl, M., Shor, P. W., & Zeng, B. (2011) · Journal of Mathematical Physics · doi:10.1063/1.3534799 · Cited by 22 · see citing works
Learning stabilizer states by Bell samplingPreprint
Montanaro, A. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1707.04012 · Cited by 21 · see citing works
“By a result of [5] (see [11] for an alternative proof), up to an overall phase every stabilizer state |ψ〉 can be written in the form”
BackgroundGraph concatenation for quantum codesPreprint
Beigi, S., Chuang, I. L., Grassl, M., Zeng, B., & Shor, P. W. (2010) · DSpace@MIT (Massachusetts Institute of Technology) · Cited by 21 · see citing works
Clifford Dressed Time-Dependent Variational Principle
Mello, A. F., Santini, A., Lami, G., Nardis, J. D., & Collura, M. (2025) · Physical Review Letters · doi:10.1103/physrevlett.134.150403 · Cited by 22 · see citing works
“The algorithm allowing this is known as tableau algorithm, as it consists in storing and evolving efficiently a compact table representing the Pauli strings that stabilize the state [22, 23].”
Commuting quantum circuits: efficiently classical simulations versus hardness results
Ni, X., & den Nest, M. V. (2013) · Quantum Information and Computation · doi:10.26421/qic13.1-2-5 · Cited by 20 · see citing works
Extracting GHZ states from linear cluster states
de Jong, J., Hahn, F., Tcholtchev, N., Hauswirth, M., & Pappa, A. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.013330 · Cited by 19 · see citing works
A monomial matrix formalism to describe quantum many-body states
den Nest, M. V. (2011) · New Journal of Physics · doi:10.1088/1367-2630/13/12/123004 · Cited by 19 · see citing works
Efficient Learning of Quantum States Prepared With Few Fermionic Non-Gaussian Gates
Mele, A. A., & Herasymenko, Y. (2025) · PRX Quantum · doi:10.1103/prxquantum.6.010319 · Cited by 17 · see citing works
Stabilizer rank and higher-order Fourier analysis
Labib, F. (2022) · Quantum · doi:10.22331/q-2022-02-09-645 · Cited by 16 · see citing works
“It is well known that stabilizer states are quadratic forms defined on affine subspaces [7, 11, 14].”
BackgroundDecoding holographic codes with an integer optimization decoder
Harris, R. J., Coupe, E., McMahon, N. A., Brennen, G. K., & Stace, T. M. (2020) · Physical Review A · doi:10.1103/physreva.102.062417 · Cited by 18 · see citing works
Computing quopit Clifford circuit amplitudes by the sum-over-paths technique
Koh, D. E., Penney, M. D., & Spekkens, R. W. (2017) · Quantum Information and Computation · doi:10.26421/qic17.13-14-1 · Cited by 16 · see citing works
Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems
Bermejo-Vega, J., Lin, C. Y.-Y., & den Nest, M. V. (2016) · Quantum Information and Computation · doi:10.26421/qic16.5-6-1 · Cited by 16 · see citing works
“Keywords: classical simulations, stabilizer formalism, normalizer circuits, infinitedimensional quantum systems Communicated by: S Braunstein & B Terhal”
BackgroundLocal invariants of stabilizer codes
den Nest, M. V., Dehaene, J., & Moor, B. D. (2004) · Physical Review A · doi:10.1103/physreva.70.032323 · Cited by 16 · see citing works
Self-citationThe Classification of Clifford Gates over Qubits
Grier, D., & Schaeffer, L. (2022) · Quantum · doi:10.22331/q-2022-06-13-734 · Cited by 15 · see citing works
“In general, stabilizer states are of the form (unnormalized) ∑ x∈A(−1)i|x〉 where A is an affine space over F2, q(x) is a quadratic form, and l(x) is a linear form [12, 22].”
BackgroundWeaving, T., Ralli, A., Kirby, W. M., Tranter, A., Love, P. J., & Coveney, P. V. (2023) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.2c00910 · Cited by 14 · see citing works
Approximate stabilizer rank and improved weak simulation of Clifford-dominated circuits for qudits
Huang, Y., & Love, P. J. (2019) · Physical Review A · doi:10.1103/physreva.99.052307 · Cited by 15 · see citing works
“To prove this canonical form holds true for all qubit stabilizer states, one only need to make sure that every state in this form is the eigenstate of a stabilizer operator, as shown in [12].”
BackgroundClassification of small triorthogonal codes
Nezami, S., & Haah, J. (2022) · Physical Review A · doi:10.1103/physreva.106.012437 · Cited by 15 · see citing works
Depth-Optimal Synthesis of Clifford Circuits with SAT Solvers
Peham, T., Brandl, N., Kueng, R., Wille, R., & Burgholzer, L. (2023) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce57702.2023.00095 · Cited by 15 · see citing works
“representation of a finite symplectic group [38]–[40] with additional structure.”
BackgroundEfficient Inner-product Algorithm for Stabilizer StatesPreprint
García, H. J., Markov, I. L., & Cross, A. W. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1210.6646 · Cited by 15 · see citing works
Hostens, E., Dehaene, J., & Moor, B. D. (2006) · Physical Review A · doi:10.1103/physreva.74.062318 · Cited by 13 · see citing works
Self-citationLocal unitary classification for sets of generalized Bell states
Wu, B., Jiang, J., Zhang, J., Tian, G., & Sun, X. (2018) · Physical Review A · doi:10.1103/physreva.98.022304 · Cited by 13 · see citing works
Invariants of the local Clifford group
den Nest, M. V., Dehaene, J., & Moor, B. D. (2005) · Physical Review A · doi:10.1103/physreva.71.022310 · Cited by 14 · see citing works
“It is indeed well known that n-qubit (local) Clifford operations can be represented elegantly by a certain class of 2n × 2n linear operators over F2 [1][4] and this binary picture makes the (local) Clifford group particularly manageable in the following.”
BackgroundSelf-citationThe axiomatic and the operational approaches to resource theories of magic do not coincide
Heimendahl, A., Heinrich, M., & Gross, D. (2022) · Journal of Mathematical Physics · doi:10.1063/5.0085774 · Cited by 13 · see citing works
Principle of majorization: Application to random quantum circuits
Vallejos, R. O., de Melo, F., & Carlo, G. G. (2021) · Physical Review A · doi:10.1103/physreva.104.012602 · Cited by 13 · see citing works
A Non-Commuting Stabilizer FormalismPreprint
Ni, X., Buerschaper, O., & den Nest, M. V. (2014) · Cited by 15 · see citing works
Unitary partitioning and the contextual subspace variational quantum eigensolver
Ralli, A., Weaving, T., Tranter, A., Kirby, W. M., Love, P. J., & Coveney, P. V. (2023) · Physical Review Research · doi:10.1103/physrevresearch.5.013095 · Cited by 12 · see citing works
Entanglement Purification with Quantum LDPC Codes and Iterative Decoding
Rengaswamy, N., Raveendran, N., Raina, A., & Vasic, B. V. (2024) · Quantum · doi:10.22331/q-2024-01-24-1233 · Cited by 12 · see citing works
“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 [40], [41].”
MethodHardware-tailored diagonalization circuits
Miller, D., Fischer, L. E., Levi, K., Kuehnke, E. J., Sokolov, I. O., Barkoutsos, P. K., Eisert, J., & Tavernelli, I. (2024) · npj Quantum Information · doi:10.1038/s41534-024-00901-1 · Cited by 21 · see citing works
“In particular, L is non-empty if and only if it is possible to diagonalize P 1 , … , P m with a circuit based on Γ as in Fig.”
Quantum many-body simulations with PauliStrings.jl
Loizeau, N., Peacock, J. C., & Sels, D. (2025) · SciPost Physics Codebases · doi:10.21468/scipostphyscodeb.54 · Cited by 12 · see citing works
“[12].”
Pllaha, T., Tirkkonen, O., & Calderbank, A. R. (2022) · IEEE Transactions on Information Theory · doi:10.1109/tit.2022.3186872 · Cited by 12 · see citing works
“Indeed, we show that BSSCs are nothing else but stabilizer states [23], and their sparsity is determined by the diagonal portion of the corresponding stabilizer group; see Corollaries 2 and 3.”
BackgroundEntropic uncertainty relations and the stabilizer formalism
Niekamp, S., Kleinmann, M., & Gühne, O. (2012) · Journal of Mathematical Physics · doi:10.1063/1.3678200 · Cited by 12 · see citing works
Quantum linear network coding for entanglement distribution in restricted architectures
de Beaudrap, N., & Herbert, S. (2020) · Quantum · doi:10.22331/q-2020-11-01-356 · Cited by 12 · see citing works
“It is worth observing that, while our proposed formalism was conceptualised from the starting point of classical network codes (as emphasised in Corollary 6), the manner in which the state is tracked bears some resemblance to the quadratic representation of the Stabliser formalism as described by Dehaene and de Moor […”
MethodClassical Simulability of Quantum Circuits with Shallow Magic Depth
Zhang, Y., & Zhang, Y. (2025) · PRX Quantum · doi:10.1103/prxquantum.6.010337 · Cited by 12 · see citing works
Local equivalence, surface-code states, and matroids
Sarvepalli, P. K., & Raussendorf, R. (2010) · Physical Review A · doi:10.1103/physreva.82.022304 · Cited by 11 · see citing works
“If we repeatedly use the [[2, 1, 1]] code, then we obtain distance two counterexamples for any n ≥ 28.”
BackgroundMethodLocal equivalence of complete bipartite and repeater graph states
Tzitrin, I. (2018) · Physical Review A · doi:10.1103/physreva.98.032305 · Cited by 11 · see citing works
On the geometry of stabilizer states
García, H. J., Markov, I. L., & Cross, A. W. (2014) · Quantum Information and Computation · doi:10.5555/2638682.2638691 · Cited by 11 · see citing works
On maximum-likelihood decoding with circuit-level errors
Pryadko, L. P. (2020) · Quantum · doi:10.22331/q-2020-08-06-304 · Cited by 10 · see citing works
“[36] Jeroen Dehaene and Bart De Moor, “Clifford group, sta-”
BackgroundLower Bounds on Stabilizer Rank
Peleg, S., Shpilka, A., & Volk, B. L. (2022) · Quantum · doi:10.22331/q-2022-02-15-652 · Cited by 10 · see citing works
“As shown in [8, 26], stabilizer functions indeed correspond to stabilizer states up to normalization (which has no effect on the stabilizer rank).”
BackgroundThe equivalence of two approaches to the design of entanglement distillation protocolsPreprint
Hostens, E., Dehaene, J., & Moor, B. D. (2004) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0406017 · Cited by 10 · see citing works
Self-citationQuantum-classical separations in shallow-circuit-based learning with and without noises
Zhang, Z., Gong, W., Li, W., & Deng, D.-L. (2024) · Communications Physics · doi:10.1038/s42005-024-01783-7 · Cited by 9 · see citing works
“https://doi.org/10.1038/s42005-024-01783-7 Quantum-classical separations in shallow-circuit-based learning with and without noises”
InfluentialGarcía, A., Romero, J. L., & Klimov, A. B. (2010) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/43/38/385301 · Cited by 9 · see citing works
Serially Concatenated Unity-Rate Codes Improve Quantum Codes Without Coding-Rate Reduction
Babar, Z., Nguyen, H. V., Botsinis, P., Alanis, D., Chandra, D., Ng, S. X., & Hanzo, L. (2016) · IEEE Communications Letters · doi:10.1109/lcomm.2016.2593874 · Cited by 9 · see citing works
“More explicitly, for an n-qubit Pauli operator P , which belongs to the Pauli group Gn , an n-qubit Clifford operator V ensures that VPV † ∈ Gn [11].”
BackgroundComplete Flow-Preserving Rewrite Rules for MBQC Patterns with Pauli Measurements
McElvanney, T., & Backens, M. (2023) · Electronic Proceedings in Theoretical Computer Science · doi:10.4204/eptcs.394.5 · Cited by 9 · see citing works
“It has long been known [13, 26] that an n-qubit stabilizer state can be written (up to normalisation) as”
BackgroundKerdock Codes Determine Unitary 2-Designs
Can, T., Rengaswamy, N., Calderbank, R., & Pfister, H. D. (2020) · IEEE Transactions on Information Theory · doi:10.1109/tit.2020.3015683 · Cited by 9 · see citing works
“These eigenvectors are called stabilizer states in the quantum information literature [28] (also see end of Section III).”
- 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 › Computer Science › Artificial Intelligence › 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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
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 ›
Are you an author, or do you see a problem with this page? Request a review
