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Clifford group, stabilizer states, and linear and quadratic operations over GF(2)
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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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“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].”
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Babar, Z., Botsinis, P., Alanis, D., Ng, S. X., & Hanzo, L. (2015) · IEEE Access · doi:10.1109/access.2015.2405533 · Cited by 42
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Fast and Effective Techniques for T-Count Reduction via Spider Nest IdentitiesPreprint
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Learning Quantum Circuits of Some T Gates
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“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].”
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Leone, L., Oliviero, S. F. E., Lloyd, S., & Hamma, A. (2024) · Physical Review A · doi:10.1103/physreva.109.022429 · Cited by 53
Classical simulation complexity of extended clifford circuits
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Stabilizer Notation for Spekkens’ Toy Theory
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One T Gate Makes Distribution Learning Hard
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“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”
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“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.”
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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
“(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.”
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A non-commuting stabilizer formalism
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Quantum Entanglement: Separability, Measure, Fidelity of Teleportation, and Distillation
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Fast Stabiliser Simulation with Quadratic Form Expansions
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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
“A closely related algorithm was given earlier by Dehaene and De Moor [56].”
Hashing protocol for distilling multipartite Calderbank-Shor-Steane states
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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
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“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
Graph concatenation for quantum codes
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Learning stabilizer states by Bell samplingPreprint
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Stabilizer rank and higher-order Fourier analysis
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“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
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Computing quopit Clifford circuit amplitudes by the sum-over-paths technique
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Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems
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“Keywords: classical simulations, stabilizer formalism, normalizer circuits, infinitedimensional quantum systems Communicated by: S Braunstein & B Terhal”
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Self-citationThe Classification of Clifford Gates over Qubits
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“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 15
Approximate stabilizer rank and improved weak simulation of Clifford-dominated circuits for qudits
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BackgroundClassification of small triorthogonal codes
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Depth-Optimal Synthesis of Clifford Circuits with SAT Solvers
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“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
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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
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
“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
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 15
A Non-Commuting Stabilizer FormalismPreprint
Ni, X., Buerschaper, O., & den Nest, M. V. (2014) · Cited by 15
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
Entanglement Purification with Quantum LDPC Codes and Iterative Decoding
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“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 27
“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
“[12].”
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“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
Quantum linear network coding for entanglement distribution in restricted architectures
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“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 17
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
“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
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
On maximum-likelihood decoding with circuit-level errors
Pryadko, L. P. (2020) · Quantum · doi:10.22331/q-2020-08-06-304 · Cited by 10
“[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
“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
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 10
“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
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
“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
“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 10
“These eigenvectors are called stabilizer states in the quantum information literature [28] (also see end of Section III).”
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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