Alternative names: Clifford-Weil group.
Description
The Clifford group on \(n\) qubits is a subgroup of the unitary group \(U(2^n)\) that is the normalizer of the Pauli group, that forms a unitary 3-design, and that is closely related to the automorphism group of BW lattices. The group features prominently in quantum information, with the rest of the entry given in that context.
Clifford group: The \(n\)-qubit Clifford group consists of the Pauli group as well as elements that permute Pauli operators amongst themselves. Up to any phases and Pauli strings, the group is equivalent to the symplectic group \(Sp(2n,\mathbb{Z}_2)\). A Clifford circuit is a quantum circuit, defined on some qubit geometry and consisting of only Clifford gates. See Refs. [2–6] for generators, relations, and normal form.
The Clifford group cannot be expressed as a semidirect product of the Pauli and symplectic groups [7]. Restricting the group to real-valued elements yields the real Clifford group, and including complex conjugation yields the extended Clifford group [8].
Single-qubit Clifford gates, together with Paulis, realize a group with \(192\) elements. Modding out phases yields the \(48\)-element \(2O\) binary octahedral subgroup of \(SU(2)\). Further modding out the Pauli group, which corresponds to the quaternion group \(Q\), yields the permutation group \(S_3\), which consists of permutations of the three non-identity single-qubit Pauli matrices.
The two-qubit Clifford group, modded out by the Pauli group and phases, is isomorphic to \(S_6\), and its subgroups have been classified [9].
The commutant of four-fold [10,11] and higher transversal representations of the Clifford group consists of qubit permutations and projections onto the \([[2m,2m-2,2]]\) error-detecting code [12].
Clifford-group elements can be sampled efficiently [13].
Cousins
- Qubit stabilizer code— Stabilizer states can be mapped into the first lattice shell of a BW lattice over a cyclotomic field, while the Clifford group is related to the symmetry group of the lattice [14].
- Kerdock code— Kerdock codes correspond to cluster states, and the corresponding Clifford-group automorphisms of this set form a particular group [15] that is a unitary 2-design on \(U(2^n)\) [16]. As such, cluster states form complex projective on 2-designs \(\mathbb{C}P^{2^n}\). These are useful in matrix-vector multiplication [17].
- Cluster-state code— Kerdock codes correspond to cluster states, and the corresponding Clifford-group automorphisms of this set form a particular group [15] that is a unitary 2-design on \(U(2^n)\) [16]. As such, cluster states form complex projective on 2-designs \(\mathbb{C}P^{2^n}\). These are useful in matrix-vector multiplication [17].
- Complex projective space code— Stabilizer states on \(n\) qubits form 3-designs on complex projective spaces \(\mathbb{C}P^{2^n}\) [18], while the Clifford group is a unitary 3-design on \(U(2^n)\) [19,20]. The \([[2m,2m-2,2]]\) code for \(2m\) being a multiple of four obstructs the Clifford group from being a 4-design [10].
- \([[2m,2m-2,2]]\) error-detecting code— Stabilizer states on \(n\) qubits form 3-designs on complex projective spaces \(\mathbb{C}P^{2^n}\) [18], while the Clifford group is a unitary 3-design on \(U(2^n)\) [19,20]. The \([[2m,2m-2,2]]\) code for \(2m\) being a multiple of four obstructs the Clifford group from being a 4-design [10].
- Barnes-Wall (BW) lattice— Stabilizer states can be mapped into the first lattice shell of a BW lattice over a cyclotomic field, while the Clifford group is related to the symmetry group of the lattice [14].
- Real-Clifford subgroup-orbit code— The automorphism group of BW lattices is a subgroup of index 2 of a real Clifford group [21,22] (see [1,14] for an explanation).
- Haar-random qubit code— Approximating the random projections through \(t\)-designs is necessary in order to make the Haar-random qubit protocol practical. Replacing with random Clifford gates is especially convenient since the Clifford group forms a unitary 2-design and produces stabilizer codes.
- Qubit code— Computing using Clifford gates only can be efficiently simulated on a classical computer, according to the Gottesman-Knill theorem [23,24]. There is a canonical form for Clifford circuits [25,26] and many algorithms for simulating them [27–30]. Universal quantum computing can be achieved using Clifford gates and a single type of non-Clifford gate, such as the \(T\) gate [31]. More generally, the Solovay-Kitaev theorem [32,33] states that any subset of gates the generates a dense subgroup of the full \(n\)-qubit gate group can be used to construct any gate to arbitrary accuracy (see [34][35; Appx. 3]).
Member of code lists
Primary Hierarchy
Parents
Stabilizer states on \(n\) qubits form 3-designs on complex projective spaces \(\mathbb{C}P^{2^n}\) [18], while the Clifford group is a unitary 3-design on \(U(2^n)\) [19,20]. The \([[2m,2m-2,2]]\) code for \(2m\) being a multiple of four obstructs the Clifford group from being a 4-design [10].
Clifford group
References
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- [2]
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- E. Kubischta and I. Teixeira, “Classification of the Subgroups of the Two-Qubit Clifford Group”, (2024) arXiv:2409.14624
- [10]
- H. Zhu, R. Kueng, M. Grassl, and D. Gross, “The Clifford group fails gracefully to be a unitary 4-design”, (2016) arXiv:1609.08172
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- [18]
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- E. Knill, private communication, ca. 1998.
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- A. J. Malcolm et al., “Computing Efficiently in QLDPC Codes”, (2025) arXiv:2502.07150
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- [35]
- “The Solovay–Kitaev theorem”, Quantum Computation and Quantum Information 617 (2012) DOI
Page edit log
- Victor V. Albert (2025-10-28) — most recent
Cite as:
“Clifford group”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/clifford_group