nLab Clifford group

This entry is about the notion in quantum information theory. For subgroups of a Clifford algebra see instead there.

Idea

In quantum information theory, by the Clifford group on $n$-qbits (for $n \in \mathbb{N}_{\geq 1}$) one means the normalizer subgroup, inside the unitary group $U(2n)$, of the group of $n$fold $\mathbb{C}$-tensor products of linear operators from the Pauli group.

An element of the Clifford group, understood as a unitary operator on the finite-dimensional Hilbert space $\mathbb{C}^{2n}$, is also called a Clifford quantum gate or just Clifford gate, for short.

The Gottesman-Knill theorem states that quantum circuits which are built only from Clifford gates (“stabilizer circuits”) may be efficiently simulated on classical computers. Conversely this means that for a quantum computer to exhibit quantum advantage it must realize quantum gates which are non-Clifford gates.

References

General

The origin of the attention paid to the normalizer subgroup of the Pauli group inside the unitary group is (p 5 of):

• Daniel Gottesman: A Theory of Fault-Tolerant Quantum Computation, Phys. Rev. A 57 (1998) 127 [arXiv:quant-ph/9702029, doi:10.1103/PhysRevA.57.127]

Contrary to common claims, the term “Clifford group” for this normalizer subgroup seems to have emerged only in the following years.

Review:

• Jiri Tolar: On Clifford groups in quantum computing, J. Phys.: Conf. Series 1071 (2018) 012022 [arXiv:1810.10259, doi:10.1088/1742-6596/1071/1/012022]

• Kieran Mastel: The Clifford theory of the $n$-qubit Clifford group [arXiv:2307.05810]

See also:

• Wikipedia: Clifford group

• Wikipedia: Gottesman-Knill theorem

• Daniel Grier, Luke Schaeffer: The Classification of Clifford Gates over Qubits, Quantum 6 (2022) 734 [arXiv:1603.03999, doi:10.22331/q-2022-06-13-734]

• George Biswas: Exploring Classical Simulation of Quantum Circuits of Clifford Gates through Simple Examples and Intuitive Insights [arXiv:2405.13590]

Realizing protected non-Clifford gates

The practical issue of realizing fault-tolerant non-Clifford gates (either via quantum error correction or via topological error protection):

• Marek Narożniak et al.: Quantum gates for Majoranas zero modes in topological superconductors in one-dimensional geometry, Phys. Rev. B 103 (2021) 205429 [doi:10.1103/PhysRevB.103.205429]

• Ali Hamed Safwan, Raditya Weda Bomantara: Generating non-Clifford gate operations through exact mapping between Majorana fermions and $\mathbb{Z}_4$ parafermions [arXiv:2411.18736]

• Wang Yifei et al.: Efficient fault-tolerant implementations of non-Clifford gates with reconfigurable atom arrays, npj Quantum Information 10 136 (2024) [doi:10.1038/s41534-024-00945-3]

• Louis Golowich: Improved Fault-Tolerant Non-Clifford Gates (or: How to Multiply Quantumly), talk at IAS (March 03, 2025) [ias.edu, youtu.be]

• Margarita Davydova et al.: Universal fault tolerant quantum computation in 2D without getting tied in knots [arXiv:2503.15751]

• Microsoft Quantum: Non-Clifford operations, appendix D of: Roadmap to fault tolerant quantum computation using topological qubit arrays [arXiv:2502.12252, inSpire:2890977]