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arXiv:2307.05810 · Journal of Mathematical Physics

The Clifford theory of the n-qubit Clifford group

Kieran Mastel

Published Feb 1, 2026• 4 citations
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Abstract

The n-qubit Pauli group and its normalizer, the n-qubit Clifford group, have applications in quantum error correction and device characterization. Recent applications have used the representation theory of the Clifford group. We apply the tools of (the coincidentally named) Clifford theory to examine the representation theory of the Clifford group using the much simpler representation theory of the Pauli group. We find an unexpected correspondence between irreducible characters of the n-qubit Clifford group and those of the (n+1)-qubit Clifford group.

DOI
10.1063/5.0311547

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The n-qubit Pauli group and its normalizer, the n-qubit Clifford group, have applications in quantum error correction and device characterization. Recent applications have used the representation theory of the Clifford group. We apply the tools of (the coincidentally named) Clifford theory to examine the representation theory of the Clifford group using the much simpler representation theory of the Pauli group. We find an unexpected correspondence between irreducible characters of the n-qubit Clifford group and those of the (n+1)-qubit Clifford group.
The q-qubit Clifford group, that is, the normalizer of the q-qubit Pauli group in U(2q), is a fundamental structure in quantum information with a wide variety of applications. We characterize all irreducible subrepresentations of the two-copy representation φ⊗2 of the Clifford group on the two-fold tensor product of the space of linear operators M2q⊗2. In the companion paper [Helsen et al., e-print arXiv:1701.04299 (2017)], we apply this result to improve the statistics of randomized benchmarking, a method for characterizing quantum systems.
Based on a presentation of Cn and the help of [The GAP Group 2022, https://www.gap-system.org] we construct the character table of the Clifford group Cn for n = 1, 2, 3. As an application, we can efficiently decompose the (higher power of) tensor product of the matrix representation in those cases. Our results recover some known results in Helsen et al. [J. Math. Phys. 59, 072201 (2018)] and Wallman and Flammia [New J. Phys. 16(10), 103032 (2014)] and reveal some new phenomena. We prove that when n ≥ 3, (1) the trivial character is the only linear character for Cn and hence Cn equals to its commutator subgroup, (2) the n-qubit Pauli group Pn is the only proper non-trivial normal subgroup of Cn, (3) the matrix representation M2n is a faithful representation for Cn. As a byproduct, we give a presentation of the finite symplectic group Sp(2n, 2) in terms of generators and relations.
The Clifford group plays a central role in quantum information science. It is the building block for many error-correcting schemes and matches the first three moments of the Haar measure over the unitary group-a property that is essential for a broad range of quantum algorithms, with applications in pseudorandomness, learning theory, benchmarking, and entanglement distillation. At the heart of understanding many properties of the Clifford group lies the Clifford commutant: the set of operators that commute with k -fold tensor powers of Clifford unitaries. Previous understanding of this commutant has been limited to relatively small values of k , constrained by the number of qubits n . In this work, we develop a complete theory of the Clifford commutant. Our first result provides an explicit orthogonal basis for the commutant and computes its dimension for arbitrary n and k . We also introduce an alternative and easy-to-manipulate basis formed by isotropic sums of Pauli operators. We show that this basis is generated by products of permutations, which generate the unitary group commutant, and at most three other operators. Additionally, we develop a graphical calculus allowing a diagrammatic manipulation of elements of this basis. These results enable a wealth of applications: among others, we characterize all measurable magic measures and identify optimal strategies for stabilizer property testing, whose success probability also offers an operational interpretation to stabilizer entropies. Finally, we show that these results also generalize to multi-qudit systems with prime local dimension.
Clifford group {\it dipoles} Cn±\mathcal{C}_n^{\pm}Cn±​ are subgroups of the nnn-qubit Clifford group Cn\mathcal{C}_nCn​ defined as (non-split) extensions of the extraspecial subgroups E2n+1±E_{2n+1}^{\pm}E2n+1±​ of the Pauli error group Pn\mathcal{P}_nPn​. The subgroups Cn±:=E2n+1±.Ω±(2n,2)\mathcal{C}_n^{\pm}:=E_{2n+1}^{\pm}.\Omega^{\pm}(2n,2)Cn±​:=E2n+1±​.Ω±(2n,2) involve pairs of orthogonal subgroups Ω±(2n,2)\Omega^{\pm}(2n,2)Ω±(2n,2) of the symplectic group Sp(2n,2)Sp(2n,2)Sp(2n,2). They are of special relevance to the geometry of quantum error correction, and encompass several concepts proposed so far, including complex reflection groups, BN-pairs in the Clifford group, and some unitary designs.
The representation theory of the Clifford group is playing an increasingly prominent role in quantum information theory, including in such diverse use cases as the construction of protocols for quantum system certification, quantum simulation, and quantum cryptography. In these applications, the tensor powers of the defining representation seem particularly important. The representation theory of these tensor powers is understood in two regimes. 1. For odd qudits in the case where the power t is not larger than the number of systems n: Here, a duality theory between the Clifford group and certain discrete orthogonal groups can be used to make fairly explicit statements about the occurring irreps (this theory is related to Howe duality and the η-correspondence). 2. For qubits: Tensor powers up to t = 4 have been analyzed on a case-by-case basis. In this paper, we provide a unified framework for the duality approach that also covers qubit systems. To this end, we translate the notion of rank of symplectic representations to representations of the qubit Clifford group, and generalize the η correspondence between symplectic and orthogonal groups to a correspondence between the Clifford and certain orthogonal-stochastic groups. As a sample application, we provide a protocol to efficiently implement the complex conjugate of a black-box Clifford unitary evolution.