A complete theory of the Clifford commutant

Lennart Bittel; Jens Eisert; Lorenzo Leone; Antonio A. Mele; Salvatore F.E. Oliviero

doi:10.22331/q-2026-07-22-2171

A complete theory of the Clifford commutant

Lennart Bittel¹, Jens Eisert^1,2, Lorenzo Leone¹, Antonio A. Mele¹, and Salvatore F.E. Oliviero^3,1

¹Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
²Helmholtz-Zentrum Berlin für Materialien und Energie, Berlin, Germany
³Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, IT-56126 Pisa, Italy

Published:	2026-07-22, volume 10, page 2171
Editor:	Aaron Goldberg
Eprint:	arXiv:2504.12263v2
Doi:	https://doi.org/10.22331/q-2026-07-22-2171
Citation:	Quantum 10, 2171 (2026).

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Abstract

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.

► BibTeX data

@article{Bittel2026completetheoryof,
  doi = {10.22331/q-2026-07-22-2171},
  url = {https://doi.org/10.22331/q-2026-07-22-2171},
  title = {A complete theory of the {C}lifford commutant},
  author = {Bittel, Lennart and Eisert, Jens and Leone, Lorenzo and Mele, Antonio A. and Oliviero, Salvatore F.E.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2171},
  month = jul,
  year = {2026}
}

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The above citations are from SAO/NASA ADS (last updated successfully 2026-07-22 12:50:47). The list may be incomplete as not all publishers provide suitable and complete citation data.

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