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Center for Nonlinear Studies |
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.ArXiv link: https://arxiv.org/abs/2504.12263.
Bio:
Antonio Anna Mele is a PhD student in Jens Eisert’s group at Freie Universität Berlin. His research has focused on quantum learning theory and randomized protocols, the mathematical structure of Clifford and fermionic/bosonic Gaussian circuits, and the complexity of noisy quantum circuits. He earned his Bachelor’s degree in Physics from the University of Pisa and a joint Master’s degree from the University of Trento and SISSA in Trieste. During his PhD, he was a research intern at the 2024 Quantum Computing Summer School at Los Alamos National Laboratory, working under the supervision of Marco Cerezo (CCS3).
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