A complete theory of the Clifford commutant

TL;DR

This work delivers a complete, practical theory of the Clifford group’s k-th order commutant for arbitrary n and k. It constructs an orthogonal basis via independent graph-based Pauli monomials, identifies a minimal generating set—permutations plus three Pauli-sum generators—and introduces a graphical calculus to manipulate these operators. The framework unifies and extends prior results, provides explicit dimension formulas, and applies to Haar-Clifford averages, magic-state resource theory, and stabilizer-state property testing, with a natural generalization to qudits. The results illuminate how all measurable magic monotones arise as expectations of Pauli-polynomial operators in the Clifford commutant and yield concrete tools for high-order design-type analyses and stabilizer-based protocols.

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.

Theorems & Definitions (150)

• Definition : Primitive Pauli monomials. Informal of \ref{['def:primitivepaulimonomials']}
• Theorem : Algebraic structure of the commutant. Informal version of \ref{['th:algebraicstructurecommutantofclifford']}
• Definition : Independent graph-based Pauli monomials
• Theorem : Basis of the commutant. Informal version of \ref{['th:fullcommutantnk', 'cor:dimcom']}
• Theorem : Magic-state resource theory and the commutant. Informal summary of \ref{['Sec:magicstateresourcetheory']}
• Lemma 1: Cardinality of $\mathrm{Sym}_0(\mathbb{F}_2^{m\times m})$ Mathf2Dim
• Lemma 2: Number of subspaces with even Hamming weight
• proof
• Definition 3: Twirling (super-)operator over a group $\mathcal{G}_n$
• Definition 4: $k$-th order commutant of a group $\mathcal{G}_n$