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.
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.
Abstract Schur–Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the t -fold tensor powers U⊗t U ⊗ t of all unitaries U∈U(d) U ∈ U ( d ) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries . The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications: We resolve an open problem in quantum property testing by showing that “stabilizerness” is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group. We find that tensor powers of stabilizer states have an increased symmetry group. Conversely, we provide corresponding de Finetti theorems , showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well). We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana ) – a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem . We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.
We introduce the Clifford entropy, a measure of how close an arbitrary unitary is to a Clifford unitary, which generalizes the stabilizer entropy for states. We show that this quantity vanishes if and only if a unitary is Clifford, is invariant under composition with Clifford unitaries, and is subadditive under tensor products. Rewriting the Clifford entropy in terms of the stabilizer entropy of the corresponding Choi state allows us to derive an upper bound: that this bound is not tight follows from considering the properties of symmetric informationally complete sets. Nevertheless we are able to numerically estimate the maximum in low dimensions, comparing it to the average over all unitaries, which we derive analytically. Finally, harnessing a concentration of measure result, we show that as the dimension grows large, with probability approaching unity, the ratio between the Clifford entropy of a Haar random unitary and that of a fixed magic gate gives a lower bound on the depth of a doped Clifford circuit which realizes the former in terms of the latter. In fact, numerical evidence suggests that this result holds reliably even in low dimensions. We conclude with several directions for future research.
The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being the ubiquitous Pauli group, via the Weyl (i.e., Pauli) expansion of unitaries at these levels. In particular, we characterize the support of the standard Clifford operations on the Pauli group. Since conjugation of a Pauli by a third level unitary produces traceless Hermitian Cliffords, we characterize their Pauli support as well. Semi-Clifford unitaries are known to have ancilla savings in the teleportation model, and we explore their Pauli support via symplectic transvections. Finally, we show that, up to multiplication by a Clifford, every third level unitary commutes with at least one Pauli matrix. This can be used inductively to show that, up to a multiplication by a Clifford, every third level unitary is supported on a maximal commutative subgroup of the Pauli group. Additionally, it can be easily seen that the latter implies the generalized semi-Clifford conjecture, proven by Beigi and Shor (2010). We discuss potential applications in quantum error correction and the design of flag gadgets.
In quantum information theory and statistical physics, symmetries of multiple copies, or replicas, of a system play a pivotal role. For unitary ensembles, these symmetries are encoded in the replicated commutant: the algebra of operators commuting with the ensemble across k replicas. Determining the commutant is straightforward for the full unitary group, but remains a major obstacle for structured, computationally relevant circuit families. We solve this problem for matchgate circuits, which prepare fermionic Gaussian states on n qubits. Using a Majorana fermion representation, we show that operators coupling different system copies generate the orthogonal Lie algebra so(k), endowing the space of invariants with rich and tractable structure. This underlying symmetry decomposes the matchgate commutant into irreducible sectors, which we completely resolve via a Gelfand--Tsetlin construction. We provide an explicit orthonormal basis of the matchgate commutant for all k and n, together with a formula for its dimension that grows polynomially in n. Furthermore, we characterize the commutant of the Clifford--matchgate subgroup, showing that restricting to signed permutations of Majorana modes yields a commutant that qualitatively diverges from the matchgate case for k≥4 replicas. Ultimately, our orthonormal basis turns algebraic classification into a working toolbox. Using it, we derive closed-form expressions for matchgate twirling channels and a fermionic analogue of Weingarten calculus, the projector encoding all moments of the Gaussian state orbit, state and unitary frame potentials, the average nonstabilizerness of fermionic Gaussian states, a systematic hierarchy of non-Gaussianity measures, and a fermionic de Finetti theorem.
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.
Magic states enable universal, fault-tolerant quantum computation within the stabilizer framework. Their non-stabilizerness supplies the resource needed to bypass the Eastin-Knill theorem while allowing fault-tolerant distillation. Although many measures of magic exist, not every nonzero-magic state is known to be distillable, and many of currently known distillable states are special cases of Clifford-stabilizer states, defined as pure states uniquely stabilized by finite Clifford subgroups. We develop a general framework for group-covariant functionals on Hermitian operators, introducing the notions of G-stabilizer spaces, states, and codes for arbitrary finite subgroups G⊂U(H). We define analytic families of G-covariant functionals and prove that any G-invariant pure state is extremal for a broad class of derived functionals, including symmetric, max-type, and Rényi-type functionals, whenever the underlying family is G-covariant. This extremality holds for purity-preserving variations orthogonal to the stabilized subspace. Specializing to the Pauli and Clifford groups, we recover the extremality structure of canonical magic measures such as mana, stabilizer Rényi entropies, generalized Rényi entropies, and stabilizer fidelity, and show that Clifford-stabilizer states extremize them. We classify such states for qubits, qutrits, ququints, and two-qubit systems, identifying new candidates for magic distillation. We further propose an inefficient distillation protocol for a two-qubit magic state with stabilizer fidelity exceeding standard benchmarks, and conjecture that SIC-POVM fiducial states are Clifford-stabilizer states.