Abstract Stabilizer states serve as ‘classical objects’ in the stabilizer formalism of quantum theory, and play an important role in quantum error correction, fault-tolerant quantum computation, and quantum communication. They provide an efficient description of many basic features of quantum theory and exhibit a rich structure. For prime dimensional systems, they may be defined by two quite different yet equivalent ways: The first is via stabilizer groups (maximal Abelian subgroups of the discrete Heisenberg–Weyl group). The second is via the orbits of the Clifford group acting on any computational basis state. However, in a general dimensional system, this equivalence breaks down, and consequently, it is desirable to clarify the difference and relation between the above two approaches to stabilizer states. In this work, we show that these two approaches are equivalent if and only if the system dimension is square-free (i.e. has no square factor). Furthermore, we completely reveal the relation between the Clifford orbits and stabilizer states in an arbitrary dimensional system, derive the explicit expressions of the Clifford orbits and determine their cardinalities.
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 The standard stabilizer formalism provides a setting to show that quantum computations restricted to operations within the Clifford group are classically efficiently simulable: this is the content of the well-known Gottesman–Knill theorem. This work analyzes the mathematical structure behind this theorem to find possible generalizations and derivation of constraints required for constructing a non-trivial generalized Clifford group. We prove that if the closure of the stabilizing set is dense in the set of SU(d) transformations, then the associated Clifford group is trivial, consisting only of local gates and permutations of subsystems. This result demonstrates the close relationship between the density of the stabilizing set and the simplicity of the corresponding Clifford group. We apply the analysis to investigate stabilization with binary observables for qubits and find that the formalism is equivalent to the standard stabilization for a low number of qubits. Based on the observed patterns, we conjecture that a large class of generalized stabilizer states are equivalent to the standard ones. Our results provide better insights into the structure of Gottesman–Knill-type results, consequently allowing us to draw a sharper line between quantum and classical computation.
The Clifford hierarchy plays a crucial role in the stabilizer formalism of quantum error correction and quantum computation. Apart form the zeroth level (the discrete Heisenberg–Weyl group) and the first level (the Clifford group), all other levels of the Clifford hierarchy are not groups. However, the diagonal gates at all levels do form groups, and it is desirable to characterize their generators and structures. In this paper, we study the diagonal gates at the second level of the Clifford hierarchy. For this, we introduce the notion of a
T -gate in an arbitrary dimension, generalizing the corresponding notion in prime dimensions. By the use of the
T -gate, we are able to completely characterize the group structures of the diagonal gates at the second level of the Clifford hierarchy in any (not necessarily prime) dimension. It turns out that the classification depends crucially on the number-theoretic nature of the dimension. The results highlight the special role of the first two primes,
2 and
3 , in the prime factorization of the dimension. The
T -gate in an arbitrary dimension, apart from its key role as a generator of the diagonal gates, may have independent interest and further applications in quantum theory.
Stabilizer states, which are also known as the Clifford states, have been commonly utilized in quantum information, quantum error correction, and quantum circuit simulation due to their simple mathematical structure. In this work, we apply stabilizer states to tackle quantum many-body ground state problems and introduce the concept of stabilizer ground states. We establish an equivalence formalism for identifying stabilizer ground states of general Pauli Hamiltonians. Moreover, we develop an exact and linear-scaled algorithm to obtain stabilizer ground states of 1D local Hamiltonians and thus free from discrete optimization. This proposed equivalence formalism and linear-scaled algorithm are not only applicable to finite-size systems, but also adaptable to infinite periodic systems. The scalability and efficiency of the algorithms are numerically benchmarked on different Hamiltonians. Finally, we demonstrate that stabilizer ground states are promising tools for not only qualitative understanding of quantum systems, but also cornerstones of more advanced classical or quantum algorithms.
The term Clifford group was introduced in 1998 by D. Gottesmann in his investigation of quantum error-correcting codes. The simplest Clifford group in multiqubit quantum computation is generated by a restricted set of unitary Clifford gates - the Hadamard, π /4-phase and controlled-X gates. Because of this restriction the Clifford model of quantum computation can be efficiently simulated on a classical computer (the Gottesmann-Knill theorem). However, this fact does not diminish the importance of the Clifford model, since it may serve as a suitable starting point for a full-fledged quantum computation. In the general case of a single or composite quantum system with finite-dimensional Hilbert space the finite Weyl-Heisenberg group of unitary operators defines the quantum kinematics and the states of the quantum register. Then the corresponding Clifford group is defined as the group of unitary operators leaving the Weyl-Heisenberg group invariant. The aim of this contribution is to show that our comprehensive results on symmetries of the Pauli gradings of quantum operator algebras - covering any single as well as composite finite quantum systems - directly correspond to Clifford groups defined as quotients with respect to U(1).
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.