We study the stabiliser formalism for qudits of arbitrary dimension $d$. In the free case, we show that the basic theorem of the stabiliser formalism remains valid: if the stabiliser subgroup $H$ is free as a $Z/dZ$-module and contains no non-trivial scalars, then the protected space $V^H$ is naturally identified with the state space of a smaller number of qudits of the same dimension, and the quotient $N(H)/H$ is identified with the Pauli group on a smaller number of qudits.
We then remove the freeness assumption and describe the resulting structure in general. In this case, the protected space is identified with a tensor product of qudit spaces of possibly smaller dimensions, and the quotient $N(H)/H$ is described by a corresponding product of qudit Pauli groups, possibly of smaller dimensions, over a common center. We also characterise the shifted free case, which is exactly the situation in which $N(H)/H$ is again an ordinary qudit Pauli group. Our approach is algebraic and uniform, and applies in particular to the qudit Kitaev model and to its shifted and twisted variants.
We study the stabiliser formalism for qudits of arbitrary dimension d. In the free case, we show that the basic theorem of the stabiliser formalism remains valid: if the stabiliser subgroup H is free as a Z/dZ-module and contains no non-trivial scalars, then the protected space VH is naturally identified with the state space of a smaller number of qudits of the same dimension, and the quotient N(H)/H is identified with the Pauli group on a smaller number of qudits.
We then remove the freeness assumption and describe the resulting structure in general. In this case, the protected space is identified with a tensor product of qudit spaces of possibly smaller dimensions, and the quotient N(H)/H is described by a corresponding product of qudit Pauli groups, possibly of smaller dimensions, over a common center. We also characterise the shifted free case, which is exactly the situation in which N(H)/H is again an ordinary qudit Pauli group. Our approach is algebraic and uniform, and applies in particular to the qudit Kitaev model and to its shifted and twisted variants.
Qudits with local dimension d>2 can have unique structure and uses that qubits ( d=2 ) cannot. Qudit Pauli operators provide a very useful basis of the space of qudit states and operators. We study the structure of the qudit Pauli group for any, including composite, d in several ways. To cover composite values of d , we work with modules over commutative rings, which generalize the notion of vector spaces over fields. For any specified set of commutation relations, we construct a set of qudit Paulis satisfying those relations. We also study the maximum size of sets of Paulis that mutually non-commute and sets that non-commute in pairs. Finally, we give methods to find near minimal generating sets of Pauli subgroups, calculate the sizes of Pauli subgroups, and find bases of logical operators for qudit stabilizer codes. Useful tools in this study are normal forms from linear algebra over commutative rings, including the Smith normal form, alternating Smith normal form, and Howell normal form of matrices. Possible applications of this work include the construction and analysis of qudit stabilizer codes, entanglement assisted codes, parafermion codes, and fermionic Hamiltonian simulation.
Galois qudits are q-dimensional quantum systems whose choice of Pauli group encodes the arithmetic of some finite field Fq. They differ from the more familiar modular qudit, which are the same quantum system but whose choice of Pauli group are the clock and shift operators, which encode the arithmetic of integer addition and multiplication modulo q. Galois qudits are a useful mathematical construct that allow us to leverage the mathematical tools that are native to the larger qudit while only physically building smaller qudits. In particular, a Galois qudit of dimension q=2s is exactly the same thing as a collection of s qubits, not only in its Hilbert space, but also in its Pauli group, and Clifford hierarchy. This formalism has found a lot of utility recently in constructing quantum error-correcting codes over qubits with useful properties.
In this review, we build on existing literature to collect and formalise facts and proofs about Galois qudits over binary extension fields. We define them and their Clifford hierarchies, describe what it means to measure their Pauli operators, describe their stabiliser tableaux, formally define qudit-to-qubit mappings, and finally describe quantum Reed-Solomon codes.
Recent work has shown that quantum double models can be used to circumvent the Bravyi-Koenig bound. In particular, this is achieved through code-switching protocols in which non-Pauli stabilizer phases appear as intermediate codes. We introduce the XCd(k) stabilizer formalism, which generalizes the Pauli stabilizer formalism by combining Pauli-X operators with diagonal operators drawn from a fixed level k of the Clifford hierarchy on prime-dimensional qudits. This yields a natural class of beyond-Pauli stabilizer models, including many twisted quantum double phases. We classify a large class of these XCd(k) stabilizer states, showing that they are uniform superpositions over affine-linear subspaces with diagonal phases. In particular, when the stabilizers of a state require operators from level k of the Clifford hierarchy, we show that the diagonal gates required to produce the phases in the wavefunction sit one level higher. We further show that every 2D twisted quantum double Dω(Zpn) with ω a product of type-III cocycles and p an odd prime admits a realization using only Clifford (k=2) stabilizers, and we conjecture that phases where ω contains a type-I or type-II cocycle require level-p Clifford-hierarchy stabilizers for p-dimensional qudits. Finally, we demonstrate that a pair of qudit toric codes may be deterministically prepared in a logical magic state by measuring XCd(k) stabilizers. Together, these results provide a concrete stabilizer framework for topological codes beyond Pauli stabilizer models and their logical gates.
In this paper, we determine the complete ground-state spaces for a broad class of generalized stabilizer quantum models defined over CW-decompositions of non-compact manifolds of arbitrary dimensions and topological type. We introduce a C∗-algebraic framework in which the quantum spin algebra $\A$ is realized as the twisted group C∗-algebra of an infinite discrete Pauli module P endowed with a symplectic structure. We establish a correspondence between stabilizer models and isotropic subgroups of P: Every Lagrangian extension of such a subgroup determines a C∗-diagonal of $\A$, whose Weyl groupoid encodes the dynamics through a continuous 1-cocycle. This realization yields an explicit family of extremal ground-states for every admissible Lagrangian model, together with their decomposition into sectors. When the sector space is T0, this gives a complete characterization of the ground-state spaces. We also prove that every Lagrangian stabilizer model admits a unique frustration-free ground state with a spectral gap. Our framework applies in arbitrary dimension, including higher abelian quantum doubles, homological codes, and a broad class of stabilizer models.
We study incomplete sets of orthogonal stabiliser states that cannot be extended by a further stabiliser state orthogonal to all of its members. Such sets are the stabiliser analogue of unextendible product bases (UPBs), and we thus call them unextendible stabiliser bases (USBs). Leveraging the symplectic geometry underlying the n-qudit Pauli group in prime local dimension, we explicitly construct USBs for systems of four qubits and three qudits of odd prime local dimension. Moreover, we show that these are the respective minimal qubit, respectively qudit numbers for which such bases exist, and that USBs exist for all n≥4 qubit and for all n≥3 odd-prime-dimensional qudit systems. Finally, we compare the resource-theoretic aspects of USBs with those of UPBs. We establish that, analogous to the case of UPBs, the orthogonal complement of every unextendible stabiliser set is a stabiliser-free subspace, and its normalised projector is necessarily magic; moreover, it is bound magic in odd prime dimension, yet need not be for qubits. We also show that USB unextendibility alone imposes no uniform quantitative obstruction to discrimination by stabiliser operations.
In this work, we compute the number of [[n,k]]d stabilizer codes made up of d -dimensional qudits, for arbitrary positive integers d . In a seminal work by Gross \cite{Gross2006} the number of [[n,k]]d stabilizer codes was computed for the case when d is a prime (or the power of a prime, i.e., d=pm , but when the qudits are Galois-qudits). The proof in \cite{Gross2006} is inapplicable to the non-prime case. For our proof, we introduce a group structure to [[n,k]]d codes, and use this in conjunction with the Chinese remainder theorem to count the number of [[n,k]]d codes. Our work overlaps with \cite{Gross2006} when d is a prime and in this case our results match exactly, but the results differ for the more generic case. Despite that, the overall order of magnitude of the number of stabilizer codes scales agnostic of whether the dimension is prime or non-prime. This is surprising since the method employed to count the number of stabilizer states (or more generally stabilizer codes) depends on whether d is prime or not. The cardinality of stabilizer states, which was so far known only for the prime-dimensional case (and the Galois qudit prime-power dimensional case) plays an important role as a quantifier in many topics in quantum computing. Salient among these are the resource theory of magic, design theory, de Finetti theorem for stabilizer states, the study and optimisation of the classical simulability of Clifford circuits, the study of quantum contextuality of small-dimensional systems and the study of Wigner-functions. Our work makes available this quantifier for the generic case, and thus is an important step needed to place results for quantum computing with non-prime dimensional quantum systems on the same pedestal as prime-dimensional systems.