In this work we study the structure and cardinality of maximal sets of commuting and anticommuting Paulis in the setting of the abelian Pauli group. We provide necessary and sufficient conditions for anticommuting sets to be maximal, and present an efficient algorithm for generating anticommuting sets of maximum size. As a theoretical tool, we introduce commutativity maps, and study properties of maps associated with elements in the cosets with respect to anticommuting minimal generating sets. We also derive expressions for the number of distinct sets of commuting and anticommuting abelian Paulis of a given size.
In this work we study the structure and cardinality of maximal sets of commuting and anticommuting Paulis in the setting of the abelian Pauli group. We provide necessary and sufficient conditions for anticommuting sets to be maximal, and present an efficient algorithm for generating anticommuting sets of maximum size. As a theoretical tool, we introduce commutativity maps, and study properties of maps associated with elements in the cosets with respect to anticommuting minimal generating sets. We also derive expressions for the number of distinct sets of commuting and anticommuting abelian Paulis of a given size.
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.
A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) N-qubit Pauli graph is shown to be pseudo-geometric, i. e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle Q(4, 3), the dual ofW(3).
A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding {\it Pauli graph} are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) N-qubit Pauli graph is shown to be pseudo-geometric, i.\,e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle Q(4,3), the dual of W(3).
Any quantum computation consists of a sequence of unitary evolutions described by a finite set of Hamiltonians. When this set is taken to consist of only products of Pauli operators, we show that the minimal such set generating su(2^{N}) contains 2N+1 elements. We provide a number of examples of such generating sets and furthermore provide an algorithm for producing a sequence of rotations corresponding to any given Pauli rotation, which is shown to have optimal complexity. We also observe that certain sets generate su(2^{N}) at a faster rate than others, and we show how this rate can be optimized by tuning the fraction of anticommuting pairs of generators. Finally, we briefly comment on implications for measurement-based and trapped ion quantum computation as well as the construction of fault-tolerant gate sets.
There exists a large class of groups of operators acting on Hilbert spaces, where commutativity of group elements can be expressed in the geometric language of symplectic polar spaces embedded in the projective spaces PG(n,p), n being odd and p a prime. Here, we present a result about commuting and non-commuting group elements based on the existence of so-called Moebius pairs of n-simplices, i. e., pairs of n-simplices which are \emph{mutually inscribed and circumscribed} to each other. For group elements representing an n-simplex there is no element outside the centre which commutes with all of them. This allows to express the dimension n of the associated polar space in group theoretic terms. Any Moebius pair of n-simplices according to our construction corresponds to two disjoint families of group elements (operators) with the following properties: (i) Any two distinct elements of the same family do not commute. (ii) Each element of one family commutes with all but one of the elements from the other family. A three-qubit generalised Pauli group serves as a non-trivial example to illustrate the theory for p=2 and n=5.