Related papers: Cayley graphs and complexity geometry
We define a range of new coarse geometric invariants based on various graph-theoretic measures of complexity for finite graphs, including: treewidth, pathwidth, cutwidth and bandwidth. We prove that, for bounded degree graphs, these…
Due to their elegant and simple nature, unitary Cayley graphs have been an active research topic in the literature. These graphs are naturally connected to several branches of mathematics, including number theory, finite algebra,…
In the first part of this review we introduce the basics theory behind geometric phases and emphasize their importance in quantum theory. The subject is presented in a general way so as to illustrate its wide applicability, but we also…
Quantum complexity measures the difficulty of obtaining a given state starting from a typically unentangled state. In this work, we show that complexity, when defined through the minimization of a Riemannian cost functional over the…
The kinematical setting of spherically symmetric quantum geometry, derived from the full theory of loop quantum gravity, is developed. This extends previous studies of homogeneous models to inhomogeneous ones where interesting field theory…
The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this…
The formalism for histories-based generalized quantum mechanics developed in two earlier papers is applied to the treatment of histories (of particles or fields or more general objects) in curved spacetimes (which need not admit foliation…
A fundamental problem in quantum information is to describe efficiently multipartite quantum states. An efficient representation in terms of graphs exists for several families of quantum states (graph, cluster, stabilizer states),…
We propose models of quantum neural networks through Clifford algebras, which are capable of capturing geometric features of systems and to produce entanglement. Due to their representations in terms of Pauli matrices, the Clifford algebras…
A connected graph is called \emph{geodetic} if there is a unique geodesic between each pair of vertices. In this paper we prove that if a finitely generated group admits a Cayley graph which is geodetic, then the group must be virtually…
The concept of quantum complexity has far-reaching implications spanning theoretical computer science, quantum many-body physics, and high energy physics. The quantum complexity of a unitary transformation or quantum state is defined as the…
We undertake a study of the notion of a quantum graph over arbitrary finite-dimensional $C^*$-algebras $B$ equipped with arbitrary faithful states. Quantum graphs are realised principally as either certain operators on $L^2(B)$, the quantum…
In this expository paper we present a brief introduction to the geometrical modeling of some quantum computing problems. After a brief introduction to establish the terminology, we focus on quantum information geometry and ZX-calculus,…
The similarity transformations of quantum orthogonal groups are developed and FRT theory is reformulated to the Cartesian basis. The quantum orthogonal Cayley-Klein groups are introduced as the algebra functions over an associative algebra…
Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…
We reconstruct a quantum group associated with any Lie algebra together with its representation theory from twisted homologies of generalized configuration spaces of disks. Along the way it brings new combinatorics to the theory, but our…
We present some observations on a restricted variant of unitary Cayley graphs modulo n, and the implications for a decomposition of elements of symplectic operators over the integers modulo n. We define quadratic unitary Cayley graphs G_n,…
`How do our ideas about quantum mechanics affect our understanding of spacetime?' This familiar question leads to quantum gravity. The complementary question is also important: `How do our ideas about spacetime affect our understanding of…
The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many…
Any canonical quantum theory can be understood to arise from the compatibility of the statistical geometry of distinguishable observations with the canonical Poisson structure of Hamiltonian dynamics. This geometric perspective offers a…