Related papers: Cayley graphs and complexity geometry
This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various `nonpositive…
We present a generally covariant approach to quantum mechanics in which generalized positions, momenta and time variables are treated as coordinates on a fundamental "phase-spacetime." We show that this covariant starting point makes…
We expound a concise construction of finite groups and groupoids whose Cayley graphs satisfy graded acyclicity requirements. Our acyclicity criteria concern cyclic patterns formed by coset-like configurations w.r.t. subsets of the generator…
A generalized Noether's theorem and the operational determination of a physical geometry in quantum physics are used to motivate a quantum geometry consisting of relations between quantum states that are defined by a universal group. Making…
Science is rich in abstract concepts that capture complex processes in astonishingly simple ways. A prominent example is the reduction of molecules to simple graphs. This work introduces a design principle for parametrized quantum circuits…
An algorithm for quantum computing Hamiltonian cycles of simple, cubic, bipartite graphs is discussed. It is shown that it is possible to evolve a quantum computer into an entanglement of states which map onto the set of all possible paths…
Introduced the quantitative measure of the structural complexity of the graph (complex network, etc.) based on a procedure similar to the renormalization process, considering the difference between actual and averaged graph structures on…
Geometric semigroup theory is the systematic investigation of finitely-generated semigroups using the topology and geometry of their associated automata. In this article we show how a number of easily-defined expansions on finite semigroups…
The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that…
Quantum graphity is a background independent model for emergent locality, spatial geometry and matter. The states of the system correspond to dynamical graphs on N vertices. At high energy, the graph describing the system is highly…
Given a unitary operator in a finite dimensional complex Hilbert space, its unitary reduction to a subspace is defined. The application to quantum graphs is discussed. It is shown how the reduction allows to generate the scattering matrices…
Owing to their interesting spectral properties, the synthetic crystals over lattices other than regular Euclidean lattices, such as hyperbolic and fractal ones, have attracted renewed attention, especially from materials and meta-materials…
We interpret quantum computing as a geometric evolution process by reformulating finite quantum systems via Connes' noncommutative geometry. In this formulation, quantum states are represented as noncommutative connections, while gauge…
Clustering is a fundamental property of complex networks and it is the mathematical expression of a ubiquitous phenomenon that arises in various types of self-organized networks such as biological networks, computer networks or social…
We review quantum causal histories starting with their interpretations as a quantum field theory on a causal set and a quantum geometry. We discuss the difficulties that background independent theories based on quantum geometry encounter in…
The problem of constructing a quantum theory of gravity is considered from a novel viewpoint. It is argued that any consistent theory of gravity should incorporate a relational character between the matter constituents of the theory. In…
Using tangent bundle geometry we construct an equivalent reformulation of classical field theory on flat spacetimes which simultaneously encodes the perspectives of multiple observers. Its generalization to curved spacetimes realizes a new…
Quantum Graphity (QG) is a model of emergent geometry in which space is represented by a dynamical graph. The graph evolves under the action of a Hamiltonian from a high-energy pre-geometric state to a low-energy state in which geometry…
Quantum geometry defines the phase and amplitude distances between quantum states. The phase distance is characterized by the Berry curvature and thus relates to topological phenomena. The significance of the full quantum geometry,…
We present simple graph-theoretic characterizations of Cayley graphs for left-cancellative monoids, groups, left-quasigroups and quasigroups. We show that these characterizations are effective for the end-regular graphs of finite degree.