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A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from the first graph onto the second can be found in time poly (n).
We describe the structure of the $n$-qubit Clifford group $\mathcal{C}_n$ via Cayley graphs, whose vertices represent group elements and edges represent generators. In order to obtain the action of Clifford gates on a given quantum state, we introduce a quotient procedure. Quotienting the Cayley graph by the stabilizer subgroup of a state gives a reduced graph which depicts the state's Clifford orbit. Using this protocol for $\mathcal{C}_2$, we reproduce and generalize the reachability graphs. Since the procedure is state-independent, we extend our study to non-stabilizer states, including the W and Dicke states. Furthermore, our new construction provides a more precise understanding of state evolution under Clifford circuit action.
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Given non-trivial group G and a non-empty subset S of G such that not containing the identity element of G and S=S{sup −1}:={s"−"1∣s∈S}, the Cayley graph Γ=Cay(G,S) is the graph whose vertex set V(Γ) is G, and edge set E(Γ) is {{g,sg}∣g∈G,s∈S}. The Cayley graphs are presently being considered as models architecture for interconnection networks. When there is an uncertainty on vertices or edges, fuzzy graph has special importance. In this paper, we develop the concept of Cayley fuzzy graphs on the fuzzy groups and investigate some basic properties of Cayley fuzzy graphs. We introduce some sufficient conditions on the fuzzy graphs that under it’s, a fuzzy graph is a Cayley fuzzy graph and discuss some operations of the fuzzy graphs on the Cayley fuzzy graphs, prove that the product of two Cayley fuzzy graphs is a Cayley fuzzy graph, for some products. Also we provide an application of Cayley fuzzy graphs, and use Dijkstra’s Algorithm for finding the fuzzy shortest path in a Cayley fuzzy graph.
Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new homological quantum error correcting codes. They are low density parity check codes with linear rate and distance n{sup ε}. Their rate is evaluated via Euler characteristic arguments and their distance using Z{sub 2}-systolic geometry. This construction answers a question of Zémor [“On Cayley graphs, surface codes, and the limits of homological coding for quantum error correction,” in Proceedings of Second International Workshop on Coding and Cryptology (IWCC), Lecture Notes in Computer Science Vol. 5557 (2009), pp. 259–273], who asked whether homological codes with such parameters could exist at all.
In this note is reported a collection of constructions of symmetric networks that provide the largest known values for the number of nodes that can be placed in a network of a given degree and diameter. Some of the constructions are in the range of current potential engineering significance. The constructions are Cayley graphs of linear groups obtained by experimental computation.
This paper studies deterministic movement over toroidal grids, integrating local information, bounded memory and choice at individual nodes. The research is motivated by recent work on deterministic random walks, and applications in multi-agent systems. Several results regarding passing tokens through toroidal grids are discussed, as well as some open questions.
In a previous paper (Funct. Anal. Appl., 3 (2015), 205–209), we defined quantum Markov states. Here we recall this definition and present a proof of the results from that paper (which are given there without proofs). We give a definition of a quantum hidden Markov state generated by a function of a quantum Markov process and show how it is related to other definitions of such states. Our definitions work for quantum Markov fields on ℤ{sup N} and on graphs. We consider an example with the Cayley tree.