Related papers: Quantum graph walks II: Quantum walks on graph cov…
We give the first specific conjectures on how frequently graphs satisfy sufficient conditions for being uniquely characterized by spectral information. These conjectures arise from a theoretical framework that we developed based on…
We study how quantum walks can be used to find structural anomalies in graphs via several examples. Two of our examples are based on star graphs, graphs with a single central vertex to which the other vertices, which we call external…
In this paper we introduce a novel polynomial-time algorithm to compute graph invariants based on the modified random walk idea on graphs. However not proved to be a full graph invariant by now, our method gives the right answer for the…
We introduce the quantum stochastic walk (QSW), which determines the evolution of generalized quantum mechanical walk on a graph that obeys a quantum stochastic equation of motion. Using an axiomatic approach, we specify the rules for all…
Quantum walks provide a framework for understanding and designing quantum algorithms that is both intuitive and universal. To leverage the computational power of these walks, it is important to be able to programmably modify the graph a…
Let $G$ be a graph with $n$ vertices, and let $L(G)$ and $Q(G)$ be the Laplacian matrix and signless Laplacian matrix of $G$, respectively. The polynomial $\pi(L(G);x)={\rm per}(xI-L(G))$ (resp. $\pi(Q(G);x)={\rm per}(xI-Q(G))$) is called…
The physics of quantum walks on graphs is formulated in Hamiltonian language, both for simple quantum walks and for composite walks, where extra discrete degrees of freedom live at each node of the graph. It is shown how to map between…
We introduce a model of a quantum walk on a graph in which a particle jumps between neighboring nodes and interacts with independent spins sitting on the edges. Entanglement propagates with the walker. We apply this model to the case of a…
We present two new proofs of the the important q-commuting property holding among certain pairs of quantum minors of an n x n q-generic matrix. The first uses elementary quasideterminantal arithmetic; the second involves paths in an…
Graph representations are a powerful concept for solving complex problems across natural science, as patterns of connectivity can give rise to a multitude of emergent phenomena. Graph-based approaches have proven particularly fruitful in…
In this paper we study a class of discrete quantum walks, known as bipartite walks. These include the well-known Grover's walks. Any discrete quantum walk is given by the powers of a unitary matrix $U$ indexed by arcs or edges of the…
We study "holomorphic quadratic differentials" on graphs. We relate them to the reactive power in an LC circuit, and also to the chromatic polynomial of a graph. Specifically, we show that the chromatic polynomial $\chi$ of a graph $G$, at…
We evaluate the variance of coefficients of the characteristic polynomial for binary quantum graphs using a dynamical approach. This is the first example where a spectral statistic can be evaluated in terms of periodic orbits for a system…
Coherent evolution governs the behaviour of all quantum systems, but in nature it is often subjected to influence of a classical environment. For analysing quantum transport phenomena quantum walks emerge as suitable model systems. In…
The quantum walk is a powerful tool to develop quantum algorithms, which usually are based on searching for a vertex in a graph with multiple marked vertices, Ambainis's quantum algorithm for solving the element distinctness problem being…
As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex $u$ induces the same regular partition around $u$, where all vertices of each cell…
We consider matrices of the form $qD+A$, with $D$ being the diagonal matrix of degrees, $A$ being the adjacency matrix, and $q$ a fixed value. Given a graph $H$ and $B\subseteq V(G)$, which we call a coalescent pair $(H,B)$, we derive a…
We study the spectral determinant of the Laplacian on finite graphs characterized by their number of vertices V and of bonds B. We present a path integral derivation which leads to two equivalent expressions of the spectral determinant of…
We introduce a simple diagrammatic approach for estimating how a randomly walking quantum particle searches on a graph in continuous-time, which involves sketching small weighted graphs with self-loops and considering degenerate…
It is demonstrated that in gate-based quantum computing architectures quantum walk is a natural mathematical description of quantum gates. It originates from field-matter interaction driving the system, but is not attached to specific qubit…