Quantum walks generated by the adjacency matrix or the Laplacian are known to exhibit low transfer fidelity on general graphs. In this paper, we study continuous-time quantum walks governed by the generalized Laplacian operator L_k = A+kD, where A is the adjacency matrix, D is the degree matrix, and k is a real-valued parameter. Recent work of Duda, McLaughlin, and Wong showed that in the single-excitation Heisenberg (XYZ) spin model, one can realize walks generated by this family of operators on signed weighted graphs. Motivated by earlier studies on vertex-weighted graphs, we demonstrate that for certain graphs, tuning the parameter k can significantly enhance the fidelity of state transfer between endpoints.
Quantum walks generated by the adjacency matrix or the Laplacian are known to exhibit low transfer fidelity on general graphs. In this paper, we study continuous-time quantum walks governed by the generalized Laplacian operator L_k = A+kD, where A is the adjacency matrix, D is the degree matrix, and k is a real-valued parameter. Recent work of Duda, McLaughlin, and Wong showed that in the single-excitation Heisenberg (XYZ) spin model, one can realize walks generated by this family of operators on signed weighted graphs. Motivated by earlier studies on vertex-weighted graphs, we demonstrate that for certain graphs, tuning the parameter k can significantly enhance the fidelity of state transfer between endpoints.
A single excitation in a quantum spin network described by the Heisenberg model can affect a variety of continuous-time quantum walks on unweighted graphs, including those governed by the discrete Laplacian, adjacency matrix, and signless Laplacian. In this paper, we show that the Heisenberg model can affect these three quantum walks on signed weighted graphs, as well as a generalized Laplacian equal to the discrete Laplacian plus a real-valued multiple of the degree matrix, for which the standard Laplacian, adjacency matrix, and signless Laplacian are special cases. We explore the algorithmic consequence of this generalized Laplacian quantum walk when searching a weighted barbell graph consisting of two equal-sized, unweighted cliques connected by a single signed weighted edge or bridge, with the search oracle constituting an external magnetic field in the spin network. We prove that there are two weights for the bridge (which could both be positive, both negative, or one of each, depending on the multiple of the degree matrix) that allow amplitude to cross from one clique to the other---except for the standard and signless Laplacians that respectively only have one negative or positive weight bridge---boosting the success probability from 0.5 to 0.820 or 0.843 for each weight. Moreover, one of the weights leads to a two-stage algorithm that further boosts the success probability to 0.996.
Quantum walks are the quantum analogues of classical random walks or Markov chains. They are universal models of quantum computing, and they underpin a variety of quantum algorithms. We prove that a continuous-time quantum walk effected by a generalized Laplacian, which can arise in spin chains, can solve a computational problem more quickly than typical quantum walks governed by the standard Laplacian or adjacency matrix. This generalized Laplacian consists of the standard Laplacian plus a real-valued multiple of the degree matrix, and we prove that as the magnitude of the multiple of the degree matrix is increased, its corresponding quantum walk can search the complete bipartite graph with multiple marked vertices in time that approaches the optimal. This raises the potential for the generalized Laplacian quantum walk to be a useful method for developing additional faster quantum algorithms.
The join X ∨ Y of two graphs X and Y is the graph obtained by joining each vertex of X to each vertex of Y . We explore the behaviour of a continuous quantum walk on a join graph with positive edge weights having the adjacency matrix or Laplacian matrix as its associated Hamiltonian, where the underlying graphs are assumed to be regular when dealing with the adjacency matrix. We characterize strong cospectrality, periodicity and perfect state transfer (PST) in a join graph. We also determine conditions in which strong cospectrality, periodicity and PST are preserved in the join. Under certain conditions, we show that there are graphs with no PST that exhibit PST when joined with another graph. We also show that | | U M ( X ∨ Y , t ) u , v | − | U M ( X , t ) u , v | | ≤ 2 | V ( X ) | for all vertices u and v of X , where U M ( X ∨ Y , t ) and U M ( X , t ) denote the transition matrices of X ∨ Y and X respectively relative to the adjacency or Laplacian matrix. We demonstrate that the bound 2 | V ( X ) | is tight for infinite families of graphs.
Quantum computing is believed to provide many advantages over traditional computing, particularly considering the speed at which computations can be performed. One of the challenges that needs to be resolved in order to construct a quantum computer is the transmission of information from one part of the computer to another. Quantum walks, the quantum analogues of classical random walks, provide one potential method for resolving this challenge. \n \nIn this thesis, we use techniques from algebraic graph theory and number theory to analyze the mathematical model for continuous time quantum walks on graphs. For the continuous time quantum walk model, we define a transition operator, which is a function of a Hamiltonian. We focus on the cases where the adjacency matrix or the Laplacian of a graph act as the Hamiltionian. We mainly consider quantum walks on paths as a model for spin chains, which are the underlying basis of a quantum communication protocol. \n \nFor communication to be efficient, we desire states to be transferred with high fidelity, a measure of the amount of similarity between the transmitted state and the received state. At the maximum fidelity of 1, we say we have achieved perfect state transfer. Examples of perfect state transfer are relatively rare, so the concept of pretty good state transfer was introduced as a natural relaxation, which exists if fidelities arbitrarily close to 1 are obtained. \n \nOur first main result is to characterize pretty good state transfer on paths. Previously, pretty good state transfer on paths was considered mainly for the end vertices, though results for both models indicated that if there was pretty good state transfer between the end vertices, then there was pretty good state transfer between internal vertices equidistant from each end. We complete the characterization by demonstrating, for the adjacency matrix model, a family of paths where pretty good state transfer exists between internal vertices but not between end vertices, and verifying that no other example exists. For the Laplacian model, we show that there are no paths with pretty good state transfer between internal vertices but not between the end vertices. \n \nOur second main result considers initial states involving multiple vertices. Under the adjacency matrix model, we provide necessary and sufficient conditions for pretty good state transfer in a particular family of paths in terms of the eigenvalue support of the initial state. We also discuss recent results on fractional revival, which is another form of multiple qubit state transfer.
Let G be a graph and let t be a positive real number. Then the evolution of the continuous quantum walk defined on G is described by the transition matrix U(t)=exp(itH).The matrix H is called Hamiltonian. So far the most studied quantum walks are the ones whose Hamiltonians are the adjacency matrices of the underlying graphs and initial states are vertex states e_a, with e_a being the characteristic vector of vertex a. \n \nThis thesis focuses on Laplacian edge state transfer, that is, the quantum walks whose initial states are edge states e_a-e_b and Hamiltonians are the Laplacians of the underlying graphs. So far the research about perfect state transfer only involves vertex states and Laplacian edge state transfer has not been studied before. We extend the known results of perfect vertex state transfer and periodicity of vertex states to Laplacian edge state transfer. \n \nWe prove two useful closure properties for perfect Laplacian edge state transfer. One is that complementation preserves perfect edge state transfer. The other is that if G has perfect Laplacian edge state transfer at time τ and H has perfect Laplacian vertex state transfer also at time τ, then with some mild assumption on the pairs of vertex states and edge states that have perfect state transfer, the Cartesian product G □ H also admits perfect edge state transfer. Those two properties provide us new ways to construct graphs with perfect Laplacian edge state transfer. We also observe one phenomenon that happens in Laplacian edge state transfer which never happens in vertex state transfer: if there is perfect state transfer from e_a-e_b to e_α-e_β and also from e_b-e_c to e_β-e_γ at the same time t in G, then G admits perfect state transfer from e_a-e_c to e_α-e_γ at time t. \n \nWe give characterizations of perfect Laplacian edge state transfer in cycles, paths and complete bipartite graphs K_{2,4n}. We study perfect state transfer and periodicity on edge states with special spectral features. We also consider the case when the unsigned Laplacian is Hamiltonian and initial states are plus states of the form e_a+e_b. In this case, we characterize perfect state transfer in paths, cycles and bipartite graphs. \nWe close this thesis by a list of open questions.
We propose a class of continuous-time quantum walk models on graphs induced by a certain class of discrete-time quantum walk models with a parameter [Formula: see text]. Here the graph treated in this paper can be applied to both finite and infinite cases. The induced continuous-time quantum walk is an extended version of the (free) discrete-Schrödinger equation driven by the normalized Laplacian: the element of the weighted Hermitian takes not only a scalar value but also a matrix value depending on the underlying discrete-time quantum walk. We show that each discrete-time quantum walk with an appropriate setting of the parameter [Formula: see text] in the long time limit identifies with its induced continuous-time quantum walk and gives the running time for the discrete-time to approximate the induced continuous-time quantum walk with a small error [Formula: see text]. We also investigate the detailed spectral information on the induced continuous-time quantum walk.