Related papers: Quantum Walks on Regular Graphs and Eigenvalues
Quantum and random walks have been shown to be equivalent in the following sense: a time-dependent random walk can be constructed such that its vertex distribution at all time instants is identical to the vertex distribution of any…
We consider the continuous-time quantum walk defined on the adjacency matrix of a graph. At each instant, the walk defines a mixing matrix which is doubly-stochastic. The average of the mixing matrices contains relevant information about…
In network science, graphs obtained by taking the Kronecker or tensor power of the adjacency matrix of an initiator graph are used to construct complex networks. In this paper, we analytically prove sufficient conditions under which such…
In this paper, we consider a spectral analysis of discrete time quantum walks on the path. For isospectral coin cases, we show that the time averaged distribution and stationary distributions of the quantum walks are described by the pair…
Quantum walks determined by the coin operator on graphs have been intensively studied. The typical examples of coin operator are the Grover and Fourier matrices. The periodicity of the Grover walk is well investigated. However, the…
A certain signed adjacency matrix of the hypercube, which Hao Huang used last year to resolve the sensitivity conjecture, is closely related to the unique, 4-cycle free, 2-fold cover of the hypercube. We develop a framework in which this…
The most elementary quantum walk is characterized by a 2-dimensional unitary coin flip matrix, which can be parameterized by 4 real variables. The influence of the choice of the coin flip matrix on the time evolution operator is analysed in…
We study the spectral analysis and the scattering theory for time evolution operators of position-dependent quantum walks. Our main purpose of this paper is construction of generalized eigenfunctions of the time evolution operator. Roughly…
In this paper we focus our attention on a particle that follows a unidirectional quantum walk, an alternative version of the nowadays widespread discrete-time quantum walk on a line. Here the walker at each time step can either remain in…
This work describes a new algorithm for creating a superposition over the edge set of a graph, encoding a quantum sample of the random walk stationary distribution. The algorithm requires a number of quantum walk steps scaling as…
We define coined Quantum Walks on the infinite rooted binary tree given by unitary operators $U(C)$ on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix $C\in U(3)$, and study their spectral properties.…
We investigate the use of discrete-time quantum walks to sample from an almost-uniform distribution, in the absence of any external source of randomness. Integers are encoded on the vertices of a cycle graph, and a quantum walker evolves…
Quantum walks underlie an important class of quantum computing algorithms, and represent promising approaches in various simulations and practical applications. Here we design stroboscopically monitored quantum walks and their subsequent…
This paper studies long range random walks on ${\mathbb{Z}_q}^d$. $X_{t+1} = X_t + Z_t \mod q$, with $(Z_t)$ independent and identically distributed. Multiple entries of $Z_t$ can be non-zero in a transition. An emphasis is on finding the…
Incorporating higher-order interactions in information processing enables us to build more accurate models, gain deeper insights into complex systems, and address real-world challenges more effectively. However, existing methods, such as…
We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion…
We introduce the concept of group state transfer on graphs, summarize its relationship to other concepts in the theory of quantum walks, set up a basic theory, and discuss examples. Let $X$ be a graph with adjacency matrix $A$ and consider…
Quantum walks are referred to as quantum analogs to random walks in mathematics. They have been studied as quantum algorithms in quantum information for quantum computers. There are two types of quantum walks. One is the discrete-time…
We propose categories of $1$-dimensional and multi-dimensional quantum walks. In the categories, an object is a quantum walk, and a morphism is an intertwining operator between two quantum walks. The new framework enables us to discuss…
We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains, that is, classical Markov chains with added memory. We show that these can simulate quantum walks, allowing us…