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A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. Hitting times for discrete quantum walks on graphs give an average time before the walk…
We identify a key difference between quantum search by discrete- and continuous-time quantum walks: a discrete-time walk typically performs one walk step per oracle query, whereas a continuous-time walk can effectively perform multiple walk…
Continuous-time quantum walks (CTQWs) play a crucial role in quantum computing, especially for designing quantum algorithms. However, how to efficiently implement CTQWs is a challenging issue. In this paper, we study implementation of CTQWs…
The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph $H$ is present in an input graph $G$, has been the subject of considerable study. However, even for relatively simple subgraphs, such as paths…
One approach to the development of quantum search algorithms is the quantum walk. A spatial search can be effected by the continuous-time evolution of a single quantum particle on a graph containing a marked site. In many physical…
In this tutorial, which contains some original results, we bridge the fields of quantum computing algorithms, conservation laws, and many-body quantum systems by examining three algorithms for searching an unordered database of size $N$…
Continuous-time quantum walks (CTQWs) provide a valuable model for quantum transport, universal quantum computation and quantum spatial search, among others. Recently, the empowering role of new degrees of freedom in the Hamiltonian…
Quantum computing promises to improve the information processing power to levels unreachable by classical computation. Quantum walks are heading the development of quantum algorithms for searching information on graphs more efficiently than…
In this survey paper we give an intuitive treatment of the discrete time quantization of classical Markov chains. Grover search and the quantum walk based search algorithms of Ambainis, Szegedy and Magniez et al. will be stated as quantum…
A randomly walking quantum particle evolving by Schr\"odinger's equation searches for a unique marked vertex on the "simplex of complete graphs" in time $\Theta(N^{3/4})$. In this paper, we give a weighted version of this graph that…
The spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) The walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a…
Quantum walks, both discrete (coined) and continuous time, on a general graph of N vertices with undirected edges are reviewed in some detail. The resource requirements for implementing a quantum walk as a program on a quantum computer are…
Local search algorithms use the neighborhood relations among search states and often perform well for a variety of NP-hard combinatorial search problems. This paper shows how quantum computers can also use these neighborhood relations. An…
While the quantum query complexity of $k$-distinctness is known to be $O\left(n^{3/4-1/4(2^k-1)}\right)$ for any constant $k \geq 4$, the best previous upper bound on the time complexity was $\widetilde{O}\left(n^{1-1/k}\right)$. We give a…
Quantum walks (QW) are of crucial importance in the development of quantum information processing algorithms. Recently, several quantum algorithms have been proposed to implement network analysis, in particular to rank the centrality of…
We consider the problem of searching a d-dimensional lattice of N sites for a single marked location. We present a Hamiltonian that solves this problem in time of order sqrt(N) for d>2 and of order sqrt(N) log(N) in the critical dimension…
Mixing properties of discrete-time quantum walks on two-dimensional grids with torus-like boundary conditions are analyzed, focusing on their connection to the complexity of the corresponding abstract search algorithm. In particular, an…
The problem of finding a marked node in a graph can be solved by the spatial search algorithm based on continuous-time quantum walks (CTQW). However, this algorithm is known to run in optimal time only for a handful of graphs. In this work,…
Quantum computation using continuous-time evolution under a natural hardware Hamiltonian is a promising near- and mid-term direction toward powerful quantum computing hardware. We investigate the performance of continuous-time quantum walks…
Quantum walks have been shown to be fruitful tools in analysing the dynamic properties of quantum systems. This article proposes to use quantum walks as an approach to Quantum Neural Networks (QNNs). QNNs replace binary McCulloch-Pitts…