A Quantum Random Walk Search Algorithm

Abstract

Quantum random walks on graphs have been shown to display many interesting properties, including exponentially fast hitting times when compared with their classical counterparts. However, it is still unclear how to use these novel properties to gain an algorithmic speed-up over classical algorithms. In this paper, we present a quantum search algorithm based on the quantum random walk architecture that provides such a speed-up. It will be shown that this algorithm performs an oracle search on a database of $N$ items with $O(\sqrt{N})$ calls to the oracle, yielding a speed-up similar to other quantum search algorithms. It appears that the quantum random walk formulation has considerable flexibility, presenting interesting opportunities for development of other, possibly novel quantum algorithms.

13 pages, 3 figures

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• Quantum Search of Spatial Regions
• Complexity for one-dimensional discrete time quantum walk circuits
• Correcting for Potential Barriers in Quantum Walk Search
• Approaching unstructured search from function bilateral symmetry detection - A quantum algorithm
• Return probability of quantum and correlated random walks
• Orthogonal projections of hypercubes
• Characterization of anomalous diffusion in one-dimensional quantum walks
• Faster Search of Clustered Marked States with Lackadaisical Quantum Walks
• Generation of topologically useful entangled states
• Lackadaisical quantum walks with multiple marked vertices
• Exceptional configurations of quantum walks with Grover's coin
• Dissipation in fermionic two-body continuous-time quantum walk under the steepest entropy ascent formalism
• Hybrid atom-molecule quantum walks in one-dimensional optical lattice
• State transfer in Grover walks on unitary and quadratic unitary Cayley graphs over finite commutative rings
• Strong dispersion property for the quantum walk on the hypercube
• Convergence of coined quantum walks on d-dimensional Euclidean space
• Realization of the probability laws in the quantum central limit theorems by a quantum walk
• Key graph properties affecting transport efficiency of flip-flop Grover percolated quantum walks
• Dynamical Localization of Quantum Walks in Random Environments
• Low Rank Approximation in Simulations of Quantum Algorithms
• Continuous-time quantum walks on a defective lattice: Boosting the spreading of delocalized states through Parrondo's strategy
• Universal scaling hypothesis of quantum spatial search in complex networks
• Quantum annealing and condensed matter physics
• Transition in Splitting Probabilities of Quantum Walks
• Reversion of Quantum Walks via interventions on coin space
• Neighborhood-History Quantum Walk
• Graph identification by quantum walks
• Absorption-based qubit estimation in discrete-time quantum walks
• Projection Theorem for Discrete-Time Quantum Walks
• Three-state quantum walk on the Cayley Graph of the Dihedral Group
• Spectral quantization of discrete random walks on half-line, and orthogonal polynomials on the unit circle
• Quantum search of matching on signed graphs
• Asymptotic reduced density matrix of discrete-time quantum walks