scholarly journals Searching via Nonlinear Quantum Walk on the 2D-Grid

Algorithms ◽  
2020 ◽  
Vol 13 (11) ◽  
pp. 305
Author(s):  
Giuseppe Di Molfetta ◽  
Basile Herzog
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Optimal Choice ◽  
Marked Vertex ◽  
Linear Dispersion ◽  
Linear Dispersion Relation ◽  
Nonlinear Phase ◽  
Computational Advantage ◽  
Nonlinear Quantum ◽  
Amplitude Amplification

We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer, rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional grid, keeping the same computational advantage with respect to the classical algorithms. For this purpose, we have considered the free lattice Hamiltonian, with linear dispersion relation introduced by Childs and Ge The numerical simulations showed that the walker finds the marked vertex in O(N1/4log3/4N) steps, with probability O(1/logN), for an overall complexity of O(N1/4log5/4N), using amplitude amplification. We also proved that there exists an optimal choice of the walker parameters to avoid the time measurement precision affecting the complexity searching time of the algorithm.

scholarly journals Continuous limits of linear and nonlinear quantum walks

2019 ◽  
Vol 32 (04) ◽  
pp. 2050008 ◽  
Author(s):  
Masaya Maeda ◽  
Akito Suzuki
Keyword(s):  
Quantum Walk ◽  
Fixed Time ◽  
Quantum Walks ◽  
Continuous Limit ◽  
Time Interval ◽  
Fixed Time Interval ◽  
Nonlinear Dirac Equation ◽  
Nonlinear Quantum ◽  
Special Case ◽  
Linear Quantum

In this paper, we consider the continuous limit of a nonlinear quantum walk (NLQW) that incorporates a linear quantum walk as a special case. In particular, we rigorously prove that the walker (solution) of the NLQW on a lattice [Formula: see text] uniformly converges (in Sobolev space [Formula: see text]) to the solution to a nonlinear Dirac equation (NLD) on a fixed time interval as [Formula: see text]. Here, to compare the walker defined on [Formula: see text] and the solution to the NLD defined on [Formula: see text], we use Shannon interpolation.

scholarly journals Quantum search on Hanoi network

2019 ◽  
Vol 17 (07) ◽  
pp. 1950060
Author(s):  
Pulak Ranjan Giri ◽  
Vladimir Korepin
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Periodic Lattice ◽  
Quantum Search ◽  
Main Structure ◽  
Running Time ◽  
One Dimensional ◽  
Target State ◽  
Amplitude Amplification ◽  
Walk Algorithm

Hanoi network (HN) has a one-dimensional periodic lattice as its main structure with additional long-range edges, which allow having efficient quantum walk algorithm that can find a target state on the network faster than the exhaustive classical search. In this paper, we use regular quantum walks and lackadaisical quantum walks, respectively, to search for a target state. From the curve fitting of the numerical results for HN of degrees three and four, we find that their running time for the regular quantum walks are followed by amplitude amplification scales as [Formula: see text] and [Formula: see text], respectively. For the search by lackadaisical quantum walks, the running time scales are as [Formula: see text] and [Formula: see text], respectively.

Correcting for potential barriers in quantum walk search

2015 ◽  
Vol 15 (15&16) ◽  
pp. 1365-1372
Author(s):  
Andris Ambainis ◽  
Thomas G. Wong
Keyword(s):  
Potential Energy ◽  
Complete Graph ◽  
Quantum Particle ◽  
Quantum Walk ◽  
Energy Barriers ◽  
Marked Vertex ◽  
Potential Barriers ◽  
Amplitude Amplification

A randomly walking quantum particle searches in Grover’s Θ(√ N) iterations for a marked vertex on the complete graph of N vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a “coin” flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with N. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the amplification recovers the Θ(√ N) runtime.

scholarly journals Analysis of lackadaisical quantum walks

2020 ◽  
Vol 20 (13&14) ◽  
pp. 1138-1153
Author(s):  
Peter Hoyer ◽  
Zhan Yu
Keyword(s):  
Random Walk ◽  
Success Probability ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Transitive Graph ◽  
Marked Vertex ◽  
Johnson Graphs ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

The lackadaisical quantum walk is a quantum analogue of the lazy random walk obtained by adding a self-loop to each vertex in the graph. We analytically prove that lackadaisical quantum walks can find a unique marked vertex on any regular locally arc-transitive graph with constant success probability quadratically faster than the hitting time. This result proves several speculations and numerical findings in previous work, including the conjectures that the lackadaisical quantum walk finds a unique marked vertex with constant success probability on the torus, cycle, Johnson graphs, and other classes of vertex-transitive graphs. Our proof establishes and uses a relationship between lackadaisical quantum walks and quantum interpolated walks for any regular locally arc-transitive graph.

Global Optimization With Quantum Walk Enhanced Grover Search

2014 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Global Optimization ◽  
Optimization Problems ◽  
Early Stage ◽  
Hybrid Approach ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Tunneling Effect ◽  
Grover’S Algorithm ◽  
Grover's Algorithm ◽  
Grover Search

One of the significant breakthroughs in quantum computation is Grover’s algorithm for unsorted database search. Recently, the applications of Grover’s algorithm to solve global optimization problems have been demonstrated, where unknown optimum solutions are found by iteratively improving the threshold value for the selective phase shift operator in Grover rotation. In this paper, a hybrid approach that combines continuous-time quantum walks with Grover search is proposed. By taking advantage of quantum tunneling effect, local barriers are overcome and better threshold values can be found at the early stage of search process. The new algorithm based on the formalism is demonstrated with benchmark examples of global optimization. The results between the new algorithm and the Grover search method are also compared.

Higher-dimensional open quantum walk in environment of quantum Bernoulli noises

2021 ◽  
pp. 2250001
Author(s):  
Ce Wang
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Initial State ◽  
Quantum Random Walks ◽  
Open Quantum ◽  
Higher Dimensional ◽  
Gauss Type ◽  
Open Quantum Random Walks ◽  
Limit Probability ◽  
Creation Operators

Open quantum walks (OQWs) (also known as open quantum random walks) are quantum analogs of classical Markov chains in probability theory, and have potential application in quantum information and quantum computation. Quantum Bernoulli noises (QBNs) are annihilation and creation operators acting on Bernoulli functionals, and can be used as the environment of an open quantum system. In this paper, by using QBNs as the environment, we introduce an OQW on a general higher-dimensional integer lattice. We obtain a quantum channel representation of the walk, which shows that the walk is indeed an OQW. We prove that all the states of the walk are separable provided its initial state is separable. We also prove that, for some initial states, the walk has a limit probability distribution of higher-dimensional Gauss type. Finally, we show links between the walk and a unitary quantum walk recently introduced in terms of QBNs.

Exploring scalar quantum walks on Cayley graphs

2008 ◽  
Vol 8 (1&2) ◽  
pp. 68-81
Author(s):  
O.L. Acevedo ◽  
J. Roland ◽  
N.J. Cerf
Keyword(s):  
Evolution Operator ◽  
Cayley Graphs ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Constructive Method ◽  
Quantum Evolution ◽  
Necessary Condition ◽  
Internal Space ◽  
Special Cases ◽  
Scalar Quantum

A quantum walk, \emph{i.e.}, the quantum evolution of a particle on a graph, is termed \emph{scalar} if the internal space of the moving particle (often called the coin) has dimension one. Here, we study the existence of scalar quantum walks on Cayley graphs, which are built from the generators of a group. After deriving a necessary condition on these generators for the existence of a scalar quantum walk, we present a general method to express the evolution operator of the walk, assuming homogeneity of the evolution. We use this necessary condition and the subsequent constructive method to investigate the existence of scalar quantum walks on Cayley graphs of groups presented with two or three generators. In this restricted framework, we classify all groups -- in terms of relations between their generators -- that admit scalar quantum walks, and we also derive the form of the most general evolution operator. Finally, we point out some interesting special cases, and extend our study to a few examples of Cayley graphs built with more than three generators.

Sign in / Sign up

Export Citation Format

Share Document