scholarly journals Characterizing the Relationship Between Unitary Quantum Walks and Non-Homogeneous Random Walks

2021 ◽  
Author(s):  
Matheus Guedes de Andrade ◽  
Franklin De Lima Marquezino ◽  
Daniel Ratton Figueiredo
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Random Walks ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Time Dependent ◽  
Matrix Sequence ◽  
The Relationship

Quantum walks on graphs are ubiquitous in quantum computing finding a myriad of applications. Likewise, random walks on graphs are a fundamental building block for a large number of algorithms with diverse applications. While the relationship between quantum and random walks has been recently discussed in specific scenarios, this work establishes a formal equivalence between the two processes on arbitrary finite graphs and general conditions for shift and coin operators. It requires empowering random walks with time heterogeneity, where the transition probability of the walker is non-uniform and time dependent. The equivalence is obtained by equating the probability of measuring the quantum walk on a given node of the graph and the probability that the random walk is at that same node, for all nodes and time steps. The first result establishes procedure for a stochastic matrix sequence to induce a random walk that yields the exact same vertex probability distribution sequence of any given quantum walk, including the scenario with multiple interfering walkers. The second result establishes a similar procedure in the opposite direction. Given any random walk, a time-dependent quantum walk with the exact same vertex probability distribution is constructed. Interestingly, the matrices constructed by the first procedure allows for a different simulation approach for quantum walks where node samples respect neighbor locality and convergence is guaranteed by the law of large numbers, enabling efficient (polynomial-time) sampling of quantum graph trajectories. Furthermore, the complexity of constructing this sequence of matrices is discussed in the general case.

scholarly journals A limit theorem for a splitting distribution of a quantum walk

2018 ◽  
Vol 16 (03) ◽  
pp. 1850023
Author(s):  
Takuya Machida
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Random Walks ◽  
Discrete Time ◽  
Limit Distribution ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Unitary Evolution ◽  
Time Quantum ◽  
Long Time

Discrete-time quantum walks are considered a counterpart of random walks and their study has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

scholarly journals Efficient circuits for quantum walks

2010 ◽  
Vol 10 (5&6) ◽  
pp. 420-434
Author(s):  
C.-F. Chiang ◽  
D. Nagaj ◽  
P. Wocjan
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Problem Size ◽  
Integrable Probability ◽  
The Difference ◽  
General Method

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions \cite{GroverRudolph}. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk \cite{Szegedy}, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents \cite{Vigoda, Vazirani} and sampling from binary contingency tables \cite{Stefankovi}. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

scholarly journals Multiple transitions between normal and hyperballistic diffusion in quantum walks with time-dependent jumps

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Marcelo A. Pires ◽  
Giuseppe Di Molfetta ◽  
Sílvio M. Duarte Queirós
Keyword(s):  
Probability Distribution ◽  
Uniform Distribution ◽  
Quantum Walk ◽  
Step Length ◽  
Quantum Walks ◽  
Time Dependent ◽  
Asymptotic Approach ◽  
Diffusive Regime ◽  
Dynamical Regimes ◽  
Functional Forms

AbstractWe extend to the gamut of functional forms of the probability distribution of the time-dependent step-length a previous model dubbed Elephant Quantum Walk, which considers a uniform distribution and yields hyperballistic dynamics where the variance grows cubicly with time, σ2 ∝ t3, and a Gaussian for the position of the walker. We investigate this proposal both locally and globally with the results showing that the time-dependent interplay between interference, memory and long-range hopping leads to multiple transitions between dynamical regimes, namely ballistic → diffusive → superdiffusive → ballistic → hyperballistic for non-hermitian coin whereas the first diffusive regime is quelled for implementations using the Hadamard coin. In addition, we observe a robust asymptotic approach to maximal coin-space entanglement.

Spreading behavior of quantum walks induced by random walks

2017 ◽  
Vol 17 (5&6) ◽  
pp. 399-414
Author(s):  
Yusuke Higuchi ◽  
Etsuo Segawa
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Finite Length ◽  
Limit Distribution ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Round Trip ◽  
Spreading Behavior

In this paper, we consider the quantum walk on Z with attachment of one-length path periodically. This small modification to Z provides localization of the quantum walk. The eigenspace causing this localization is generated by finite length round trip paths. We find that the localization is due to the eigenvalues of an underlying random walk. Moreover we find that the transience of the underlying random walk provides a slow down of the pseudo velocity of the induced quantum walk and a different limit distribution from the Konno distribution.

Return probability of quantum walks with final-time dependence

2011 ◽  
Vol 11 (9&10) ◽  
pp. 761-773
Author(s):  
Yusuke Ide ◽  
Norio Konno ◽  
Takuya Machida ◽  
Etsuo Segawa
Keyword(s):  
Random Walk ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Time Dependent ◽  
One Dimension ◽  
Final Time ◽  
Counting Approach ◽  
Return Probability ◽  
Time Quantum ◽  
Path Counting

We analyze final-time dependent discrete-time quantum walks in one dimension. We compute asymptotics of the return probability of the quantum walk by a path counting approach. Moreover, we discuss a relation between the quantum walk and the corresponding final-time dependent classical random walk.

scholarly journals Quantum walks and elliptic integrals

2010 ◽  
Vol 20 (6) ◽  
pp. 1091-1098 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Random Walk ◽  
Generating Function ◽  
Elliptic Integral ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Elliptic Integrals ◽  
Two Dimensional ◽  
One Dimensional ◽  
Similar Expression ◽  
Return Probability

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

scholarly journals One dimensional quantum walks with memory

2010 ◽  
Vol 10 (5&6) ◽  
pp. 509-524
Author(s):  
M. Mc Gettrick
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Walks ◽  
One Dimensional ◽  
With Memory ◽  
Previous Step

We investigate the quantum versions of a one-dimensional random walk, whose corresponding Markov Chain is of order 2. This corresponds to the walk having a memory of one previous step. We derive the amplitudes and probabilities for these walks, and point out how they differ from both classical random walks, and quantum walks without memory.

Transition probability matrices for correlated random walks

1980 ◽  
Vol 17 (01) ◽  
pp. 253-258 ◽  
Author(s):  
R. B. Nain ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Difference Equations ◽  
Generating Functions ◽  
Transition Probability ◽  
Correlated Random Walk ◽  
One Dimensional ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.

Exploring activity-driven network with biased walks

2017 ◽  
Vol 28 (09) ◽  
pp. 1750111
Author(s):  
Yan Wang ◽  
Ding Juan Wu ◽  
Fang Lv ◽  
Meng Long Su
Keyword(s):  
Random Walk ◽  
Diffusion Process ◽  
Random Walks ◽  
Stationary Distribution ◽  
Significant Role ◽  
Search Strategy ◽  
Transition Probability ◽  
Edge Weighting ◽  
Coverage Function ◽  
Analytical Expressions

We investigate the concurrent dynamics of biased random walks and the activity-driven network, where the preferential transition probability is in terms of the edge-weighting parameter. We also obtain the analytical expressions for stationary distribution and the coverage function in directed and undirected networks, all of which depend on the weight parameter. Appropriately adjusting this parameter, more effective search strategy can be obtained when compared with the unbiased random walk, whether in directed or undirected networks. Since network weights play a significant role in the diffusion process.

RANDOM WALKS AND DIMENSIONS OF RANDOM TREES

2010 ◽  
Vol 13 (04) ◽  
pp. 677-689
Author(s):  
MOKHTAR H. KONSOWA
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Trees ◽  
The Relationship

We study the relationship between the type of the random walk on some random trees and the structure of those trees in terms of fractal and resistance dimensions. This paper generalizes some results of Refs. 8–10.

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