Differences between Quantum Walks and Classical Random Walks in Limit Distributions

2011 ◽  
Author(s):  
Yutaka Shikano
Keyword(s):  
Random Walks ◽  
Quantum Walks ◽  
Limit Distributions

scholarly journals Area of Brownian Motion with Generatingfunctionology

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michel Nguyên Thê
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Generating Functions ◽  
Limit Distributions ◽  
Dyck Paths ◽  
Different Types ◽  
International Audience

International audience This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.

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.

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 Relation between random walks and quantum walks

2015 ◽  
Vol 91 (5) ◽  
Author(s):  
Stefan Boettcher ◽  
Stefan Falkner ◽  
Renato Portugal
Keyword(s):  
Random Walks ◽  
Quantum Walks

scholarly journals Properties of quantum stochastic walks from the asymptotic scaling exponent

2018 ◽  
Vol 18 (3&4) ◽  
pp. 181-197
Author(s):  
Krzysztof Domino ◽  
Adam Glos ◽  
Mateusz Ostaszewski ◽  
Lukasz Pawela ◽  
Przemyslaw Sadowski
Keyword(s):  
Probability Distribution ◽  
Random Walks ◽  
Quantum Walks ◽  
The Other ◽  
Scaling Exponent ◽  
Continuous Change ◽  
Ballistic Regime ◽  
Second Moment ◽  
The Mean ◽  
Asymptotic Scaling

This work focuses on the study of quantum stochastic walks, which are a generalization of coherent, \ie unitary quantum walks. Our main goal is to present a measure of a coherence of the walk. To this end, we utilize the asymptotic scaling exponent of the second moment of the walk \ie of the mean squared distance covered by a walk. As the quantum stochastic walk model encompasses both classical random walks and quantum walks, we are interested how the continuous change from one regime to the other influences the asymptotic scaling exponent. Moreover this model allows for behavior which is not found in any of the previously mentioned model -- a model with global dissipation. We derive the probability distribution for the walker, and determine the asymptotic scaling exponent analytically, showing that ballistic regime of the walk is maintained even at large dissipation strength.

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 A PATH INTEGRAL APPROACH FOR DISORDERED QUANTUM WALKS IN ONE DIMENSION

2005 ◽  
Vol 05 (04) ◽  
pp. L529-L537 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Random Walks ◽  
Path Integral ◽  
Quantum Walks ◽  
Random Fluctuation ◽  
One Dimension ◽  
Integral Approach ◽  
Path Integral Approach ◽  
Present Letter

The present letter gives a rigorous way from quantum to classical random walks by introducing an independent random fluctuation and then taking expectations based on a path integral approach.

scholarly journals Limit distributions of two-dimensional quantum walks

2008 ◽  
Vol 77 (6) ◽  
Author(s):  
Kyohei Watabe ◽  
Naoki Kobayashi ◽  
Makoto Katori ◽  
Norio Konno
Keyword(s):  
Quantum Walks ◽  
Two Dimensional ◽  
Limit Distributions
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