INTERACTING FOCK SPACES AND THE MOMENTS OF THE LIMIT DISTRIBUTIONS FOR QUANTUM RANDOM WALKS

2013 ◽  
Vol 16 (01) ◽  
pp. 1350003 ◽  
Author(s):  
CHUL KI KO ◽  
HYUN JAE YOO
Keyword(s):  
Spectral Analysis ◽  
Random Walks ◽  
Vacuum Expectation ◽  
High Dimensional ◽  
Fock Spaces ◽  
Quantum Random Walks ◽  
One Dimensional ◽  
Limit Distributions ◽  
Time Quantum ◽  
Adjoint Operators

We investigate the limit distributions of the discrete time quantum random walks on lattice spaces via a spectral analysis of concretely given self-adjoint operators. We discuss the interacting Fock spaces associated with the limit distributions. Thereby, we represent the moments of the limit distribution by vacuum expectation of the monomials of the Fock operator. We get formulas not only for one-dimensional walks but also for high-dimensional walks.

Limit theorems and absorption problems for quantum radom walks in one dimension

2002 ◽  
Vol 2 (Special) ◽  
pp. 578-595
Author(s):  
N. Konno
Keyword(s):  
Random Walks ◽  
Limit Theorems ◽  
The Other ◽  
One Dimension ◽  
Unitary Matrices ◽  
Quantum Random Walks ◽  
One Dimensional ◽  
The Difference ◽  
The One

In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of one-dimensional quantum random walks determined by $2 \times 2$ unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intensively by Ambainis et al. in 2001. The difference between both types of quantum random walks is also clarified.

EVOLUTION OF CONTINUOUS-TIME QUANTUM RANDOM WALKS ON CIRCLES

2005 ◽  
Vol 05 (01) ◽  
pp. L73-L83 ◽  
Author(s):  
NORIO INUI ◽  
KOICHIRO KASAHARA ◽  
YOSHINAO KONISHI ◽  
NORIO KONNO
Keyword(s):  
Standard Deviation ◽  
Random Walks ◽  
Discrete Time ◽  
Continuous Time ◽  
Quantum Random Walks ◽  
Time Quantum ◽  
Mixing Property ◽  
Temporal Standard Deviation

This work deals with both instantaneous uniform mixing property and temporal standard deviation for continuous-time quantum random walks on circles in order to study their fluctuations comparing with discrete-time quantum random walks, and continuous- and discrete-time classical random walks.

scholarly journals Quantum random walks in one dimension via generating functions

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Andrew Bressler ◽  
Robin Pemantle
Keyword(s):  
Random Walks ◽  
Generating Functions ◽  
Nearest Neighbor ◽  
Function Analysis ◽  
One Dimension ◽  
Quantum Random Walks ◽  
Linear Speed ◽  
One Dimensional ◽  
Asymptotic Term ◽  
International Audience

International audience We analyze nearest neighbor one-dimensional quantum random walks with arbitrary unitary coin-flip matrices. Using a multivariate generating function analysis we give a simplified proof of a known phenomenon, namely that the walk has linear speed rather than the diffusive behavior observed in classical random walks. We also obtain exact formulae for the leading asymptotic term of the wave function and the location probabilities.

QUANTUM PARRONDO'S GAMES CONSTRUCTED BY QUANTUM RANDOM WALKS

2013 ◽  
Vol 12 (04) ◽  
pp. 1350024 ◽  
Author(s):  
MIN LI ◽  
YONG-SHENG ZHANG ◽  
GUANG-CAN GUO
Keyword(s):  
Random Walks ◽  
Discrete Time ◽  
Quantum Walks ◽  
Quantum Random Walks ◽  
Parrondo’S Paradox ◽  
Time Quantum

We construct a Parrondo's game using discrete-time quantum walks (DTQWs). Two losing games are represented by two different coin operators. By mixing the two coin operators UA(αA, βA, γA) and UB(αB, βB, γB), we may win the game. Here, we mix the two games in position instead of time. With a number of selections of the parameters, we can win the game with sequences ABB, ABBB, etc. If we set βA = 45°, γA = 0, αB = 0, βB = 88°, we find game 1 with [Formula: see text], [Formula: see text] will win and get the most profit. If we set αA = 0, βA = 45°, αB = 0, βB = 88° and game 2 with [Formula: see text], [Formula: see text] will win most. Game 1 is equivalent to game 2 with changes in sequences and steps. But at large enough steps, the game will lose at last. Parrondo's paradox does not exist in classical situation with our model.

scholarly journals Limit Theorems for a Generalized Feller Game

2013 ◽  
Vol 50 (1) ◽  
pp. 54-63 ◽  
Author(s):  
Keisuke Matsumoto ◽  
Toshio Nakata
Keyword(s):  
Random Walks ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Convergence In Distribution ◽  
Stable Law ◽  
One Dimensional ◽  
Limit Distributions ◽  
Polynomial Size ◽  
Large Numbers ◽  
Geometric Size

In this paper we study limit theorems for the Feller game which is constructed from one-dimensional simple symmetric random walks, and corresponds to the St. Petersburg game. Motivated by a generalization of the St. Petersburg game which was investigated by Gut (2010), we generalize the Feller game by introducing the parameter α. We investigate limit distributions of the generalized Feller game corresponding to the results of Gut. Firstly, we give the weak law of large numbers for α=1. Moreover, for 0<α≤1, we have convergence in distribution to a stable law with index α. Finally, some limit theorems for a polynomial size and a geometric size deviation are given.

scholarly journals Comparing Algorithms for Graph Isomorphism Using Discrete- and Continuous-Time Quantum Random Walks

2013 ◽  
Vol 10 (7) ◽  
pp. 1653-1661 ◽  
Author(s):  
Kenneth Rudinger ◽  
John King Gamble ◽  
Eric Bach ◽  
Mark Friesen ◽  
Robert Joynt ◽  
...  
Keyword(s):  
Random Walks ◽  
Continuous Time ◽  
Graph Isomorphism ◽  
Quantum Random Walks ◽  
Time Quantum
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