Two models of quantum random walk

Open Physics ◽  
2003 ◽  
Vol 1 (4) ◽  
Author(s):  
Jozef Košík
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Upper Bound ◽  
Explicit Solution ◽  
New Method ◽  
Speed Of Convergence ◽  
Quantum Random Walk ◽  
Dimensional Lattice ◽  
Special Model ◽  
One Dimensional

AbstractWe present an overview of two models of quantum random walk. In the first model, the discrete quantum random walk, we present the explicit solution for the recurring amplitude of the quantum random walk on a one-dimensional lattice. We also introduce a new method of solving the problem of random walk in the most general case and use it to derive the hitting amplitude for quantum random walk on the hypercube. The second is a special model based on a local interaction between neighboring spin-1/2 particles on a one-dimensional lattice. We present explicit results for the relevant quantities and obtain an upper bound on the speed of convergence to limiting probability distribution.

Phase transition in one-dimensional random walk with partially reflecting boundaries

1985 ◽  
Vol 17 (03) ◽  
pp. 594-606 ◽  
Author(s):  
Ora E. Percus
Keyword(s):  
Random Walk ◽  
Transition Probability ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Boundary Case ◽  
Reflecting Boundaries ◽  
The Mean ◽  
The One ◽  
The Relationship ◽  
Asymptotic Forms

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p). The probability distribution (Pn (m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of P n(m) result, depending on the relationship between transition probability and the reflection probability.

Phase transition in one-dimensional random walk with partially reflecting boundaries

1985 ◽  
Vol 17 (3) ◽  
pp. 594-606 ◽  
Author(s):  
Ora E. Percus
Keyword(s):  
Random Walk ◽  
Transition Probability ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Boundary Case ◽  
Reflecting Boundaries ◽  
The Mean ◽  
The One ◽  
The Relationship ◽  
Asymptotic Forms

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p).The probability distribution (Pn(m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of Pn(m) result, depending on the relationship between transition probability and the reflection probability.

scholarly journals The Maximum of a Symmetric Next Neighbor Walk on the Nonnegative Integers

2014 ◽  
Vol 51 (01) ◽  
pp. 162-173
Author(s):  
Ora E. Percus ◽  
Jerome K. Percus
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Generating Functions ◽  
Two Dimensional ◽  
Symmetric Random Walk ◽  
One Dimensional ◽  
Dimensional Distribution

We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the two-dimensional probability distribution P{S n = x, max1≤j≤n S n = a} of being at position x after n steps, while the maximal location that the walker has achieved during these n steps is a. We also obtain the familiar (marginal) one-dimensional distribution for S n = x, but more importantly that for max1≤j≤n S j = a asymptotically at fixed a 2 / n. We are able to compute and compare the expectations and variances of the two one-dimensional distributions, finding that they have qualitatively similar forms, but differ quantitatively in the anticipated fashion.

One-dimensional quantum random walk for fermions and bosons

1995 ◽  
Vol 52 (4) ◽  
pp. 3381-3389 ◽  
Author(s):  
Salvador Godoy ◽  
Francisco Espinosa
Keyword(s):  
Random Walk ◽  
Quantum Random Walk ◽  
One Dimensional
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