# Framework for discrete-time quantum walks and a symmetric walk on a binary tree

## Abstract

We formulate a framework for discrete-time quantum walks, motivated by classical random walks with memory. We present a specific representation of the classical walk with memory 2, on which this is based. The framework has no need for coin spaces, it imposes no constraints on the evolution operator other than unitarity, and is unifying of other approaches. As an example we construct a symmetric discrete-time quantum walk on the semi-infinite binary tree. The generating function of the amplitude at the root is computed in closed form, as a function of time and the initial level n in the tree, and we find the asymptotic and a full numerical solution for the amplitude. It exhibits a sharp interference peak and a power-law tail, as opposed to the exponentially decaying tail of a broadly peaked distribution of the classical symmetric random walk on a binary tree. The probability peak is orders of magnitude larger than it is for the classical walk (already at small n). The quantum walk shows a polynomial algorithmic speedup in n over the classical walk, which we conjecture to be of the order 2/3, based on strong trends in data.

- Authors:

- Department of Physics, Oregon State University, Corvallis, Oregon 97331 (United States)
- Department of Mathematics, Oregon State University, Corvallis, Oregon 97331 (United States)
- School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, Oregon 97331 (United States)

- Publication Date:

- OSTI Identifier:
- 22068656

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. A

- Additional Journal Information:
- Journal Volume: 84; Journal Issue: 3; Other Information: (c) 2011 American Institute of Physics; Country of input: Syrian Arab Republic; Journal ID: ISSN 1050-2947

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 74 ATOMIC AND MOLECULAR PHYSICS; AMPLITUDES; ASYMPTOTIC SOLUTIONS; DISTRIBUTION; GRAPH THEORY; INTERFERENCE; NUMERICAL SOLUTION; POLYNOMIALS; PROBABILITY; QUANTUM MECHANICS; RANDOMNESS; SYMMETRY; TIME DEPENDENCE; UNITARITY

### Citation Formats

```
Dimcovic, Zlatko, Rockwell, Daniel, Milligan, Ian, Burton, Robert M., Kovchegov, Yevgeniy, and Nguyen, Thinh.
```*Framework for discrete-time quantum walks and a symmetric walk on a binary tree*. United States: N. p., 2011.
Web. doi:10.1103/PHYSREVA.84.032311.

```
Dimcovic, Zlatko, Rockwell, Daniel, Milligan, Ian, Burton, Robert M., Kovchegov, Yevgeniy, & Nguyen, Thinh.
```*Framework for discrete-time quantum walks and a symmetric walk on a binary tree*. United States. doi:10.1103/PHYSREVA.84.032311.

```
Dimcovic, Zlatko, Rockwell, Daniel, Milligan, Ian, Burton, Robert M., Kovchegov, Yevgeniy, and Nguyen, Thinh. Thu .
"Framework for discrete-time quantum walks and a symmetric walk on a binary tree". United States. doi:10.1103/PHYSREVA.84.032311.
```

```
@article{osti_22068656,
```

title = {Framework for discrete-time quantum walks and a symmetric walk on a binary tree},

author = {Dimcovic, Zlatko and Rockwell, Daniel and Milligan, Ian and Burton, Robert M. and Kovchegov, Yevgeniy and Nguyen, Thinh},

abstractNote = {We formulate a framework for discrete-time quantum walks, motivated by classical random walks with memory. We present a specific representation of the classical walk with memory 2, on which this is based. The framework has no need for coin spaces, it imposes no constraints on the evolution operator other than unitarity, and is unifying of other approaches. As an example we construct a symmetric discrete-time quantum walk on the semi-infinite binary tree. The generating function of the amplitude at the root is computed in closed form, as a function of time and the initial level n in the tree, and we find the asymptotic and a full numerical solution for the amplitude. It exhibits a sharp interference peak and a power-law tail, as opposed to the exponentially decaying tail of a broadly peaked distribution of the classical symmetric random walk on a binary tree. The probability peak is orders of magnitude larger than it is for the classical walk (already at small n). The quantum walk shows a polynomial algorithmic speedup in n over the classical walk, which we conjecture to be of the order 2/3, based on strong trends in data.},

doi = {10.1103/PHYSREVA.84.032311},

journal = {Physical Review. A},

issn = {1050-2947},

number = 3,

volume = 84,

place = {United States},

year = {2011},

month = {9}

}