# Limit Theorem for a Time-Dependent Coined Quantum Walk on the Line

@inproceedings{Machida2009LimitTF, title={Limit Theorem for a Time-Dependent Coined Quantum Walk on the Line}, author={T. Machida and N. Konno}, booktitle={IWNC}, year={2009} }

We study time-dependent discrete-time quantum walks on the one-dimensional lattice. We compute the limit distribution of a two-period quantum walk defined by two orthogonal matrices. For the symmetric case, the distribution is determined by one of two matrices. Moreover, limit theorems for two special cases are presented.

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#### References

SHOWING 1-10 OF 15 REFERENCES

One-dimensional discrete-time quantum walks on random environments

- Physics, Computer Science
- Quantum Inf. Process.
- 2009

This work considers discrete-time nearest-neighbor quantum walks on random environments in one dimension and presents both quenched and annealed weak limit theorems for the quantum walk. Expand

Connecting the discrete- and continuous-time quantum walks

- Physics
- 2006

Recently, quantized versions of random walks have been explored as effective elements for quantum algorithms. In the simplest case of one dimension, the theory has remained divided into the… Expand

Weak limits for quantum random walks.

- Mathematics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2004

A general weak limit theorem for quantum random walks in one and more dimensions is formulated and proved with X(n)/n converges weakly as n--> infinity to a certain distribution which is absolutely continuous and of bounded support. Expand

Quantum walks in higher dimensions

- Mathematics, Physics
- 2002

We analyse the quantum walk in higher spatial dimensions and compare classical and quantum spreading as a function of time. Tensor products of Hadamard transformations and the discrete Fourier… Expand

Quantum Random Walks in One Dimension

- Physics, Computer Science
- Quantum Inf. Process.
- 2002

The results show that the behavior of quantum random walk is striking different from that of the classical ramdom walk. Expand

A Path Integral Approach for Disordered Quantum Walks in One Dimension

- Mathematics, Physics
- 2004

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.

Aperiodic quantum random walks.

- Physics, Medicine
- Physical review letters
- 2004

We generalize the quantum random walk protocol for a particle in a one-dimensional chain, by using several types of biased quantum coins, arranged in aperiodic sequences, in a manner that leads to a… Expand

Quantum walk with a time-dependent coin

- Physics
- 2006

We introduce quantum walks with a time-dependent coin, and show how they include, as a particular case, the generalized quantum walk recently studied by Wojcik et al. [Phys. Rev. Lett. 93, 180601… Expand

Quasiperiodic dynamics of a quantum walk on the line.

- Physics, Medicine
- Physical review letters
- 2004

The dynamics of a generalization of a quantum coin walk on the line is studied, which is a natural model for a diffusion modified by quantum or interference effects, and shows that its dynamics can be viewed as a discrete version of Bloch oscillations. Expand

A new type of limit theorems for the one-dimensional quantum random walk

- Physics, Mathematics
- 2002

In this paper we consider the one-dimensional quantum random walk X^{varphi} _n at time n starting from initial qubit state varphi determined by 2 times 2 unitary matrix U. We give a combinatorial… Expand