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One-dimensional three-state quantum walk.

Norio Inui¹, Norio Konno, Etsuo Segawa

¹Graduate School of Engineering, University of Hyogo, 2167, Shosha, Himeji, Hyogo, 671-2280, Japan. inui@eng.u-hyogo.ac.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|December 31, 2005

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This study explores a three-state quantum walk, revealing particles can become trapped near the origin. Unlike simpler models, the probability of return to the origin remains significant over time.

Area of Science:

Quantum mechanics
Condensed matter physics
Statistical mechanics

Background:

Quantum walks are quantum analogues of classical random walks.
Hadamard walks are a common type of quantum walk with discrete coin operations.
Previous studies often focused on two-state quantum walks.

Purpose of the Study:

Investigate a generalized one-dimensional Hadamard walk with three inner states.
Analyze the particle's wave function and spatial probability distribution.
Compare the behavior to two-state Hadamard walks.

Main Methods:

Rigorous calculation of the wave function for arbitrary initial qubit states.
Analysis of the spatial probability distribution over time.
Mathematical modeling of a three-state quantum system.

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Main Results:

The three-state quantum walk allows movement left, right, or staying in place.
The probability of finding the particle at the origin does not converge to zero for most initial states.
Particles exhibit a high probability of being trapped near the origin after many time steps.

Conclusions:

The three-state quantum walk exhibits unique trapping behavior not seen in two-state systems.
This suggests potential applications in quantum computing and simulation where localization is desired.
The generalized Hadamard walk offers a richer model for quantum transport phenomena.