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Quantum random walks and piecewise deterministic evolutions.
1Fakultät für Physik and BiBoS, Universität Bielefeld, D-33619 Bielefeld, Germany.
Physical Review Letters
|April 20, 2004
Summary
Quantum random walks (QRW) driven by a Hadamard coin follow a hyperbolic equation. This allows simple analytical descriptions of the quantum random walk
Area of Science:
- Quantum mechanics
- Statistical physics
- Probability theory
Background:
- Quantum random walks (QRW) are quantum analogues of classical random walks.
- Understanding the behavior of QRWs in continuous space and time is crucial for quantum computing and simulation.
Purpose of the Study:
- To analyze the probability density of a Hadamard-driven QRW in the continuous limit.
- To derive analytical expressions for the QRW's position and variance.
Main Methods:
- Analysis of the hyperbolic evolution equation governing the QRW probability density.
- Investigation of the role of spatially inhomogeneous transition rates and nonlinear drift terms.
Main Results:
- The QRW probability density in the continuous limit follows a hyperbolic evolution equation.
- The QRW position is analytically tractable despite a nonlinear drift term.
- The derived variance exhibits quadratic time dependence, characteristic of QRWs.
Conclusions:
- The study provides a simplified analytical framework for understanding Hadamard-driven QRWs in continuous space-time.
- The findings offer insights into the unique statistical properties of quantum walks compared to classical ones.