Two-component Dirac-like Hamiltonian for generating quantum walk on one-, two- and three-dimensional lattices

Area of Science:

• Quantum mechanics
• Condensed matter physics
• Quantum information science

Background:

• Discrete-time quantum walks are a fundamental model in quantum computation.
• Quantum walks on lattices are used to simulate various quantum phenomena.
• Dirac-like Hamiltonians describe relativistic quantum particles.

Purpose of the Study:

• To derive a general two-component Dirac-like Hamiltonian for discrete-time quantum walks on multi-dimensional lattices.
• To investigate the role of coin operations in different lattice dimensions.
• To establish a framework for simulating and controlling quantum systems.

Main Methods:

• Utilizing the unitary operator for two-state discrete-time quantum walks.
• Employing Pauli basis states as position translation states.
• Extending the quantum walk model from 1D to 2D and 3D lattices.

Main Results:

• A general two-component Dirac-like Hamiltonian was obtained for quantum walks on 1D, 2D, and 3D lattices.
• Three distinct Hamiltonians were derived for 1D lattices using different Pauli basis pairs.
• The necessity of external coin operations was shown to decrease in higher dimensions, serving instead as a control resource.

Conclusions:

• The derived two-component Hamiltonian provides a unified framework for quantum walks on various lattices.
• This framework facilitates the simulation, control, and study of quantum systems governed by Dirac-like Hamiltonians.
• The findings offer new perspectives on quantum walk dynamics in higher dimensions.