Path-sum solution of the Weyl quantum walk in 3 + 1 dimensions.

Area of Science:

• Quantum mechanics
• High-energy physics
• Condensed matter physics

Background:

• The Weyl quantum walk is a discrete-time model for particle evolution in 3+1 dimensions.
• It is defined on a Cayley graph and, in a specific regime, mimics Weyl's equation.
• It was recently identified as the unique homogeneous and isotropic unitary evolution on a specific Cayley graph.

Purpose of the Study:

• To provide the general solution for the Weyl quantum walk in the position representation.
• To derive the analytical expression for the propagator (transition amplitude) between nodes on the graph.
• To explore the quantum nature of the walk through path interference.

Main Methods:

• Utilizing the binary encoding of admissible paths on the Cayley graph.
• Leveraging the semigroup structure of the walk's transition matrices.
• Developing an analytical expression for the propagator in the position representation.

Main Results:

• The general solution for the Weyl quantum walk evolution is obtained.
• An analytical expression for the propagator is derived, detailing transitions between graph nodes.
• The solution highlights the role of quantum interference in the walk's behavior.

Conclusions:

• The study provides a complete description of the Weyl quantum walk dynamics.
• The findings offer insights into quantum information processing and fundamental physics.
• The work contributes to understanding quantum dynamics on complex graph structures.