Spectral Property of Magnetic Quantum Walk on Hypercube

Summary

The paper introduces a model of magnetic quantum walk on a general hypercube using quantum Bernoulli noises and investigates its spectral properties, showing that the point-spectrum and approximate-spectrum of the evolution operator are independent of the magnetic potential.

Highlights

• Introduced a model of magnetic quantum walk on a general hypercube using quantum Bernoulli noises.
• Investigated the spectral properties of the evolution operator W(ν).
• Showed that the point-spectrum and approximate-spectrum of W(ν) are independent of the magnetic potential ν.
• Proved that the algebraic sums Uσ of the coin operator system C are completely independent of the magnetic potential ν.
• Demonstrated that the walk W(ν) has spectral stability with respect to the magnetic potential ν.
• Compared the walk in this paper with the one in [14] and highlighted the generalization of technical propositions and theorems.
• Suggested that the magnetic quantum walk can be analyzed from a viewpoint of probability distribution.

Key Insights

• The use of quantum Bernoulli noises allows for the introduction of a magnetic potential to the quantum walk on a hypercube, enabling the study of its spectral properties.
• The independence of the point-spectrum and approximate-spectrum of W(ν) from the magnetic potential ν suggests that the walk has spectral stability, which is an important property for quantum systems.
• The generalization of technical propositions and theorems from [14] highlights the robustness of the approach and its potential for application to other quantum systems.
• The comparison with the walk in [14] demonstrates the flexibility of the model and its ability to accommodate different magnetic potentials.
• The suggestion that the magnetic quantum walk can be analyzed from a viewpoint of probability distribution opens up new avenues for research and potential applications.
• The use of algebraic sums Uσ of the coin operator system C provides a powerful tool for analyzing the spectral properties of the evolution operator W(ν).
• The independence of the algebraic sums Uσ from the magnetic potential ν underscores the stability of the walk's spectral properties.

Mindmap

Citation

Wang, C. (2024). Spectral Property of Magnetic Quantum Walk on Hypercube (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.18401