Spectral theory of sampled quantum-walk averages

Develop a spectral theory for averages of continuous-time quantum-walk mixing matrices obtained by sampling observation times from probability laws, including the spectral-moment structure of such averages.

Background

The paper studies uniform average mixing for continuous-time quantum walks on strongly regular graphs by representing averaged transition probabilities through cosine moments of a time law. It notes that this spectral-moment perspective is part of a broader framework in which a probability law is viewed as a sampling distribution.

The cited open problem asks for a general spectral theory of these averages. The present paper resolves the uniform-average-mixing existence question for connected strongly regular graphs, but does not develop a general spectral theory beyond that setting.

References

The general spectral-moment viewpoint is developed in , where a law is called a sampling distribution, and a spectral theory of such averages is posed as an open problem in Open Problem~4.6.

— Moment Methods for Uniform Average Mixing on Strongly Regular Graphs  (2610.10017 - Kang, 7 Oct 2026) in Section 1, Scope and main results