Complete theoretical characterization of quantum-walk vs. adjacency spectra

Establish a complete theoretical characterization of the relationship between the spectrum of the coined quantum walk operator (equivalently, the quantum walk characteristic polynomial) and the classical adjacency spectrum across graph families, determining precisely for which graphs the quantum walk spectrum strictly refines the adjacency spectrum and under what conditions they coincide.

Background

Prior work demonstrated empirically and in special cases that the quantum walk spectrum can provide a graph invariant strictly stronger than the classical adjacency spectrum. However, a general theory delineating exactly when and how this strengthening occurs has not been fully established.

The present paper settles a significant special case by proving completeness for strongly regular graphs of prime order with degree k ≥ 6, but the global, all-graphs characterization alluded to in the introduction remains to be fully determined.

References

These works show experimentally and in special cases that the quantum walk spectrum refines the classical spectrum, but a complete theoretical characterization has remained open.

— The Quantum Walk Characteristic Polynomial Distinguishes All Strongly Regular Graphs of Prime Orde  (2604.01507 - Roldan, 2 Apr 2026) in Section 1 (Introduction)