Krylov complexity is not a measure of distance between states or operators

• DOI: 10.48550/arxiv.2311.04093
• Publication Date: 2024-04-22
• Authors: Sergio E. Aguilar...; Andrew Rolph
• Keywords: High Energy Physics - Theory; Quantum Physics; 0303 health sciences; 03 medical and health sciences; High Energy Physics - Theory (hep-th); FOS: Physical sciences; Quantum Physics (quant-ph)

We ask whether Krylov complexity is mutually compatible with the circuit and Nielsen definitions of complexity. We show that the Krylov complexities between three states fail to satisfy the triangle inequality and so cannot be a measure of distance: there is no possible metric for which Krylov complexity is the length of the shortest path to the target state or operator. We show this explicitly in the simplest example, a single qubit, and in general. Published by the American Physical Society 2024