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

Author:

Aguilar-Gutierrez Sergio E.1ORCID,Rolph Andrew23ORCID

Affiliation:

1. Institute for Theoretical Physics

2. University of Amsterdam

3. Vrije Universiteit Brussel (VUB) and The International Solvay Institutes

Abstract

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

Funder

Universiteit van Amsterdam

Fonds Wetenschappelijk Onderzoek

Nederlandse Organisatie voor Wetenschappelijk Onderzoek

Vrije Universiteit Brussel

Delta Institute for Theoretical Physics

International Centre for Theoretical Physics

Uitvoeringsinstituut Werknemersverzekeringen

Publisher

American Physical Society (APS)

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