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

TL;DR

This paper demonstrates that Krylov complexity does not satisfy the triangle inequality and cannot serve as a true measure of distance between quantum states or operators, challenging its interpretation as a metric.

Contribution

The authors prove that Krylov complexity fails to meet the criteria of a metric, specifically the triangle inequality, in simple and general quantum systems.

Findings

Krylov complexity violates the triangle inequality.

It cannot be interpreted as a distance measure.

This holds for both simple and general quantum systems.

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.