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

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.

Figure 1

The time evolution of a pure state on the Bloch sphere. The poles of the sphere are the energy eigenstates $|E_0⟩$ and $|E_1⟩$. The time evolution of an arbitrary pure state generates the red curves. By definition, Krylov complexity gives us the complexity between states related by time evolution, on the same red curve, and we show that it is inconsistent with a measure of distance because it fails to satisfy the triangle inequality (2).