A relation between Krylov and Nielsen complexity

TL;DR

This paper uncovers a mathematical relation between Krylov and Nielsen complexities, linking quantum chaos and quantum computation approaches to quantifying quantum evolution complexity.

Contribution

It establishes a connection between Krylov and Nielsen complexities, showing how Krylov complexity's time average bounds Nielsen complexity with a tailored penalty schedule.

Findings

Time average of Krylov complexity expressed as a trace of a matrix

Upper bound on Nielsen complexity derived from Krylov basis

Mathematical link between two different complexity measures

Abstract

Krylov complexity and Nielsen complexity are successful approaches to quantifying quantum evolution complexity that have been actively pursued without much contact between the two lines of research. The two quantities are motivated by quantum chaos and quantum computation, respectively, while the relevant mathematics is as different as matrix diagonalization algorithms and geodesic flows on curved manifolds. We demonstrate that, despite these differences, there is a relation between the two quantities. Namely, the time average of Krylov complexity of state evolution can be expressed as a trace of a certain matrix, which also controls an upper bound on Nielsen complexity with a specific custom-tailored penalty schedule adapted to the Krylov basis.