Explicit Connections Between Krylov and Nielsen Complexity

TL;DR

This paper establishes a direct link between Krylov and Nielsen complexity measures, showing their equivalence under certain conditions and providing evidence of their correspondence in the SYK model.

Contribution

It introduces a method to relate Krylov and Nielsen complexity by aligning their basis and metrics, and demonstrates this connection explicitly in the SYK model.

Findings

Krylov complexity equals the squared length of a specific geodesic in Nielsen geometry.

The length provides an upper bound on Nielsen complexity, saturated by minimal geodesics.

In the SYK model, Krylov and Nielsen complexities are closely related for certain precursor ranges.

Abstract

We establish a direct correspondence between Krylov and Nielsen complexity by choosing the Krylov basis to be part of the elementary gate set of Nielsen geometry and selecting a Nielsen complexity metric compatible with the Krylov metric. Up to normalization, the Krylov complexity of a Hermitian operator then equals the length squared of a straight-line trajectory on the manifold of unitaries that connects the identity operator with a precursor operator. The corresponding length provides an upper bound on Nielsen complexity that saturates whenever the straight line is a minimal geodesic. While for general systems we can only establish saturation in the limit of small precursors, we provide evidence that in the Sachdev-Ye-Kitaev (SYK) model there is a precise correspondence between Krylov complexity and (the square of) Nielsen complexity for a finite range of precursors.