Emergence of Krylov complexity through quantum walks: An exploration of the quantum origins of complexity

TL;DR

This paper explores how Krylov complexity naturally arises from quantum walks on graphs, providing analytical results for models like SYK and hypercube graphs, and comparing Krylov and circuit complexities in black hole contexts.

Contribution

It establishes a connection between quantum walks and Krylov complexity, enabling analytical computation of Krylov complexity for SYK and hypercube graphs, and compares it with circuit complexity in black hole models.

Findings

Analytic computation of Lanczos coefficients for SYK model.

Complete characterization of Krylov complexity for hypercube graphs.

Krylov complexity can saturate faster than predicted by random unitary circuits.

Abstract

In this work we study the relationship between quantum random walks on graphs and Krylov/spread complexity. We show that the latter's definition naturally emerges through a canonical method of reducing a graph to a chain, on which we can identify the usual Krylov structure. We use this identification to construct families of graphs corresponding to special classes of systems with known complexity features and conversely, to compute Krylov complexity for graphs of physical interest. The two main outcomes are the analytic computation of the Lanczos coefficients for the SYK model for an arbitrary number $q$ of interacting fermions and the complete characterization of Krylov complexity for the hypercube graph in any number of dimensions. The latter serves as the starting point for an in-depth comparison between Krylov and circuit complexities as they purportedly arise in the context of black holes. We find that while under certain conditions Krylov complexity follows the growth and saturation pattern ascribed to such systems, the timescale at which saturation happens can generally be shorter than what is predicted by random unitary circuits, due to the effects of quantum speed-ups commonly occurring when comparing quantum and classical random walks.