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

TL;DR

This work builds a formal bridge between quantum walks on graphs and Krylov/spread complexity by reducing graph dynamics to a Krylov chain and extracting the Lanczos data. It yields analytic Lanczos coefficients for the SYK model at arbitrary $q$ and a complete Krylov description of the hypercube, enabling direct comparisons with circuit complexity in holographic settings. The results reveal that Krylov complexity can mirror chaotic growth under certain conditions but saturates faster than random unitary circuits due to quantum speed-ups in quantum walks. By unifying graph structure, operator growth, and complexity, the paper offers a versatile framework with implications for quantum algorithms, many-body chaos, and black-hole physics.

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.

Figures (9)

• Figure 1: The graph G$_4$, which is the outcome of the fusion of two complete binary trees of order 4 at the leaves. The vertical organization of vertices illustrates the columns mentioned the main text.
• Figure 2: The graph of a cube with its vertices labeled for reference.
• Figure 3: A different version of the quantum walk on the cube, in which the initial state is an equal superposition of vertices 2 and 6. The dashed line expresses vertices 5 and 8 to be connected.
• Figure 4: The first 5 neighborhoods, shown from top to bottom, of the factorially growing graph.
• Figure 5: A squid graph of 7 neighborhoods