[论文解读] Emergence of Krylov complexity through quantum walks: An exploration of the quantum origins of complexity

Dimitrios Patramanis, Watse Sybesma|arXiv (Cornell University)|Feb 4, 2026

Quantum Computing Algorithms and Architecture被引用 0

一句话总结

This paper links continuous-time quantum walks on graphs to Krylov (spread) complexity, showing how a Krylov chain emerges from graph reductions and computing Lanczos coefficients for the SYK model and Krylov complexity for hypercubes, with a comparison to circuit complexity.

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.

研究动机与目标

• Demonstrate how Krylov complexity emerges from continuous-time quantum walks on graphs via a graph-to-chain reduction.
• Relate graph structure to Lanczos coefficients and thus to Krylov/spread complexity.
• Compute analytic Lanczos coefficients for the SYK model with arbitrary q.
• Characterize Krylov complexity for hypercube graphs in any dimension.
• Compare Krylov complexity growth to circuit complexity and discuss implications for black holes.

提出的方法

• Map a graph's CTQW to a tridiagonal Krylov (Lanczos) chain using neighborhood states to define a Krylov basis.
• Derive explicit formulas for Lanczos coefficients a_n and b_n from graph edges within/between neighborhoods (E_n, I_n).
• Express time evolution amplitudes on the chain and define Krylov complexity as the average distance on the chain, C_K = sum_n n |phi_n(t)|^2.
• Provide a general reduction scheme to obtain the Krylov chain for arbitrary graphs.
• Analyze special graph families to realize different b_n behaviors (constant, sqrt(n), sqrt(n(n-1)), n).
• Compute analytic Lanczos coefficients for the SYK model with arbitrary q and characterize Krylov complexity for hypercube graphs in any dimension.

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.

实验结果

研究问题

• RQ1 How does a continuous-time quantum walk on a graph correspond to a Krylov chain with Lanczos coefficients a_n and b_n?
• RQ2 How can one use graph structure (V_n, E_n, I_n) to bound and determine the Lanczos coefficients and hence Krylov complexity growth?
• RQ3 What are the analytic Lanczos coefficients for the SYK model with arbitrary fermion number q?
• RQ4 What is the Krylov complexity behavior of hypercube graphs in arbitrary dimensions, and how does it compare to circuit complexity in related holographic contexts?
• RQ5 Under what graph constructions and initial states do Krylov and circuit complexities exhibit similar growth and saturation patterns?

主要发现

• The average distance on the Krylov chain serves as a natural measure of Krylov/spread complexity for CTQWs.
• Lanczos coefficients a_n and b_n are given by graph-dependent expressions a_n = I_n / V_n and b_n = E_{n-1} / sqrt(V_n V_{n-1}), linking graph topology to complexity growth.
• Analytic Lanczos coefficients for the SYK model are obtained for arbitrary q, enabling explicit Krylov complexity calculations for this model.
• Krylov complexity for the hypercube graph is fully characterized in any dimension, providing a benchmark for comparison with circuit complexity.
• Under certain conditions, Krylov complexity mirrors growth and saturation patterns associated with black-hole contexts, but saturation times can be shorter due to quantum speed-ups relative to random unitary circuits.
• The framework shows that quantum speed-ups in quantum walks can alter the expected saturation times compared to classical or random-unitary analyses.

Figure 2 : The graph of a cube with its vertices labeled for reference.