Source-linked AI summary
Building Krylov complexity from circuit complexity
Chenwei Lv, Ren Zhang, Qi Zhou
TL;DR
Krylov complexity depends on an ambiguous choice of distances between basis states. The paper derives those distances from circuit geometry when dynamical symmetries exist, and shows that multiple Krylov complexities can jointly describe time-dependent operator dynamics.
Problem
The Lanczos algorithm orders Krylov basis states but does not set their distances, leaving quantitative Krylov complexity dependent on an ambiguous spacing choice.
Source: p. 1Method
The paper uses circuit-complexity geometry under dynamical symmetry to interpret Krylov complexity as operator height and determine distances between Krylov basis states.
Source: pp. 1–3Results
Fixing circuit geometry uniquely determines Krylov-basis distances, while time-dependent dynamics can require multiple Krylov complexities to determine the operator state.
Source: p. 3Takeaways & Limitations
With dynamical symmetry, Krylov complexity has a geometric meaning as operator height, and the approach can be generalized to other symmetry groups.
Source: pp. 2, 5
Abstract
from arXiv · showhide
Krylov complexity has emerged as a new probe of operator growth in a wide range of non-equilibrium quantum dynamics. However, a fundamental issue remains in such studies: the definition of the distance between basis states in Krylov space is ambiguous. Here, we show that Krylov complexity can be rigorously established from circuit complexity when dynamical symmetries exist. Whereas circuit complexity characterizes the geodesic distance in a multi-dimensional operator space, Krylov complexity measures the height of the final operator in a particular direction. The geometric representation of circuit complexity thus unambiguously designates the distance between basis states in Krylov space. This geometric approach also applies to time-dependent Liouvillian superoperators, where a single Krylov complexity is no longer sufficient. Multiple Krylov complexity may be exploited jointly to fully describe operator dynamics.