  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.
