Building Krylov complexity from circuit complexity

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.