Holo Club: Explicit Connections Between Krylov and Nielsen Complexity

Speaker: Le-Chen Qu (IFT)

Venue&Time: Grey Room 3 / 11:30

Abstract: We establish a precise correspondence between Krylov complexity and Nielsen complexity in quantum many-body systems. By identifying the Krylov basis with the generators of elementary gates in Nielsen’s geometric framework, the Krylov complexity of precursor operators acquires a geometric interpretation as the length of straight-line paths in Nielsen geometry. This construction provides an upper bound on Nielsen complexity, which becomes exact in the small-precursor limit. Employing the SYK model as a test case, we demonstrate that the straight-line trajectory satisfies the Nielsen geodesic equation and remains both locally and globally minimizing up to a critical value of $z$, beyond which conjugate points and geodesic loops arise. Within this framework, cost functions quadratic in the Krylov index are further shown to reproduce the conjectured proportionality between the complexity growth rate and the operator size.