Explicit Connections Between Krylov and Nielsen Complexity

Abstract

We establish a direct correspondence between Krylov and Nielsen complexity by choosing the Krylov basis to be part of the elementary gate set of Nielsen geometry and selecting a Nielsen complexity metric compatible with the Krylov metric. Up to normalization, the Krylov complexity of a Hermitian operator then equals the length squared of a straight-line trajectory on the manifold of unitaries that connects the identity operator with a precursor operator. The corresponding length provides an upper bound on Nielsen complexity that saturates whenever the straight line is a minimal geodesic. While for general systems we can only establish saturation in the limit of small precursors, we provide evidence that in the Sachdev-Ye-Kitaev (SYK) model there is a precise correspondence between Krylov complexity and (the square of) Nielsen complexity for a finite range of precursors.

Figures (4)

• Figure 1: Conjugate points (red dot) along the geodesic (black straight-line) are defined as the points where the perturbation satisfying the geodesic deviation equation vanishes.
• Figure 2: Absolute value of the determinant in Eq. \ref{['eq:detDelta']} as a function of the precursor size $z$ for various penalty ratios $\alpha \equiv G_{\rm even}/G_{\rm odd}$. The system size of the SYK model is fixed at $N = 8$.
• Figure 3: Jacobi fields and conjugate points on Berger sphere with odd generators $P_x=P_y=1$ and different values of the even generator $P_z$. Conjugate points appear when $\det \mathcal{J}=0$.
• Figure 4: The location of the first conjugate point on Berger spheres (with $P_x=1$).