Two-dimensional Toda–Arnoldi correspondence: Holomorphic Krylov geometry and counterdiabatic transport

This paper investigates the relationship between the Arnoldi reduction of non-Hermitian Hamiltonians and the two-dimensional Toda lattice. While Hermitian systems (via Lanczos) yield a tridiagonal matrix related to the standard Toda lattice, non-Hermitian systems typically produce an upper Hessenberg matrix. The author asks whether the diagonal and subdiagonal entries of this Hessenberg matrix form a closed integrable system and whether they possess a geometric interpretation in terms of Krylov subspace evolution.

The author employs a holomorphic deformation of a cyclic seed state vector, constructing the Arnoldi basis at each point in the complex parameter space. By analyzing the Gram determinants of these holomorphic Krylov subspaces, the study demonstrates that these determinants act as tau-functions for the two-dimensional Toda lattice, with the Arnoldi coefficients serving as the corresponding Flaschka variables.

The study establishes a formal Toda-Arnoldi correspondence. It proves that the diagonal and subdiagonal Arnoldi coefficients coincide with the Flaschka variables of the two-dimensional Toda lattice, effectively closing the Toda dynamics on this sector of the Arnoldi matrix. Furthermore, the author shows that the squared subdiagonal coefficients (the beta coefficients) directly determine the Fubini-Study metric and the Berry curvature of the holomorphic Krylov subspaces.

Along real parameter paths, the Arnoldi-frame connection provides a Hermitian tridiagonal generator that facilitates exact isospectral transport. When the Arnoldi matrix is diagonalizable with a nondegenerate spectrum, this generator functions as a counterdiabatic driving term, effectively canceling transitions between instantaneous eigenspaces and providing a shortcut to adiabaticity.

This work bridges the gap between integrable systems (Toda lattice theory) and non-Hermitian quantum dynamics. By providing a geometric interpretation of Arnoldi coefficients, it offers a new way to analyze the stability and geometric properties of Krylov subspaces. The ability to construct counterdiabatic generators using the Arnoldi-frame connection provides a practical tool for controlling non-Hermitian quantum systems, which are increasingly relevant in open-system physics and non-reciprocal quantum circuits.