Quantum chaos diagnostics for non-Hermitian systems from bi-Lanczos Krylov dynamics

In Hermitian systems, Krylov complexity has emerged as a powerful diagnostic of quantum dynamics, capable of distinguishing chaotic from integrable phases, in agreement with established probes such as spectral statistics and out-of-time-order correlators. By contrast, its role in non-Hermitian settings, relevant for modeling open quantum systems, remains less understood due to the challenges posed by complex eigenvalues and the limitations of standard approaches based on orthogonality, such as singular value decomposition. Here we demonstrate that Krylov complexity, computed via the bi-Lanczos algorithm, provides a reliable probe of quantum chaos in non-Hermitian systems, clearly discriminating chaotic and integrable regimes. Our results agree with complex spectral statistics and complex spacing ratios, underscoring the robustness of the method. Universality is supported by extensive tests in the non-Hermitian Sachdev-Ye-Kitaev model and random-matrix ensembles across multiple non-Hermitian symmetry classes, with further validation provided by the non-Hermitian random-field XXZ model as a pseudo-Hermitian system.