Krylov Complexity in Canonical Quantum Cosmology

Abstract: We explore Krylov complexity for two exactly solvable models, one in the Wheeler-DeWitt (WDW) quantum cosmology and another in loop quantum cosmology (LQC), for a spatially flat, homogeneous, and isotropic universe sourced with a massless scalar field, which serves the role of clock. While the WDW quantization of this model cannot avoid the big bang/big crunch singularity, it is replaced by a big bounce in LQC. We construct the Krylov basis analytically by applying the Lanczos algorithm and evaluate both the Krylov state and operator complexity. In regimes where the wave function of the universe is sharply peaked, our results indicate that the Krylov complexity grows quadratically with the scalar field clock for the state and operator complexities in both the WDW quantum cosmology and LQC. We further show that operator complexity is exactly twice the state complexity in these regimes. We discuss the interpretation of the global behavior of these systems by calculating the Krylov entropy for both quantum cosmological frameworks. We observe that in LQC, the Krylov complexity and entropy remain finite at the bounce, whereas in the WDW quantum cosmology, they diverge at the big bang/crunch singularity. Our work paves the way for computing Krylov complexity in more intricate quantum cosmological models, including those exhibiting phenomena such as quantum chaos.