Measurable Krylov Spaces and Eigenenergy Count in Quantum State Dynamics

In this work, we propose a quantum-mechanically measurable basis for the computation of spread complexity. Current literature focuses on computing different powers of the Hamiltonian to construct a basis for the Krylov state space and the computation of the spread complexity. We show, through a series of proofs, that time-evolved states with different evolution times can be used to construct an equivalent space to the Krylov state space used in the computation of the spread complexity. Afterwards, we introduce the effective dimension, which is upper-bounded by the number of pairwise distinct eigenvalues of the Hamiltonian. The computation of the spread complexity requires knowledge of the Hamiltonian and a classical computation of the different powers of the Hamiltonian. The computation of large powers of the Hamiltonian becomes increasingly difficult for large systems. The first part of our work addresses these issues by defining an equivalent space, where the original basis consists of quantum-mechanically measurable states. We demonstrate that a set of different time-evolved states can be used to construct a basis. We subsequently verify the results through numerical analysis, demonstrating that every time-evolved state can be reconstructed using the defined vector space. Based on this new space, we define an upper-bounded effective dimension and analyze its influence on finite-dimensional systems. We further show that the Krylov space dimension is equal to the number of pairwise distinct eigenvalues of the Hamiltonian, enabling a method to determine the number of eigenenergies the system has experimentally. Lastly, we compute the spread complexities of both basis representations and observe almost identical behavior, thus enabling the computation of spread complexities through measurements.