  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.
