  Quantum computers provide new avenues to access ground and excited state
properties of systems otherwise difficult to simulate on classical hardware.
New approaches using subspaces generated by real-time evolution have shown
efficiency in extracting eigenstate information, but the full capabilities of
such approaches are still not understood. In recent work, we developed the
variational quantum phase estimation (VQPE) method, a compact and efficient
real-time algorithm to extract eigenvalues on quantum hardware. Here we build
on that work by theoretically and numerically exploring a generalized Krylov
scheme where the Krylov subspace is constructed through a parametrized
real-time evolution, which applies to the VQPE algorithm as well as others. We
establish an error bound that justifies the fast convergence of our spectral
approximation. We also derive how the overlap with high energy eigenstates
becomes suppressed from real-time subspace diagonalization and we visualize the
process that shows the signature phase cancellations at specific eigenenergies.
We investigate various algorithm implementations and consider performance when
stochasticity is added to the target Hamiltonian in the form of spectral
statistics. To demonstrate the practicality of such real-time evolution, we
discuss its application to fundamental problems in quantum computation such as
electronic structure predictions for strongly correlated systems.
