Adequacy of Trotter-iterate subspaces for Krylov/Lanczos methods

Prove or refute that the subspace spanned by states generated by repeated short-time Trotterized evolutions U(δt)^k|Φ_init⟩ of the Anderson impurity Hamiltonian is sufficiently expressive to support accurate Lanczos/Krylov approximations of the ground state and Green’s function, including quantitative error guarantees as a function of the timestep δt and the number of iterates.

Background

The paper discusses quantum subspace expansion approaches that construct Krylov-like bases not in the full Hilbert space but in a reduced subspace formed by Trotter iterates |ψ_k⟩ = U(δt)^k|Φ_init⟩. This approach aims to mitigate optimization and measurement costs by restricting computations to a tractable subspace.

Its central working assumption is that such Trotter-iterate subspaces capture the relevant low-energy physics well enough for effective Lanczos procedures, but this remains conjectural. Establishing formal conditions and error bounds would validate the approach for impurity solvers.