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.

References

The conjecture of the method is that Trotter iterates will yield a good enough subspace of the full Hilbert space to perform the Lanczos method in.

— Dynamical mean field theory with quantum computing  (2508.00118 - Ayral, 31 Jul 2025) in Conclusion: state of the art, challenges and ways ahead (discussion of quantum subspace expansions)

How the required subspace size scales with the number of bath sites remains the central open question.

— Green's Functions from Sample-based Krylov Quantum Diagonalization: An Impurity Solver for Dynamical Mean-Field Theory  (2609.09147 - Patel et al., 8 Sep 2026) in Section Conclusion

Finally, while our classical algorithms rule out a superpolynomial quantum advantage for static properties of impurity models, the possibility for a polynomial speedup is left open. One possible route is to identify a substantially smaller subspace that has constant overlap with the ground state;~combined with quantum phase estimation, such a property, if it exists, could enable a nontrivial quantum speedup.

— Polynomial-time classical and quantum simulation of quantum impurity models  (2610.02167 - Jiang et al., 1 Oct 2026) in Section Outlook