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The Krylov subspace method is a standard approach to approximate quantumevolution, allowing to treat systems with large Hilbert spaces. Although itsapplication is general, and suitable for many-body systems, estimation of thecommitted error is involved. This makes it difficult to automate its use. Inthis paper, we solve this problem by realizing that such error can be regardedas a Loschmidt echo in a tight-binding Hamiltonian. We show that the differenttime-regimes of the approximation can be understood using simple physicalideas. More importantly, we obtain computationally cheap error bounds thatdescribe with high precision the actual error in the approximation.

Loschmidt echo approach to Krylov-subspace approximation error estimation