How can one determine the time-step for imaginary time evolution?

The real-time evolution of a quantum system described by the state $$\psi(t)$$ is formally given by,

$$\psi(t+\Delta t) = \exp^{- i \hat{H} \Delta t / \hbar} \psi(t)\tag{1}$$

For dynamical simulations of quantum systems, the time-scale $$\Delta t$$ is set by some energy scale characteristic of the system. For instance, if one is studying dynamics of a conjugated carbon system, one can choose $$\Delta t$$ to be a few orders of magnitude smaller than typical carbon-carbon bond stretching frequency (as an example, with apologies for the self-reference, see section 6.2 of this).

Now suppose one is interested in finding the ground state of some Hamiltonian using imaginary time evolution. An interesting question to ask is whether there is an equivalent way of choosing an upper bound for the time-step? If one seeks to calculate the ground state of molecular Hydrogen, does it still make sense to ignore the imaginary unit $$i$$ and set the bound using Hydrogen bond oscillation frequency?

Another approach might be to think in terms of temperature. I know imaginary time-evolution can be thought of in a temperature scale but I do not know how to connect the dots and rationalise it this way.

Edit : Adding some context to the problem as per suggestions of @NikeDattani, I'm new to imaginary time evolution algorithms and am wondering whether there is a physically inspired time-scale for the general problem. However, I'm ultimately attempting to understand how quantum imaginary time evolution works.

• +1 and welcome to our new community! It might help if you say more about which imaginary-time algorithm you want to use for finding the ground state. Are you using the PIGS algorithm? May 9, 2023 at 17:45
• Thank you so much for the kind welcome. While I'm just starting to learn about imaginary time evolution and the question I had in mind was on the general formalism, ultimately I'm trying to use the quantum imaginary time evolution algorithm. May 9, 2023 at 20:05
• Sometimes the answer is: as high as you can without producing unacceptably high Trotterization errors... The fewer the steps, the faster the calculation. May 9, 2023 at 23:52
• @Anyon thank you for the comment. This indeed is what I thought too, but it would be nice if one is able to "physically" justify the choice, as for larger systems where exact solutions are unavailable it might not be easy to determine (as far as I understand) what the unacceptable limit for the Trotter error would be. May 10, 2023 at 0:09
• Since you're in the PTCL at Oxford, have you asked David Manolopoulos this question? I think he knows a lot about imaginary time evolution, and even if I'm wrong about that, I think he would have some valuable insights for you about this calculation. Dec 1, 2023 at 20:03