3
$\begingroup$

What is the difference between (1) Trotter (2) Lie-Trotter and (3) Trotter-Suzuki approximation? Are they all different? what are the formulas and errors associated in each of these approximations in simulating a Hamiltonian evolution, let's say $\exp(-it(H_{1}+H_{2}))$?

$\endgroup$
1
  • $\begingroup$ I think Qiskit even has just Suzuki as its own method (and Trotter and TrotterSuzuki) $\endgroup$
    – Mauricio
    Jan 10, 2023 at 16:44

1 Answer 1

3
$\begingroup$

I believe they all refer to the same methods, which is to expand the unitary evolution into small steps. The most basic one is the following: $$\left[\exp\left(-\frac{it}{n}H_1\right) \exp\left(-\frac{it}{n}H_2\right) \right]^n \rightarrow \exp(-it(H_1+H_2))$$ for $n$ goes to infinity. Then, there is more trick to play where you can symmetrize this expansion, and use what called higher-order trotterization and hope to reduce the error.

On the error scaling. There is a rich literature, depending on the way the trotterization is done. Here is an example: https://arxiv.org/abs/1912.08854

Looking around here is the way people use different naming (mostly from the wiki of the Lie Product formula):

  • The Lie Product Formula is the equation given above for finite size matrix. Apparently, this formula is also sometime called the Trotter Formula
  • The Lie-Trotter formula and the Trotter-Kato theorem are more general and apply to more generic operators than matrices (unbounded linear operators)
  • The Trotter-Suzuki expansion refer more explicitly to the expansion with the Hamiltonian. Then People usually add the order of the expansion to be explicit.
$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.