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 $\exp(-it(H_{1}+H_{2}))$?
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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.
sailx
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See also Trotter-Kato Product Formulæ (Operator Theory: Advances and Applications Series, Vol. 296) by Zagrebnov Valentin A. et al https://link.springer.com/book/10.1007/978-3-031-56720-9
