Impact of ordering Hamiltonian terms for Trotterization

Question

In Trotterization, the typical Hamiltonian considered is:

$$ H = \sum_{p, q} h_{pq} a^{\dagger}_p a_q + \sum_{p, q, r, s} a^{\dagger}_p a^{\dagger}_q a_r a_s $$

Which is then converted into a sequence of gates by the Jordan Wigner transformation.

However, how do we choose the ordering of the $ pq$ and $pqrs$ terms? For example, if our Hamiltonian was:

$$ H = (a^{\dagger}_1 a_2 + a^{\dagger}_2 a_1) + (a^{\dagger}_2 a_3 + a^{\dagger}_3 a_2) + (a^{\dagger}_1 a^{\dagger}_2 a_2 a_3 + a^{\dagger}_3 a^{\dagger}_2 a_2 a_1)$$

There are at least $ 3! = 6$ ways to order the Hamiltonian, a number which quickly explodes as the size of the Hamiltonian grows.

Note that the terms don't have clean commutation rules - sometimes the terms will commute or anticommute. Is there an approach to ordering the Hamiltonian that maximizes simulation accuracy?

Answer 1

The question of how to order the terms when Trotterizing is an active area of research. It turns out to be beneficial to put commuting terms next to each other. It can also be beneficial to randomise the ordering of terms in each Trotter step (as this suppresses errors), or to even not include certain low value terms. A paper that gives many references to the work that has been done in this area is the following: Compilation by stochastic Hamiltonian sparsification.