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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?

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  • $\begingroup$ @gIS you are correct; unfortunately, Trotterization exponentiates the matrices, which incurs errors because $ e^{A + B} \neq e^A e^B $ when $A, B$ don't commute $\endgroup$
    – C. Kang
    Sep 1 '20 at 23:59
  • $\begingroup$ @Aman somewhat off topic, but I personally like Microsoft's docs $\endgroup$
    – C. Kang
    Sep 1 '20 at 23:59
  • $\begingroup$ Your Hamiltonian needs to be Hermitian, which your example is currently not. $\endgroup$
    – DaftWullie
    Sep 2 '20 at 6:51
  • $\begingroup$ @DaftWullie good point, let me revise the question to have it. I should note that I'm using molecular Hamiltonians, which are Hermitian in nature $\endgroup$
    – C. Kang
    Sep 2 '20 at 14:46
  • $\begingroup$ Am I right in thinking that when you use Jordan Wigner, your particular example translates to a triangle with $XX+YY$ couplings between each pair of qubits? In which case, symmetry would suggest that ordering shouldn't make a difference. $\endgroup$
    – DaftWullie
    Sep 2 '20 at 14:58
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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.

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