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Thinking about adiabtic quantum computation, I wonder about the following:

Can we speed up the adiabatic process if we split the target hamiltonian in commuting parts?

Given the standard setup in adiabatic QC to start with a groud state $\ket{\psi_0}$ of a known and easy to prepare system hamiltonian $H_0$, we gradually change the hamiltonian towards a target one $H_1$, where the final ground state $\ket{\psi_1}$ resembles the solution of our problem. So $$ H(t)= (1-t) H_0 + t H_1. $$ The critical thing is, to do that in nice and slowly way, such that we don't excite the system too much. Responsible for the speed is the energy gap between ground and first excited state.

Now we known, that a hamilton can be split into commuting parts, i.e. $H_1=\sum_k H_{1,k}$, with $[H_{1,a},H_{1,b}]_-=0;\forall a,b$.

I assume that we now can evolve the system in parts like: $$ H(t)= (1-t_n) H_0 + \sum_{k=0}^{n-1} \operatorname{Ramp}(t_k,t_{k+1})H_{1_k}, $$ where $\operatorname{Ramp(t_k,t_{k+1})}$ is a ramp-up function.

Can we minimize the time we need for the total adiabatic process by that approach? Maybe we could ramp up the hamiltonians in an order of decreasing gaps?

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