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I'm looking at a problem where I want to do hamiltonian simulation of an operator integral. That is, I want to implement the unitary $$\mathrm U = \exp[-i \mathrm H t]$$ where $\mathrm H$ is of the form $$\mathrm H = \int_a^b d\tau \, \mathrm A(\tau)$$ with $\mathrm A(\tau)$ a hermitian operator for every $\tau\in[a, b]$.

Question: what methods and/or complexity statements are known for such hamiltonian simulation problems?

Of course, the integral could be approximated with a Riemann sum (or something similar) after which standard Trotter-Suzuki or more complex methods could be applied. However, my guess is that this two-step approach will incur additional overhead that might be avoidable. So, do methods exist that tackle this problem more directly?

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  • $\begingroup$ Not sure if this helps, but there are well-known methods for time-dependent Hamiltonians, for ex: see (4) of arxiv.org/abs/1805.00582. $\endgroup$
    – R.G.J
    Feb 8 at 19:51

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