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### Is it possible to implement any random Hamiltonian using quantum circuit

I think you are asking whether a quantum computer can efficiently simulate the evolution of any Hamiltonian, as long as the Hamiltonian is represented by a hermitian matrix. There are some details ...
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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{...