# A Generalized Circuit for the Hamiltonian Dynamics Through the Truncated Series

In this paper, we present a fixed-quantum circuit design for the simulation of the Hamiltonian dynamics, H(t), through the truncated Taylor series method described by Berry et al. [1]. In particular, Hamiltonians which are not given as sums of unitary matrices but given as general matrices are considered and a simple divide and conquer method is presented for mapping the Hamiltonians into the circuit. The circuit is general and can be used to simulate any given matrix in the phase estimation algorithm by only changing the angle values of the quantum gates implementing the time variable t in the series. We analyze the circuit complexity and show that it requires O(N^2) number of CNOT gates for dense matrices, where N is the matrix size. We finally discuss how it can be used in adaptive processes and eigenvalue related problems along with a slight modified version of the iterative phase estimation algorithm.

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