  In this paper, we present a method for the Hamiltonian simulation in the
context of eigenvalue estimation problems which improves earlier results
dealing with Hamiltonian simulation through the truncated Taylor series. In
particular, 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. \cite{berry2015simulating}. 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. The circuit complexity depends on the number of
summation terms composing the Hamiltonian and requires $O(Ln)$ number of
quantum gates for the simulation of a molecular Hamiltonian. Here, $n$ is the
number of states of a spin orbital, and $L$ is the number of terms in the
molecular Hamiltonian and generally bounded by $O(n^4)$. We also discuss how to
use the circuit in adaptive processes and eigenvalue related problems along
with a slight modified version of the iterative phase estimation algorithm. In
addition, a simple divide and conquer method is presented for mapping a matrix
which are not given as sums of unitary matrices into the circuit. The
complexity of the circuit is directly related to the structure of the matrix
and can be bounded by $O(poly(n))$ for a matrix with $poly(n)-$sparsity.
