  The eigenvalue of a Hamiltonian, $\mathcal{H}$, can be estimated through the
phase estimation algorithm given the matrix exponential of the Hamiltonian,
$exp(-i\mathcal{H})$. The difficulty of this exponentiation impedes the
applications of the phase estimation algorithm particularly when $\mathcal{H}$
is composed of non-commuting terms. In this paper, we present a method to use
the Hamiltonian matrix directly in the phase estimation algorithm by using an
ancilla based framework: In this framework, we also show how to find the power
of the Hamiltonian matrix-which is necessary in the phase estimation
algorithm-through the successive applications. This may eliminate the necessity
of matrix exponential for the phase estimation algorithm and therefore provide
an efficient way to estimate the eigenvalues of particular Hamiltonians. The
classical and quantum algorithmic complexities of the framework are analyzed
for the Hamiltonians which can be written as a sum of simple unitary matrices
and shown that a Hamiltonian of order $2^n$ written as a sum of $L$ number of
simple terms can be used in the phase estimation algorithm with $(n+1+logL)$
number of qubits and $O(2^anL)$ number of quantum operations, where $a$ is the
number of iterations in the phase estimation. In addition, we use the
Hamiltonian of the hydrogen molecule as an example system and present the
simulation results for finding its ground state energy.
