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The eigenvalue of a Hamiltonian, $\mathcal{H}$, can be estimated through thephase estimation algorithm given the matrix exponential of the Hamiltonian,$exp(-i\mathcal{H})$. The difficulty of this exponentiation impedes theapplications of the phase estimation algorithm particularly when $\mathcal{H}$is composed of non-commuting terms. In this paper, we present a method to usethe Hamiltonian matrix directly in the phase estimation algorithm by using anancilla based framework: In this framework, we also show how to find the powerof the Hamiltonian matrix-which is necessary in the phase estimationalgorithm-through the successive applications. This may eliminate the necessityof matrix exponential for the phase estimation algorithm and therefore providean efficient way to estimate the eigenvalues of particular Hamiltonians. Theclassical and quantum algorithmic complexities of the framework are analyzedfor the Hamiltonians which can be written as a sum of simple unitary matricesand shown that a Hamiltonian of order $2^n$ written as a sum of $L$ number ofsimple 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 thenumber of iterations in the phase estimation. In addition, we use theHamiltonian of the hydrogen molecule as an example system and present thesimulation results for finding its ground state energy.

Direct Application of the Phase Estimation Algorithm to Find the Eigenvalues of the Hamiltonians