The eigenvalue of a Hamiltonian, $\mathcal{H}$, can be estimated through the phase estimation algorithm given the matrix exponential of the Hamiltonian, $\mathit{exp}(-i\mathcal{H})$. 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 $(n+1+\mathit{logL})$ number of qubits and $O(2^a\mathit{nL})$ 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.

哈密​​顿量的特征值 $\mathcal{H}$可以通过给定哈密顿量的矩阵指数的相位估计算法进行估计， $\mathit{经验值}（--\mathrm{一世}\mathcal{H}）$。这种求幂的困难阻碍了相位估计算法的应用，尤其是当$\mathcal{H}$由非通勤条款组成。在本文中，我们提出了一种使用基于ancilla的框架在相位估计算法中直接使用哈密顿矩阵的方法：在该框架中，我们还展示了如何找到哈密顿矩阵的幂，这是相位估计中必不可少的算法-通过连续的应用程序。这可以消除用于相位估计算法的矩阵指数的必要性，因此提供了一种估计特定哈密顿量特征值的有效方法。针对哈密顿量分析了框架的经典算法和量子算法的复杂性，可以将其写为简单unit矩阵的总和，并表明哈密顿量级${\mathrm{2个}}^{ñ}$写为L个简单项之和可以在相位估计算法中使用$（ñ+\mathrm{1个}+\mathit{对数}）$ 量子比特数和 $Ø（{\mathrm{2个}}^{\mathrm{一个}}\mathit{nL}）$量子操作数，其中a是相位估计中的迭代数。此外，我们以氢分子的哈密顿量为例，并给出了寻找其基态能量的模拟结果。