Power of the Sine Hamiltonian Operator for Estimating the Eigenstate Energies on Quantum Computers

Quantum computers have been shown to have tremendous potential in solving difficult problems in quantum chemistry. In this paper, we propose a new classical-quantum hybrid method, named as power of sine Hamiltonian operator (PSHO), to evaluate the eigenvalues of a given Hamiltonian (Ĥ). In PSHO, for any reference state $|{\phi }_0⟩$, the normalized energy of the ${\sin }^n\left(\hat{H}\tau \right)|{\phi }_0⟩$ state can be determined. With the increase of the power, the initial reference state can converge to the eigenstate with the largest $\left|\sin \left(E_i\tau \right)\right|$ value in the coefficients of the expansion of $|{\phi }_0⟩$, and the normalized energy of the ${\sin }^n\left(\hat{H}\tau \right)|{\phi }_0⟩$ state converges to E_i. The ground- and excited-state energies of a Hamiltonian can be determined by taking different τ values. The performance of the PSHO method is demonstrated by numerical calculations of the H₄ and LiH molecules. Compared with the current popular variational quantum eigensolver method, PSHO does not need to design the ansatz circuits and avoids the complex nonlinear optimization problems. PSHO has great application potential in near-term quantum devices.