A Novel Quantum-Classical Hybrid Algorithm for Determining Eigenstate Energies in Quantum Systems

Developing efficient quantum computing algorithms is essential for tackling computationally challenging problems across various fields. This paper presents a novel quantum algorithm, XZ24, for efficiently computing the eigen-energy spectra of arbitrary quantum systems. Given a Hamiltonian $\hat{H}$ and an initial reference state $|\psi_{\text{ref}} \rangle$, the algorithm extracts information about $\langle \psi_{\text{ref}} | \cos(\hat{H} t) | \psi_{\text{ref}} \rangle$ from an auxiliary qubit's state. By applying a Fourier transform, the algorithm resolves the energies of eigenstates of the Hamiltonian with significant overlap with the reference wavefunction. We provide a theoretical analysis and numerical simulations, showing XZ24's superior efficiency and accuracy compared to existing algorithms. XZ24 has three key advantages: 1. It removes the need for eigenstate preparation, requiring only a reference state with non-negligible overlap, improving upon methods like the Variational Quantum Eigensolver. 2. It reduces measurement overhead, measuring only one auxiliary qubit. For a system of size $L$ with precision $\epsilon$, the sampling complexity scales as $O(L \cdot \epsilon^{-1})$. When relative precision $\epsilon$ is sufficient, the complexity scales as $O(\epsilon^{-1})$, making measurements independent of system size. 3. It enables simultaneous computation of multiple eigen-energies, depending on the reference state. We anticipate that XZ24 will advance quantum system simulations and enhance applications in quantum computing.