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

TL;DR

This paper introduces a new quantum-classical hybrid algorithm, XZ24, that efficiently computes eigenstate energies of quantum systems without requiring eigenstate preparation, reducing measurement overhead and enabling simultaneous energy calculations.

Contribution

The paper presents XZ24, a novel quantum algorithm that improves efficiency and accuracy in eigenenergy estimation by eliminating eigenstate preparation and reducing measurement complexity.

Findings

XZ24 outperforms existing algorithms in efficiency and accuracy.

It requires only a single auxiliary qubit for measurements.

Measurement complexity scales favorably with system size and precision.

Abstract

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.