Quantum Jacobi-Davidson Method

TL;DR

This paper introduces the Quantum Jacobi-Davidson (QJD) and Sample-Based Quantum Jacobi-Davidson (SBQJD) methods, which significantly improve convergence speed and measurement efficiency in quantum eigenvalue problems, promising for future quantum hardware applications.

Contribution

The paper develops and demonstrates the QJD and SBQJD algorithms, novel methods that outperform existing quantum eigenvalue solvers in convergence and measurement efficiency.

Findings

QJD and SBQJD converge faster than Quantum Davidson.

SBQJD benefits from optimized reference state preparation.

Methods require fewer Pauli measurements for accurate results.

Abstract

Computing electronic structures of quantum systems is a key task underpinning many applications in photonics, solid-state physics, and quantum technologies. This task is typically performed through iterative algorithms to find the energy eigenstates of a Hamiltonian, which are usually computationally expensive and suffer from convergence issues. In this work, we develop and implement the Quantum Jacobi-Davidson (QJD) method and its quantum diagonalization variant, the Sample-Based Quantum Jacobi-Davidson (SBQJD) method, and demonstrate their fast convergence for ground state energy estimation. We assess the intrinsic algorithmic performance of our methods through exact numerical simulations on a variety of quantum systems, including 8-qubit diagonally dominant matrices, 12-qubit one-dimensional Ising models, and a 10-qubit water molecule (H$_2$O) Hamiltonian. Our results show that both QJD and SBQJD achieve significantly faster convergence and require fewer Pauli measurements compared to the recently reported Quantum Davidson method, with SBQJD further benefiting from optimized reference state preparation. These findings establish the QJD framework as an efficient general-purpose subspace-based technique for solving quantum eigenvalue problems, providing a promising foundation for sparse Hamiltonian calculations on future fault-tolerant quantum hardware.