Quantum Jacobi-Davidson Method

TL;DR

The QJD framework is established 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.

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.

Figures (8)

• Figure 1: The quantum circuit for the linear combination of unitaries (LCU) method to obtain the normalized correction vector $\ket{t}/s = \bra{0}PR^\dagger \cdot SELECT \cdot PR\ket{0}\ket{rv} = A/s\ket{rv}$, where $A$ is a non-unitary matrix, $A = \sum_{i=1}^m \alpha_iU_i$, and $s$ is a normalization factor, $s = \sum_{i=1}^m |\alpha_i|$. $q_a$ and $q_d$ denote the ancilla and data qubits, respectively. The $PR$ gate constructs the state $PR\ket{0} = \sum_{i=1}^m\sqrt{|\alpha_i|/s}\ket{i}$ on the ancilla qubits, and the $SELECT$ gate applies $SELECT\ket{i}\ket{rv} = \ket{i}U_i\ket{rv}$ on the data qubits.
• Figure 2: The quantum circuit for calculating the expectation value $\bra{rv}P_i\ket{rv}$, where $BS$ is a basis-changing gate constructed depending on the Pauli $X$, $Y$, and $Z$ gates present in $P_i$.
• Figure 3: The quantum circuit for calculating $\Re(\bra{rv}P_i\ket{r})$, where $q_a$ and $q_d$ denote the ancilla and data qubits, respectively. The states $\ket{rv}$ and $\ket{r}$ are constructed by $\ket{rv} = U_{rv}\ket{0}$ and $\ket{r} = U_{r}\ket{0}$, and $P_i$ represents a Pauli string.
• Figure 4: The flowchart of the QJD and SBQJD methods. The blue, pink, and gradient-filled boxes indicate steps to be executed on the QPU, CPU, and in a hybrid QPU+CPU environment, respectively.
• Figure 5: The comparison of the QD, QJD, and SBQJD methods for approximating the ground state energy of six diagonal-dominant matrices. From left to right: the upper row shows a decreasing overlap between the reference state and the true ground state; the lower row demonstrates an increasing number of less dominant computational bases in the reference state. $ns$ refers to the number of smallest diagonal elements in the Hamiltonian, and $npeaks$ is defined as the number of peaks in the Gaussian distribution of the reference quantum state. The suffix $\_$D indicates the use of the diagonal Hamiltonian preconditioners for the calculation. The convergence tolerance is illustrated in a horizontal dashed grey line. The curves are truncated and displayed when the energy error is within the lower bound of the $y$-axis.