Subspace-Search Quantum Imaginary Time Evolution for Excited State Computations

TL;DR

This work tackles the challenge of computing excited states on near-term quantum hardware by introducing Subspace-Search Quantum Imaginary Time Evolution (SSQITE), which fuses Subspace-Search VQE (SSVQE) with Variational Quantum Imaginary Time Evolution (VarQITE). SSQITE maintains orthogonality among evolving states while performing imaginary-time propagation, enabling simultaneous calculation of ground and multiple excited states. Benchmark results on H$_2$ and LiH show energies in chemical accuracy compared with exact values, with robustness to noise and resistance to local minima demonstrated on a toy model. The approach promises practical excited-state computations on NISQ devices and suggests avenues for extending subspace-search techniques to other quantum optimization tasks such as QIPA.

Abstract

Quantum systems in excited states are attracting significant interest with the advent of noisy intermediate scale quantum (NISQ) devices. While ground states of small molecular systems are typically explored using hybrid variational algorithms like the variational quantum eigensolver (VQE), the study of excited states has received much less attention, partly due to the absence of efficient algorithms. In this work, we introduce the subspace search quantum imaginary time evolution (SSQITE) method, which calculates excited states using quantum devices by integrating key elements of the subspace search variational quantum eigensolver (SSVQE) and the variational quantum imaginary time evolution (VarQITE) method. The effectiveness of SSQITE is demonstrated through calculations of low-lying excited states of benchmark model systems, including $\text{H}_2$ and $\text{LiH}$ molecules. A toy Hamiltonian is also employed to demonstrate that the robustness of VarQITE in avoiding local minima extends to its use in excited state algorithms. With this robustness in avoiding local minima, SSQITE shows promise for advancing quantum computations of excited states across a wide range of applications.

Figures (6)

• Figure 1: Simultaneous evolution of the energy expectation values for the three lower energy states of H$_2$ (with fixed bond-length $R=0.95 \AA$) during the first 70 integration steps of SSQITE optimization. Final energy values are highlighted on the right, and corresponding statistical errors are of the order $10^{-5}$ Ha.
• Figure 2: Comparison of the three lowest energy eigenvalues of $\text{H}_2$ determined through (a)-(b) noiseless and (c)-(d) noisy SSQITE optimization to numerically exact calculations (dashed lines) as a function of the interatomic $\text{HH}$ distance. Boxed values correspond to the final values shown highlighted in Fig. \ref{['fig:fig1o']}. The ground, first, and second excited states correspond to the X$^1\Sigma_g^+$, b$^3\Sigma_u^+$, and B$^1\Sigma_u^+$ states of H$_2$ respectively. Deviations of (b) noiseless and (d) noisy SSQITE calculations from the ground truth energy levels of the $\text{H}_2$ molecule. All noisy simulations are performed using the qiskit FakeSherbrooke backend.
• Figure 3: Variational quantum circuit ansatz with two qubits used for the SSQITE H$_2$ calculations shown in Fig. \ref{['fig:fig2o']}. The TwoLocal ansatz involves one layer of parameterized RX and RY gates, followed by a CNOT gate. This ansatz is general, in the sense that it can realize any two-qubit operation.
• Figure 4: Top: Variational quantum circuit ansatz with three qubits used for the SSQITE $\text{LiH}$ calculations shown in Fig. \ref{['fig:fig4']} based on a custom excitation preserving ansatz. Bottom: Excitation preserving subcircuit with two tunable parameters.
• Figure 5: Comparison of the three lowest energy eigenvalues of $\text{LiH}$ determined through (a)-(b) noiseless and (c)-(d) noisy simulation of SSQITE optimization to numerically exact calculations (dashed lines). The ground, first, and second excited states correspond to X$^1\Sigma^+$, a$^3\Sigma^+$, and A$^1\Sigma^+$ respectively. Note that the results from LiH differ from experimental data due to the truncated atomic orbital basis set used. Depicted are the Deviations of the (b) noiseless and (d) noisy SSQITE calculations from the ground truth energy levels of the $\text{LiH}$ molecule. All noisy simulations are performed using the qiskit FakeSherbrooke backend.