State-Specific Orbital Optimization for Enhanced Excited-States Calculation on Quantum Computers

TL;DR

This work tackles the accuracy limitations of excited-state calculations on near-term quantum computers by introducing state-specific orbital optimization, where each excited state uses its own optimally rotated orbital set. A gradient-based framework is developed to optimize orbitals via the overlap between states, with the gradient of the overlap term derived and leveraged within a VQD-based solver. The proposed SSVQD algorithm alternates between optimizing circuit parameters and orbital rotations, leading to consistently improved results over state-averaged methods on H$_2$, H$_4$, and LiH benchmarks. The approach promises enhanced accuracy for electronic structure problems on noisy quantum devices and offers pathways to hybrid state-specific/state-averaged strategies and practical extensions like frozen-core calculations.

Abstract

We propose a state-specific orbital optimization scheme for improving the accuracy of excited states of the electronic structure Hamiltonian for the use on near-term quantum computers, which can be combined with any overlap-based excited-state quantum eigensolver. We derived the gradient of the overlap term between different states generated by different orbitals with respect to the orbital rotation matrix and use the gradient-based optimization methods to optimize the orbitals. This scheme allows for more flexibility in the choice of orbitals. We implement the state-specific orbital optimization scheme with the variational quantum deflation (VQD) algorithm, and show that it achieves higher accuracy than the state-averaged orbital optimization scheme on various molecules including H4 and LiH.

Figures (5)

• Figure 1: Comparison between (a) state-averaged and (b) state-specific orbital optimization.
• Figure 2: The workflow of the optimization approach.
• Figure 3: The results of SSVQD and SAVQD on $\textbf{H}_4\;$ for the low-lying $5$ eigen-energies. The $x$-axis denotes the number of two-step iterations, while the $y$-axis shows the absolute value of the energy difference between the cc-pVDZ FCI energy and the calculated energy.
• Figure 4: The results of SSVQD and SAVQD on $\textbf{LiH}\;$ for the low lying $4$ and $5$ eigen-energies. The $x$-axis denotes the number of two-step iterations, while the $y$-axis shows the absolute value of the energy difference between the cc-pVDZ FCI energy and the calculated energy. Only the first $4$ low-lying states are shown because, when SAVQD was applied to compute the first $5$ low-lying states, the ansatz circuits converged to the $1$-st, $2$-nd, $3$-rd, $4$-th, and $6$-th states, thereby missing the $5$-th state. The convergence curve of the $5$-th state is nearly identical to that of the $4$-th state.
• Figure 5: The sketch of the state-specific and state-averaged orbital optimization. Here the $\ket{\Psi_1({\boldsymbol{\theta}}_1)}$ is the $1$-st state, which uses the orbitals $\{\psi_j^{(1)}\}_{j=1}^N$. $\ket{\Psi_2({\boldsymbol{\theta}}_2)}$ and $\ket{\Psi_3({\boldsymbol{\theta}}_3)}$ are the $2$-nd and $3$-rd states, which use the same orbitals $\{\psi_j^{(2)}\}_{j=1}^N$. The overlap between $\ket{\Psi_2({\boldsymbol{\theta}}_2)}$ and $\ket{\Psi_3({\boldsymbol{\theta}}_3)}$ can be calculated straightforwardly without the need of the external circuit like $U(\boldsymbol{u}^{(1)\dag} \boldsymbol{u}^{(2)})$.