Data Efficient Prediction of excited-state properties using Quantum Neural Networks

TL;DR

This work addresses predicting excited-state properties from ground-state information in molecular systems using a symmetry-aware quantum neural network (siQNN) partnered with a classical NN. The approach operates on the ground-state wavefunction prepared on a quantum computer, measuring only commuting Pauli-string observables to stay NISQ-friendly, and scales parameters linearly with the number of orbitals. Across LiH, H2, and H4, the siQNN-NN demonstrates strong data-efficiency, outperforming classical baselines in the low-data regime for more complex targets like transition energies and TDMs, while singling out the benefits and limitations of quantum enhancements. The study highlights a promising pathway to extract excited-state information from ground-state data, with implications for photochemistry and materials design, and outlines future directions including shot-noise considerations and scalability to larger active spaces.

Abstract

Understanding the properties of excited states of complex molecules is crucial for many chemical and physical processes. Calculating these properties is often significantly more resource-intensive than calculating their ground state counterparts. We present a quantum machine learning model that predicts excited-state properties from the molecular ground state for different geometric configurations. The model comprises a symmetry-invariant quantum neural network and a conventional neural network and is able to provide accurate predictions with only a few training data points. The proposed procedure is fully NISQ compatible. This is achieved by using a quantum circuit that requires a number of parameters linearly proportional to the number of molecular orbitals, along with a parameterized measurement observable, thereby reducing the number of necessary measurements. We benchmark the algorithm on three different molecules with three different system sizes: $H_2$ with four orbitals, LiH with five orbitals, and $H_4$ with six orbitals. For these molecules, we predict the excited state transition energies and transition dipole moments. We show that, in many cases, the procedure is able to outperform various classical models (support vector machines, Gaussian processes, and neural networks) that rely solely on classical features, by up to two orders of magnitude in the test mean squared error.

Figures (12)

• Figure 1: Schematic overview of the model used in this study. The feature set, i.e., the input to the model is given by $\mathcal{X}=(\ket{\psi_0(R)},R)$, where $\ket{\psi_{0}(R)}$ denotes the molecular ground state and $R$ is the potentially multi-dimensional distance between the atoms. The model is trained to predict various target values $y$ such as the transition energy $\Delta E$ to an excited state. The observable is a weighted sum of Pauli-strings $\mathbf{P}$, where the weights $\mathbf{w}$ are learned by a classical NN [cf. Equation \ref{['eq:observable_equation']}]. Training this model involves a pre-training of the siQNN (dashed box) followed by an end-to-end training of the whole architecture.
• Figure 2: A simple illustration of the symmetry of several electronic ground state configurations for $\mathrm{H}_{2}$ with four considered orbitals prepared on the QC using the Jordan-Wigner mapping.
• Figure 3: An example of the structure of the QNN ansatz for $\mathrm{H}_2$ with 4 orbitals and 2 electrons. Here, $\theta_{1}$--$\theta_{21}$ are trainable parameters and $q_1$--$q_8$ denote the qubits. The two-qubit gate $\mathrm{N}(\theta_i,\theta_j,\theta_k)=\exp[i(\theta_iXX+\theta_jYY+\theta_kZZ)]$ can realize every two qubit unitary optimal_two_qubit_gate and $P(\theta_n, \theta_m)$ is a two qubit entangling gate that acts as a pooling operation QCNN. When gates act on non-neighboring qubits, circles in the gates mark the qubits on which the gates are applied.
• Figure 4: The target functions for LiH in (a) and (b), for $\mathrm{H}_2$ in (c) and (d), and for $\mathrm{H}_4$ in (e) and (f) in the 6-31G basis. In (a), (c), and (e) the transition energies $\Delta E_j$ for the four energetically lowest excited singlet and triplet states, that do not exhibit avoided crossings, are shown in Hartree $\rm E_{\rm h}$. In (b), (d), and (f) the non-zero TDMs of those states are shown in units of elementary charge e and the Bohr radius $\rm{a}_0$. All values are obtained via exact diagonalization of Equation \ref{['eq:qubit_hamiltonian']}.
• Figure 5: The predictions (solid lines) of (a) $\Delta E_{T_{2}}$ and (b) $\|\boldsymbol{\mu_}{0,S_2}\|_2$ for LiH for various models trained on specific trainings datasets. In (a) four training data points (marked by crosses) and in (b) six training data points are used. The exact target functions are represented by dashed lines. In (a) the best MSE score on the test dataset is achieved by the siQNN-NN, with a value of $\qty{2.6e-7}{\hartree}^2$. The second best test MSE score is obtained by the SVR, with a value of $\qty{9.1e-5}{\hartree}^2$. In (b) the best MSE score on the test dataset is achieved by the siQNN-NN, with a value of $\qty{8.3e-4}{\hartree}^2$. The second best test MSE score is obtained by the NN, with a value of $\qty{3.7e-3}{\hartree}^2$.