Quantum Computing and Error Mitigation with Deep Learning for Frenkel Excitons

Abstract

Quantum computers, currently in the noisy intermediate-scale quantum (NISQ) era, have started to provide scientists with a novel tool to explore quantum physics and chemistry. While several electronic systems have been extensively studied, Frenkel excitons, as prototypical optical excitations, remain among the less-explored applications. Here, we first use variational quantum deflation to calculate the eigenstates of the Frenkel Hamiltonian and evaluate the observables based on the oscillator strength for each eigenstate. Furthermore, using NISQ quantum computers requires performing error mitigation techniques alongside simulations. To deal with noisy qubits, we developed a deep-learning-based framework combined with a post-selection technique to learn the noise pattern and mitigate the error. Our mitigation methods work well and outperform the conventional post-selection and remain valid on real hardware.

Figures (9)

• Figure 1: (a) Real wave function of the HOMO and LUMO of a single anthracene molecule are shown as isosurfaces (green corresponds to positive and yellow to negative values), where the isosurfaces correspond to 95 % of electrons. (b) Five anthracene molecules arranged as a 3D anthracene crystal used in our simulations, based on the experimental structure mason1964crystallography. Each molecule is labeled from index 1 to 5, where red and grey atoms represent carbon and hydrogen atoms, respectively. The interaction among three different types of nearest pairs, $V_{12}$, $V_{13}$, $V_{14}$, is taken into account in the Hamiltonian, Eq. \ref{['eq:hamiltonian']}. There are two equivalent $V_{12}$, two $V_{14}$, and four $V_{13}$ interactions in this configuration.
• Figure 2: This circuit represents an Ansatz to create the local and entangled basis of Frenkel excitons out of the initial $|00000\rangle$ ground state. The X gate flips 0 and 1. The two-qubit controlled-rotation gates, labeled by Greek letters, comprise of two rotation gates, $R_y$, around the $y$ axis with opposite angles and one CNOT gate. Post-processing gates can include post-rotation gates, post-selection gates, or other quantum gates, depending on the post-processing strategies. This circuit can be extended with respect to the number of qubits and, thus, the rank of the Frenkel Hamiltonian.
• Figure 3: Quantum simulations of Frenkel Hamiltonian with a quantum simulator with the noise model from $ibmq\_guadalupe$ device. (a) Raw results without performing any error mitigation. (b) Post-selection mitigated results. Colored lines are the results under noisy environment and dashed line are the exact ground truth.
• Figure 4: Strategy for post-selection-based deep-learning mitigation technique: (a) Process of collecting data and training feedforward neural networks. The original circuit concatenated with 2 CNOT gates is used to describe the noisy pattern of the entire circuit with post-selection gates. Random parameters are assigned in the circuit, and the circuit is measured to obtain a noisy probability distribution, which dimension will be further reduced by selecting the states within a Hamming distance (HD) equal to 1. For the training section, the reconstructed probability is concatenated with the information of the position of the CNOT gate as the input, and the labeled data for supervised learning will be calculated classically through the deterministic function derived in Appendix.\ref{['appendix_1']}. (b) Workflow of using deep learning as a post-processing tool to mitigate the optimized results. Colored and dashed lines represent simulated results and exact ground truth.
• Figure S1: Noiseless simulations of Frenkel Hamiltonian of single layer anthracene comprising 5 molecules. Dashed lines are the reference obtained from exact diagonalization.