Modern electronic structure methods face challenges in calculating eigenvalues and eigenvectors of the electronic Hamiltonian, particularly for excited states. Quantum computing offers a potential solution, with algorithms like the phase estimation algorithm promising polynomial-time calculation of Full Configuration Interaction (FCI) energies, unlike the exponential time complexity of classical methods. However, current quantum devices suffer from noise and short decoherence times, making less demanding algorithms like the Variational Quantum Eigensolver (VQE) more practical. Previous studies have explored VQE's applications in calculating ground-state dissociation profiles and potential energy surfaces. Algorithms such as qEOM-VQE and VQD have been developed for computing excited states. This study aims to apply these algorithms to industrially relevant molecules – thermally activated delayed fluorescence (TADF) emitters for OLED applications. TADF emitters offer the potential for 100% internal quantum efficiency due to the thermal excitation of non-emissive T1 excitons to emissive S1 states, a significant improvement over conventional fluorophores limited to 25%. The development of commercially viable TADF emitters requires materials with high quantum efficiency, stable properties, and appropriate emission spectra. While density functional theory (DFT) has been used in TADF emitter design, its accuracy can be limited, particularly for high-energy excited states crucial for understanding intersystem crossing. This study seeks to benchmark quantum algorithms against experimental data to improve the accuracy of excited-state simulations for TADF materials. The singlet-triplet gap (ΔEST), a crucial property for high-performance TADF emitters, will serve as a benchmark for testing the computational procedures. The focus is on a series of phenylsulfonyl-9H-carbazole (PSPCz) molecules, known for their tunable ΔEST through modification of electronic properties. The study addresses qubit reduction techniques to overcome the limitations of current quantum hardware, focusing on the HOMO-LUMO active space to reduce the qubit count to two after applying spin parity reduction.

The introduction extensively cites prior work related to quantum computing in chemistry, including the limitations of classical methods and the development of quantum algorithms such as VQE, qEOM-VQE, and VQD. It references earlier studies on the calculation of ground state and excited state energies in small molecules using these algorithms, highlighting both successes and challenges related to the noisy nature of current quantum hardware. The use of DFT methods in the design of TADF emitters is discussed, along with their limitations, specifically in calculating high-energy singlet and triplet excited states. The literature review also highlights the importance of the singlet-triplet energy gap (ΔEST) in TADF emitter design and the existing work on TADF materials, emphasizing the need for accurate computational methods to guide the design process.

The study employed a multi-step computational approach: (1) Classical structural optimization of the first triplet excited state (T1) of PSPCz, 2F-PSPCz, and 4F-PSPCz molecules using time-dependent DFT (TDDFT) with CAM-B3LYP/6-31G(d) and Tamm-Dancoff approximation (TDA). The optimized T1 geometries were used for both S1 and T1 calculations. (2) Calculation of ground state (GS) energies on quantum simulators and devices using the VQE algorithm and the Ry heuristic Ansatz (depth 1) with STO-3G and 6-31G(d) basis sets. Parity mapping was used for qubit reduction. (3) Error mitigation for ground-state energies from quantum devices using readout error mitigation and quantum state tomography. (4) Calculation of S1 and T1 excited states using qEOM-VQE and VQD algorithms on quantum simulators and devices, using the error-mitigated GS as the reference state for excited-state calculations. The SLSQP optimizer was used for statevector simulator calculations, while SPSA was employed for quantum device calculations. Readout error mitigation involved measuring expectation values of Pauli terms in the Hamiltonian and using a calibration matrix to correct for measurement noise. Quantum state tomography involved obtaining the density matrix of the ground state, finding the dominant eigenstate, and using this purified state to correct the ground-state energy and the variational parameters. For qEOM-VQE, matrix elements of the equation of motion were calculated on quantum devices, followed by classical diagonalization to obtain S1 and T1 energies. For VQD, the ground state from VQE served as the initial and reference state for calculations on the statevector simulator; for quantum devices, the GS, T1, and S0 states were used iteratively. 300 iterations were performed before applying error mitigation techniques, and the optimized wavefunction from quantum state tomography was determined using the density matrix of the unmitigated VQE result.

Calculations on the statevector simulator using qEOM-VQE and VQD with the Ry Ansatz and 6-31G(d) basis set accurately predicted ΔEST, showing good agreement with experimental values. The STO-3G basis set yielded larger ΔEST values than experimental data. Calculations on quantum devices without error mitigation resulted in significant deviations from exact energies (up to 17 mHa for qEOM-VQE and 88 mHa for VQD). The application of readout error mitigation alone provided limited improvement. However, the proposed scheme utilizing quantum state tomography significantly improved the accuracy of excited-state energy predictions from both qEOM-VQE and VQD, reducing the deviations from exact values to within 4 mHa. The ΔEST values obtained with state tomography also showed excellent agreement with the experimental results across all three PSPCz molecules. The analysis of VQD calculations on the ibmq_singapore device indicated that using a purified reference state from quantum state tomography significantly improved the accuracy of both T1 and S1 energy calculations. Unmitigated results deviated from exact values by 20 mHa (T1) and 88 mHa (S1), while purification via quantum state tomography reduced these errors to 8 mHa and 12 mHa, respectively. This highlights the impact of noise on the overlap term in VQD, emphasizing the crucial role of state tomography in mitigating these errors. Further analysis showed that using state tomography on VQD calculations yielded ΔEST values in excellent agreement with experimental data.

The results demonstrate the potential of quantum computing for accurate prediction of excited-state properties in complex molecules relevant to materials science and industrial applications. The accurate prediction of ΔEST, a key parameter for TADF emitter design, validates the use of qEOM-VQE and VQD algorithms coupled with state tomography for this purpose. The significant improvement in accuracy achieved through error mitigation highlights the critical need for such techniques in harnessing the potential of noisy quantum devices. The ability to accurately predict excited-state energies from a limited active space opens up possibilities for studying larger and more complex molecules, where the use of FCI on classical computers is computationally prohibitive. The findings demonstrate the successful implementation of VQD for calculating excited states, an extension of the VQE method, representing a significant step forward in quantum chemical calculations. The observed improvement in accuracy from using a purified ground state with state tomography suggests that this approach will be beneficial for other quantum algorithms calculating excited states.

This study demonstrates the accuracy of qEOM-VQE and VQD algorithms in predicting excited-state energies of TADF emitters, particularly when coupled with state tomography error mitigation. State tomography proves crucial for achieving high accuracy on noisy quantum devices. This work validates the use of VQD for excited state calculations and highlights the significance of state tomography for improving the reliability of quantum computation. Future research could explore the application of this approach to larger, more complex TADF emitters and other organic electronic and optical materials. Furthermore, investigating the computational cost scaling of the state tomography purification approach for larger systems by employing computationally inexpensive iterative methods, quantum principal component analysis, or variational quantum state diagonalization, will be essential for extending the applicability of this method.

The current study focuses on a specific class of TADF molecules and employs a reduced active space of HOMO and LUMO. This simplification, while necessary for the current limitations of quantum hardware, may not fully capture all electronic correlations. The accuracy of the results might be affected by the quality of the classical pre-computations used for geometry optimization and the choice of basis set. The computational cost of state tomography scales exponentially with the number of qubits, potentially limiting its application to larger systems without further advancements in error mitigation strategies. The use of an idealized theoretical model for the calculation of the molecular Hamiltonian implies that environmental factors might not be fully accounted for, influencing the results' generalizability to real-world conditions. The noisy nature of the quantum devices used might still introduce some errors despite using error mitigation techniques.