Applications of Quantum Computing for Investigations of Electronic Transitions in Phenylsulfonyl-carbazole TADF Emitters

A quantum chemistry study of the first singlet (S1) and triplet (T1) excited states of phenylsulfonyl-carbazole compounds, proposed as useful thermally activated delayed fluorescence (TADF) emitters for organic light emitting diode (OLED) applications, was performed with the quantum Equation-Of-Motion Variational Quantum Eigensolver (qEOM-VQE) and Variational Quantum Deflation (VQD) algorithms on quantum simulators and devices. These quantum simulations were performed with double zeta quality basis sets on an active space comprising the highest occupied and lowest unoccupied molecular orbitals (HOMO, LUMO) of the TADF molecules. The differences in energy separations between S1 and T1 ($\Delta E_{st}$) predicted by calculations on quantum simulators were found to be in excellent agreement with experimental data. Differences of 16 and 88 mHa with respect to exact energies were found for excited states by using the qEOM-VQE and VQD algorithms, respectively, to perform simulations on quantum devices without error mitigation. By utilizing error mitigation by state tomography to purify the quantum states and correct energy values, the large errors found for unmitigated results could be improved to differences of, at most, 3 mHa with respect to exact values. Consequently, excellent agreement could be found between values of $\Delta E_{st}$ predicted by quantum simulations and those found in experiments.


Introduction
One of the major constraints of modern electronic * Corresponding authors. E-mail: caoch@user.keio.ac.jp; gojones@us.ibm.com structure methods used for quantum chemical calculations on classical computing architecture is the difficulty of finding eigenvalues of eigenvectors of the electronic Hamiltonian. [1] The advent of quantum computing, which has demonstrated tremendous synergistic advances in both hardware and software capabilities in recent years, may provide invaluable support in the investigation of the electronic structure of molecules and materials, especially with regards to dynamical properties.
However, quantum computing is still, in many ways, a nascent technology, and quantum devices are currently hamstrung by noisiness and short decoherence times. Thus, quantum algorithms, such as the Variational Quantum Eigensolver (VQE) algorithm, [2] are currently being used to find eigenvalues for approximate Ansätze suitable for constructing relatively short circuits that can be used for quantum chemistry calculations. [3] A number of quantum computing use cases for chemistry have been investigated on quantum devices in the recent past. One of the first applications highlighted the construction of ground state dissociation profiles of hydrogen, lithium hydride and beryllium hydride via computation using an IBM Q quantum device. [3] Researchers have also performed computational studies on the mechanism of the rearrangement of the lithium superoxide dimer using larger devices. [4] In the latter case, the orbital determinants of the stationary points were extensively examined to identify a suitable orbital active space for which a reduced number of qubits could be used for computation given the current limitations of these devices. Simulations performed on classical hardware reproduced calculations involving the exact eigensolver known as Full Configuration Interaction (Full CI or FCI) for this process. However, calculations performed on quantum devices using hardware efficient Ansätze such as Ry were less capable of reproducing exact energies due to noise.
In addition to using the VQE algorithm to compute energies for the dissociation profiles of molecules and the potential energy surfaces for reactions, various algorithms, among them the Quantum Equation-of-Motion VQE (qEOM-VQE) [5] and Variational Quantum Deflation (VQD) algorithms [6], have been developed in order to compute excited states of molecules. In particular, Ollitrault et al. have demonstrated computation of the first three excited states on the dissociation profile for lithium hydride on IBM Quantum hardware. [5] Here, we take an initial step towards the application of algorithms such as qEOM-VQE and VQD to determine excitation energies of industrially relevant molecules. The subject of this study is thermally activated delayed fluorescence (TADF) emitters suitable for organic light emitting diode (OLED) applications. [7] Once the separation between the first singlet (S1) and triplet (T1) excited states (∆ !" ) is sufficiently small, non-emissive T1 excitons can be thermally excited to an emissive S1 state, providing the emission mechanism of TADF emitters. The mechanism enables OLED devices comprised of TADF emitters to potentially perform with 100% internal quantum efficiency, in contrast to devices comprising conventional fluorophores for which the quantum efficiencies are inherently limited to 25%. TADF emitters have been demonstrated as next-generation emission materials for a range of fluorophores and have been utilized in the fabrication of efficient OLEDs. [7][8][9][10][11][12] To date there have been a number of computational studies on predictions of ∆ !" to aid in the design of novel TADF materials. [12][13][14][15][16][17]  of these molecules can be finely tuned by modifying their electronic properties. [18] We have elected to perform quantum chemical investigations on the molecules, PSPCz, 2F-PSPCz, 4F-PSPCz ( Figure   1), using quantum devices and simulators. Notably, all three molecules of interest require the use of many more qubits than are currently available on useful quantum hardware or that can be reliably simulated using classical architecture. Thus, qubit reduction techniques must be applied in order to simulate the transition amplitudes of interest to this study. We have reduced the number of spatial orbitals to those that are absolutely necessary to describe the processes under investigation and have thus focused on transitions involving the HOMO and LUMO active space for each molecule. This strategy has allowed reduction of the number of qubits to just two after applying spin parity reduction.
Simulations of the qEOM-VQE and VQD algorithms were performed on classical devices using heuristic Ansätze, in order to assess the accuracy of these techniques. Particular attention was paid to comparisons with experimental data [18] which indicate that these procedures produce meaningful data for predicting the ∆ !" of TADF emitters.
Moreover, we have shown that these simulations can provide a detailed picture about how structural variation in phenylsulfonyl/carbazole molecules tunes the S1 and T1 excitations. Finally, we have found that simulations performed on a quantum device without error mitigation were much less reliable than results provided by quantum simulators. This is attributed to the fact that the quantum state of the ground state predicted on devices is a mixed state.
We have examined various strategies to solve this issue and have found that the most accurate procedure involves the use of state tomography to purify the mixed ground state obtained from the quantum device prior to application of readout error mitigation to the calculation involving the excited state. We note that since the state tomography purification approach can, in principle, be applied to other methods for computing excited states on quantum computers [19,20], and can be extended to systems requiring large number of qubits by purifying the quantum state using a computationally inexpensive iterative approach, [21] or by applying quantum principal component analysis, [22] or variational quantum state diagonalization, [23] we believe that it can be generally useful for quantum chemistry simulations of excited states on near-term quantum devices. error mitigation applied to ground state energies obtained from quantum devices, (iv) computation of the first singlet and triplet excited states (S1 and T1), using quantum simulators and quantum devices.

Four
Optimized geometries of the first triplet excited state (T1) of all PSPCz molecules were generated using the TDDFT routines contained in the Gaussian16 computational suite of programs [24] using the CAM-B3LYP/6-31G(d) method and basis set. The Tamm-Dancoff approximation (TDA) [25], which has been reported to be accurate for the calculation of triplet excited states [26], was employed for all TDDFT calculations. For the sake of simplicity, the optimized geometries of the T1 excited states were used for both the S1 and T1 excited states based on the fact that the HOMO-LUMO overlap (i.e. exchange integral) in TADF molecules is so small that the properties of the S1 and T1 states should be quite similar, except for the spin component, and the structures of these excited states should also be similar.
The HOMO and LUMO was chosen as the active space for calculating the ground state (S0), and S1 and T1 excited states because the HOMO and LUMO of the target molecules are spatially fairly wellseparated as shown in Figure 2, and both the S1 and T1 states can be well described by a transition arising from the HOMO, with atomic orbitals localized on the carbazole moiety, to the LUMO, with atomic orbitals localized on the diarylsulfone (ArSO2Ar-) moieties. Calculations were performed to obtain ground state energies on quantum simulators and devices by utilizing the VQE algorithm and the Ry heuristic Ansätze with a circuit depth equal to 1. Energies were computed using the STO-3G minimal basis set and the split-level double zeta basis set 6-31G(d). The parity mapping technique [27], which allows the number of qubits to be reduced by two, was used to map molecular spin-orbitals to qubits. The Aqua module contained in Qiskit version 0.14 [28] with an interface to the PySCF [29] program was used for all The Sequential Least SQuares Programming (SLSQP) method [30], which uses the exact gradient for energy minimization and has been found to provide accurate predictions without noise present, [31] was used for calculations using the    to perform a VQD calculation to obtain the first excited state, T1, which was in turn used as the initial state, and then both S0 and T1 were used as reference states in another VQD calculation to obtain the second excited state, S1.

A. Calculations with the statevector simulator
The energies of the S0, S1 and T1 states calculated by the Full CI, and Ry methods using qEOM-VQE and VQD algorithms with the STO-3G

B. Calculations with the qasm simulator
Although we have shown that the energies of S1 and T1 can be accurately predicted by qEOM-VQE and VQD methods on the statevector simulator, it is important to note that it is notoriously difficult to obtain similar accuracy by performing calculations on quantum devices due to the influence of device noise.
There are two major sources of noise that could affect the accuracies observed in this study. One is  are 0.999, indicating that 8192 measurement shots can satisfactorily predict these gaps.
We also found that simulations using qEOM-VQE reproduce the ∆ !" values predicted by Full CI, whereas simulations that utilize the VQD algorithm slightly overestimate these gaps. These results more sensitive to the number of measurement shots suggest that VQD is, at least in the present case, than qEOM-VQE, which may be due to the fact that VQD obtains excited state energies from a penalty term comprising the overlap between two quantum states.
This overlap could possibly be improved by increasing the number of shots by a factor of 3 to 4; however, such improvements significantly increase the computational expense of quantum simulations using the qasm simulator.  We unexpectedly found that energies obtained by the use of readout error mitigation on results obtained from the Ry Ansätze using the VQE algorithm on the ibmq_boeblingen device are almost similar to, or slightly higher than, energies obtained from calculations without readout error  We undertook an examination of the effect of error mitigation approaches on the accuracy of excited state energies predicted by the use of the Ry Ansätze with the qEOM-VQE algorithm for the carbazoles of interest to this study. Figure 10 and    Figure 13 and Table 1.
Without error mitigation, device noise causes the