Applications of quantum computing for investigations of electronic transitions in phenylsulfonyl-carbazole TADF emitters

The study addresses the challenge of accurately computing electronic excited states of molecules, particularly the singlet–triplet gap ΔEST critical for thermally activated delayed fluorescence (TADF) emitters used in OLEDs. Classical electronic structure methods face exponential scaling for exact solutions, and density functional theory (DFT) can be unreliable for high-energy singlet and triplet states relevant to intersystem crossing in TADF materials. Quantum computing offers promising alternatives: while phase estimation could, in principle, provide polynomial-time FCI energies, current noisy intermediate-scale quantum (NISQ) devices favor variational algorithms such as VQE. Extensions to excited states, including qEOM-VQE and VQD, have shown promise on small systems. This work applies these algorithms to industrially relevant phenylsulfonyl-carbazole (PSPCz) TADF emitters to determine ground and excited state energies and, crucially, to validate computed ΔEST against experimental data. The purpose is to develop and benchmark practical error mitigation schemes (notably, quantum state tomography and readout error mitigation) that enable accurate excited-state predictions on current quantum hardware and to elucidate structure–property relationships in substituted PSPCz molecules.

Prior quantum computing studies demonstrated ground-state energy calculations and simple reaction profiles on early IBM devices, though hardware-efficient ansätze suffered from noise. For excited states, qEOM-VQE and VQD have been proposed and demonstrated on small molecules (e.g., LiH). In TADF materials research, DFT (including TDDFT) has guided design but can be inaccurate for high-energy excitations and singlet–triplet gaps, especially for charge-transfer states. A known TADF design strategy is spatial separation of HOMO and LUMO across donor–acceptor moieties to minimize ΔEST. Phenylsulfonyl-9H-carbazole derivatives (PSPCz, 2F-PSPCz, 4F-PSPCz) offer tunable electronic properties and have experimental ΔEST benchmarks. However, excited-state quantum simulations on NISQ devices had not been rigorously validated against experiment for such systems, motivating this work.

- Systems: Three TADF emitters—PSPCz, 2F-PSPCz, 4F-PSPCz—where HOMO resides mainly on carbazole and LUMO on the diarylsulfone moiety. - Geometry: T1-state geometries optimized using TDDFT (CAM-B3LYP/6-31G(d)) with the Tamm–Dancoff approximation (TDA). These T1 geometries were used for both S1 and T1 due to their structural similarity in TADF systems. - Active space and mapping: HOMO–LUMO active space; configuration space includes HF, single excitations and double excitation. Particle–hole transformation, parity mapping, and two-qubit reduction yield a two-qubit spin Hamiltonian expressed as a sum of Pauli products with molecule-specific coefficients. - Basis sets: STO-3G and 6-31G(d) used to construct the active-space Hamiltonians. - Variational ansatz: Depth-1 R_y (denoted Rγ in figures) heuristic ansatz on 2 qubits, initialized to the Hartree–Fock bitstring. Circuit structure as in Fig. 7. - Algorithms: - VQE for S0 ground state energy minimization E(θ) = ⟨Ψ(θ)|H|Ψ(θ)⟩. - qEOM-VQE for excited states by constructing and measuring EOM Hamiltonian and metric matrices on the VQE ground state reference and classically diagonalizing to obtain ωn for T1 and S1. - VQD for excited states via cost functions including overlaps with previously found states to enforce orthogonality; first T1 relative to S0, then S1 relative to S0 and T1. - Simulation and hardware: - Statevector simulator (Qiskit Aer) with SLSQP optimizer for noise-free benchmarks. - IBM quantum devices: ibmq_boeblingen (qEOM-VQE) and ibmq_singapore (VQD) with SPSA optimizer for noise robustness. - Error mitigation: - Readout error mitigation: Construct a 2^n × 2^n calibration matrix M by preparing all basis states and inverting to correct measured expectation values: C_ideal = M C_noisy. - Quantum state tomography (QST): Perform full 2-qubit tomography using 16 Pauli operators grouped into 9 commuting sets; reconstruct density matrix ρ_final, extract dominant eigenstate |λ⟩ (purity λ), and correct energies via ⟨λ|H|λ⟩. Parameters θ are adjusted by fitting the ansatz state to |λ⟩. - Protocol: Run ~300 VQE iterations without mitigation to get near-ground state, then apply readout mitigation or QST to refine ground state; use that purified reference in subsequent qEOM-VQE or VQD steps. For qEOM-VQE, also apply readout mitigation to EOM matrix element measurements. - Additional details: - For simulator VQD, a spin penalty term was used; on hardware it was omitted due to noise in spin expectation values. For S1 with VQD on hardware, S0 and T1 were used as reference states in the cost function. - Software: Qiskit Aqua 0.14 interfaced with PySCF for integrals; Ignis tomography tools. - Comparison to experiment: ΔEST derived from room-temperature fluorescence and 77 K phosphorescence measurements from prior work used as benchmarks.

- Simulator accuracy: - qEOM-VQE and VQD on the statevector simulator reproduce exact diagonalization (FCI in the active space) for S0, T1, and S1 across all three molecules. - Basis set dependence of ΔEST: STO-3G yields larger gaps than 6-31G(d). Predicted ΔEST (STO-3G): PSPCz 2.54 eV (93 mHa), 2F-PSPCz 1.08 eV (41 mHa), 4F-PSPCz 0.25 eV (9 mHa). With 6-31G(d): PSPCz 0.86 eV (31 mHa), 2F-PSPCz 0.50 eV (18 mHa), 4F-PSPCz 0.0055 eV (0.2 mHa). 6-31G(d) correlates closely with experiment, with a ~0.4 eV (15 mHa) overestimation. - Device results and mitigation for qEOM-VQE (ibmq_boeblingen): - VQE ground state S0: Unmitigated energies ~13 mHa above exact; readout mitigation improves to ~6 mHa; QST improves to within ~1 mHa of exact. - Excited states via qEOM-VQE: Unmitigated deviations up to 17 mHa (largest for 4F-PSPCz); applying readout mitigation to both VQE ground state and EOM matrix elements reduces largest deviation to ~1 mHa; best protocol combines QST-purified ground state with readout-mitigated EOM measurements, yielding excited-state energies within ~3 mHa of exact. - ΔEST correlation: With the best mitigation, calculated ΔEST matches experimental trends and magnitudes well. - Device results and mitigation for VQD (ibmq_singapore): - Sensitivity to reference state purity: Using unpurified ground-state references leads to large errors, particularly for S1 (up to 88 mHa above exact). Purifying via QST reduces errors dramatically (S1 within ~12 mHa in the case study; T1 within ~8 mHa before further mitigation). - Across molecules, unmitigated deviations (T1/S1) of ~8, 12, 15 mHa (PSPCz, 2F-PSPCz, 4F-PSPCz) improve to ~0, 2, 4 mHa, respectively, with QST mitigation. - ΔEST ordering: Unmitigated VQD produces incorrect ordering (2F-PSPCz > PSPCz > 4F-PSPCz); QST-corrected VQD restores agreement with experiment (PSPCz > 2F-PSPCz > 4F-PSPCz). - Overall error summary: On hardware without mitigation, excited-state errors were ~17 mHa (qEOM-VQE) and up to 88 mHa (VQD). With QST-based purification and appropriate readout mitigation, errors are reduced to at most ~4 mHa and ΔEST agrees closely with experimental measurements.

The work demonstrates that with an appropriate active-space model (HOMO–LUMO) and 6-31G(d) basis, variational quantum algorithms (qEOM-VQE and VQD) can accurately reproduce excited-state energies and singlet–triplet gaps for realistic TADF emitters. Simulator results confirm algorithmic correctness; hardware results reveal that noise and readout errors are primary obstacles, especially affecting the purity of the VQE reference state and the accuracy of EOM matrix elements or VQD overlap terms. State tomography effectively purifies the noisy ground-state reference, correcting depolarization-induced mixedness, while readout mitigation improves measurement fidelity, particularly crucial for qEOM-VQE matrix elements. The combined protocol restores both the magnitude and ordering of ΔEST in line with experiment, validating the approach on industrially relevant molecules. Basis-set effects and the absence of dynamic correlation beyond the active space lead to systematic offsets (e.g., ~0.4 eV with 6-31G(d)), but relative trends are robust, enabling reliable structure–property insights—such as fluorination-induced tuning of S1 and T1 energies. Sensitivity near degeneracy (e.g., 4F-PSPCz) underscores the need for precise matrix element estimation; mitigation reduces these sensitivities. The methodology thus addresses the central question: can current NISQ hardware, with proper mitigation, deliver experimentally relevant excited-state properties? The results indicate yes, within a few mHa of active-space exact values and with correct ΔEST trends.

The study presents a validated workflow for computing excited states of TADF emitters on current quantum devices using HOMO–LUMO active-space Hamiltonians and variational algorithms. On simulators, qEOM-VQE and VQD reproduce exact diagonalization, and with 6-31G(d) yield ΔEST values strongly correlated with experiment. On hardware, unmitigated errors are substantial (up to ~17 mHa for qEOM-VQE and 88 mHa for VQD), but a practical scheme—purifying the VQE ground state via quantum state tomography and applying readout error mitigation (notably to qEOM matrix elements)—reduces deviations to ≤4 mHa and restores experimental ΔEST trends and magnitudes. This is, to the authors’ knowledge, the first validated excited-state study using VQD on hardware. Future work includes extending the approach to larger active spaces to capture higher singlet/triplet states, spin–orbit couplings, and dynamic correlation effects, potentially leveraging scalable state-purification techniques such as iterative purification, quantum principal component analysis, or variational quantum state diagonalization, as well as measurement reduction methods exploiting symmetries and randomized measurements.

- Active-space approximation: Restricting to HOMO–LUMO omits dynamic correlation with inactive orbitals, leading to systematic overestimation of ΔEST (e.g., ~0.4 eV with 6-31G(d)), though relative trends remain reliable. - Hardware noise: Decoherence and depolarization induce mixed ground states; accurate results require resource-intensive quantum state tomography. - Measurement overhead: QST scales as 3^n measurements; readout calibration scales as 2^n, limiting direct scalability to larger qubit counts without advanced reduction techniques. - Near-degeneracy sensitivity: When S1 and T1 are nearly degenerate (e.g., 4F-PSPCz), qEOM matrix element inaccuracies significantly affect excited-state energies unless mitigation is carefully applied. - Spin penalty omission on hardware for VQD due to noisy spin expectation values may affect state targeting compared to simulator protocols. - Experimental comparison caveat: S1 and T1 experimental energies are measured at different temperatures, potentially contributing to discrepancies (e.g., reported negative ΔEST for 4F-PSPCz).