Multistate iterative qubit coupled cluster (MS-iQCC): a quantum-inspired, state-averaged approach to ground- and excited-state energies

TL;DR

MS-iQCC presents a quantum-inspired, classically executable framework to compute ground- and excited-state energies by iteratively dressing a qubit Hamiltonian with an unrestricted pool of Pauli generators in a state-averaged, multi-state formalism. The method builds a Direct Interaction Space and uses phase-alignment to select impactful generators, yielding a compact, adaptively growing effective Hamiltonian that remains well-behaved under moderate compression and converges to chemical accuracy for all targeted states. By simultaneously optimizing multiple states and allowing multireference references, MS-iQCC avoids state-specific bias and handles strong correlation more robustly than many traditional approaches. The results on H$_4$, H$_2$O, N$_2$, and C$_2$ showcase reliable convergence of both energies and fidelities, with clear guidance on the roles of compression, phase-alignment, and model-space size. This approach provides a practical, scalable path for accurate excited-state energetics on classical hardware today and offers compact quantum-state preparations for future QPE-based quantum computing.

Abstract

We introduce the multistate iterative qubit coupled cluster (MS-iQCC) method, a quantum-inspired algorithm that runs efficiently on classical hardware and is designed to predict both ground and excited electronic states of molecules. Accurate excited-state energetics are essential for interpreting spectroscopy and chemical reactivity, but standard electronic structure methods are either too computationally expensive for larger systems or lose reliability in the presence of strong electron correlation. MS-iQCC addresses this challenge by simultaneously optimizing multiple electronic states in a single, state-averaged procedure that treats ground and excited states on equal footing. This removes the energetic bias that is introduced when excited states are computed one at a time and constrained to remain orthogonal to previously optimized states. The approach supports multireference electronic structure by allowing multideterminantal initial guesses and by adaptively building a compact exponential ansatz from a pool of qubit excitation generators. We apply MS-iQCC to H$_4$, H$_2$O, N$_2$, and C$_2$, including strongly correlated geometries, and observe robust convergence of all targeted state energies to chemically meaningful accuracy across their potential energy surfaces.

Figures (7)

• Figure 1: A step-by-step description of the MS-iQCC algorithm. The input mixed state $\hat{\rho}$ is defined in Eq. \ref{['eq: input mixed state']}. The diagonalization in step 4 marks the end of the algorithm and the resulting Hamiltonian is the $\hat{H}_{qcc}$ defined in Eq. \ref{['eq: H_qcc']}. The diagonalization can also be performed at the end of each iteration to track the convergence of state-specific energies instead of state-averaged energy. If one chooses $N_g = 1$, then step 4 becomes irrelevant and the whole procedure reduces to GS-iQCC.
• Figure 2: MS-iQCC applied to the determination of the four lowest energy eigenstates for the H$_4$ chain at an equidistant separation of $2r_e$ in the STO-3G basis set. The $\varepsilon_c$ denotes the compression threshold used. a) Errors in the state-specific energies relative to the associated FCI target energies. b) Errors in the squared wavefunction overlaps of the iQCC-rotated trial states with their corresponding target eigenstates (see Eq. \ref{['eq: overlap_error_defn']}). c) The ratio of number of terms in the effective iQCC Hamiltonian and the initial qubit-mapped Hamiltonian.
• Figure 3: The MS-iQCC procedure with $N_g = 1$ applied to the simultaneous determination of the $\text{S}_0$ and $\text{S}_1$ states of the CAS($4$e, $4$o) model of stretched H$_2$O in the $6$-$31$G basis set. The $L=4$ configurations in Eq. (\ref{['eq:h2o_model_space']}) were used to define the model space. The global optimization described in Appendix \ref{['sec:selection_opt']} was utilized for phase-alignment. a) The errors of the MS-iQCC $\text{S}_0$ and $\text{S}_1$ estimates taken with respect to the exact $\text{S}_0$ and $\text{S}_1$ solutions within the CAS($4$e, $4$o) space. b) Errors in the squared overlaps of the MS-iQCC trial states with respect to the associated target state. c) $\Delta E_{SA}$ at iteration $K$ is given by the difference between the optimized $K^{\rm{th}}$ and $(K-1)^{\rm{th}}$ MS-iQCC state-averaged energy.
• Figure 4: The same as Fig. \ref{['fig:h2o_smallbasis']}, however employing the slightly larger model space of dimension $K=6$, with extra configurations contributing to the $S_{1}$ trial state reference given in Eq. (\ref{['eq:h2o_model_space_ext']}).
• Figure 5: The MS-iQCC procedure applied to simultaneous determination of the $\text{S}_0$ and $\text{T}_1$ states of the CAS($6$e, $6$o) model of N$_2$ molecule at $r_e = 1.0975$ Å bond distance in the STO-6G basis set. A total of $L=7$ computational basis states were used, with $N_g = 5$ generators used at each iteration, and OPT strategy utilized for phase-alignment. The subplots a, b, and c correspond to energy errors, fidelity errors, and the growth factors, respectively.