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Molecular eigenstates鈥攅ncompassing both ground and excited states鈥攁re central to understanding a wide range of chemical phenomena. Ground-state eigenproperties govern molecular structure, bonding patterns, and reactivity, while higher eigenstates determine photochemical pathways, energy-transfer dynamics, spectroscopic signatures, and material functionality. Accurately computing these states remains a core challenge, especially when strong electron correlation demands multiconfigurational treatments. Classical approaches such as the Density Matrix Renormalization Group (DMRG) [1] can efficiently capture correlation in large active spaces and have been extended to the computation of multiple eigenstates [2]. However, their scalability is ultimately limited in molecular systems with significant long-range interactions. Quantum computing offers a promising alternative. In the fully fault-tolerant regime, algorithms such as Quantum Phase Estimation (QPE) provide systematically improvable accuracy with favorable asymptotic scaling. Yet current hardware faces constraints including short coherence times, substantial noise, and limited qubit counts. The emergence of early fault-tolerant devices鈥攃haracterized by higher qubit numbers and improved error resilience鈥攃reates new opportunities for quantum algorithms that bridge the gap between today鈥檚 noisy processors and future fully fault-tolerant machines. To make effective use of this intermediate regime, algorithms tailored specifically to early fault-tolerant architectures are needed. Inspired by classical eigenvalue-targeting strategies, we develop quantum methods based on Krylov-subspace techniques and Quantum Singular Value Transformation (QSVT) [3] to efficiently compute correlated molecular eigenstates [4,5]. These approaches are designed to leverage the enhanced capabilities of early fault-tolerant hardware while requiring significantly fewer resources than algorithms aimed at the fully fault-tolerant era. [1] S. R. White, Phys. Rev. Lett. 69, 2863鈥�2866 (1992)[2] N. Glaser, A. Baiardi, M. Reiher, J. Chem. Theory Comput. 19, 9329-9343 (2023)[3] A. Gily茅n, Y. Su, G. H. Low, N. Wiebe, STOC, pp. 193鈥�204 (2019)[4] M. G. J. Oliveira, N. Glaser, Phys. Rev. A 112, 052442 (2025)[5] S. Patil, N. Glaser, in preparation
