Scientists have developed a new hybrid quantum-classical algorithm to solve large-scale eigenvalue problems, a crucial requirement in nuclear many-body theory where Hamiltonian matrices often reach exceptionally large dimensions. The method combines quantum annealing and classical deflation to iteratively determine the complete eigenspectrum of both standard and generalised eigenvalue problems. This approach was benchmarked using problems originating from the Equation of Motion Phonon Method, performing calculations on actual quantum hardware and illuminating both the potential and current limitations of near-term quantum devices for tackling complex nuclear physics calculations. Hybrid quantum-classical deflation extracts complete eigenspectra for nuclear structure modelling A significant performance boost is achieved with a hybrid quantum-classical algorithm, attaining machine-precision accuracy on eigenvalue calculations within approximately 30 iterations. Classical Simulated Annealing often failed to converge or required over 450 iterations for comparable results, highlighting a clear advantage. This breakthrough addresses a key limitation in nuclear physics, where solving large eigenvalue problems, essential for modelling atomic nuclei, was previously hampered by the coherence and error correction demands of algorithms like Quantum Phase Estimation. The computational complexity of these problems scales rapidly with the number of nucleons within the nucleus, quickly exceeding the capabilities of even the most powerful classical supercomputers. Traditional methods struggle to accurately determine the energy levels and wavefunctions of these complex systems. This hinders our understanding of nuclear structure and reactions. Dr. James Maxwell and colleagues successfully extracted complete eigenspectra from problems originating from the Equation of Motion Phonon Method, utilising actual quantum hardware for the first time and demonstrating a novel computational pathway. Large-scale eigenvalue problems are commonplace in nuclear many-body theory, with Hamiltonian matrices often becoming extremely large. For instance, matrices with dimensions exceeding 1000×1000 are not uncommon in realistic nuclear structure calculations. Quantum computing presents new approaches to tackle these demanding problems, but the Quantum