Quantum simulation with sum-of-squares spectral amplification

We introduce sum-of-squares spectral amplification (SOSSA), a framework for improving quantum simulation algorithms relevant to low-energy problems. SOSSA first represents the Hamiltonian as a sum-of-squares and then applies spectral amplification to amplify the low-energy spectrum. The sum-of-squares representation can be obtained using semidefinite programming. We show that SOSSA can improve the efficiency of traditional methods in several simulation tasks involving low-energy states. Specifically, we provide fast quantum algorithms for energy and phase estimation that improve over the state-of-the-art in both query and gate complexities, complementing recent results on fast time evolution of low-energy states. To further illustrate the power of SOSSA, we apply it to the Sachdev-Ye-Kitaev model, a representative strongly correlated system, where we demonstrate asymptotic speedups by a factor of the square root of the system size. Notably, SOSSA was recently used in [G.H. Low \textit{et al.}, arXiv:2502.15882 (2025)] to achieve state-of-art costs for phase estimation of real-world quantum chemistry systems.