The Sum-of-Squares Hierarchy and Faster Quantum Simulation by Spectrum Amplification

This recorded lecture from April 1, 2025, features Nick Rubin of Google Inc. presenting groundbreaking research on quantum simulation techniques at IPAM's Optimal Transport for Density Operators workshop. Explore how spectrum amplification can significantly improve quantum computing efficiency for chemical simulations by magnifying the spectrum of low-energy states in Hamiltonians expressible as sums of squares. Discover how this approach reduces block-encoding costs to just 2ΛEgap−−−−−−√, where Egap≪Λ represents the lowest energy of the sum-of-squares Hamiltonian. Learn about a novel factorization method that creates an optimal balance between double factorization and tensor hypercontraction schemes, resulting in remarkable speedups of 4 to 195 times over current state-of-the-art methods when estimating ground-state energies for Iron-Sulfur complexes and CO2-fixation catalysts. The 46-minute presentation demonstrates how efficiently computable sum-of-squares representations can revolutionize electronic structure Hamiltonian calculations through classical simulation techniques that approximate ground-state energy from below.