Recorded 01 April 2025. Nick Rubin of Google Inc. presents "The sum-of-squares hierarchy and faster quantum simulation by spectrum amplification" at IPAM's Optimal Transport for Density Operators: Theory and Numerics Workshop. Abstract: The most advanced techniques using fault-tolerant quantum computers to estimate the ground-state energy of a chemical Hamiltonian involve compression of the Coulomb operator through tensor factorizations, enabling efficient block-encodings of the Hamiltonian. A natural challenge of these methods is the degree to which block-encoding costs can be reduced. We address this challenge through the technique of spectrum amplification, which magnifies the spectrum of the low-energy states of Hamiltonians that can be expressed as sums of squares. Spectrum amplification enables estimating ground-state energies with significantly improved cost scaling in the block encoding normalization factor Λ to just 2ΛEgap−−−−−−√ , where Egap≪Λ is the lowest energy of the sum-of-squares Hamiltonian. To achieve this, we show that sum-of-squares representations of the electronic structure Hamiltonian are efficiently computable by a family of classical simulation techniques that approximate the ground-state energy from below. In order to further optimize, we also develop a novel factorization that provides a trade-off between the two leading Coulomb integral factorization schemes-- namely, double factorization and tensor hypercontraction-- that when combined with spectrum amplification yields a factor of 4 to 195 speedup over the state of the art in ground-state energy estimation for models of Iron-Sulfur complexes and a CO2 -fixation catalyst. Learn more online at: https://www.ipam.ucla.edu/programs/workshops/workshop-i-optimal-transport-for-density-operators-theory-and-numerics/?tab=schedule