Exploring Runtime Reductions and Limits of Fault-Tolerant Quantum Chemistry Simulations

The Hamiltonian's spectral properties strongly influence the cost of simulating chemical systems on quantum computers. Existing methods like tensor hypercontraction (THC) and double factorization (DF) achieve computational savings by compressing the Hamiltonian efficiently. However, their performance is constrained by the spectral range of the Hamiltonian. First, we present findings from a numerical study examining the spectral range's lower bounds for various systems. Additionally, we leverage symmetries in electronic structure Hamiltonians to establish lower bounds within specific symmetry sectors. Secondly, we introduce symmetry-compressed double factorization (SCDF), which incorporates symmetry shifts and regularization within the DF approach. We show that SCDF reduces runtime by 50% compared to the best-performing THC method for FeMoco and P450 molecules.