Fig. 1: Scaling of the one-norm and number of unique, non-zero coefficients (NNZ) for test systems.
From: Quantum simulations of chemistry in first quantization with any basis set
a For a dense, real Hamiltonian with a fixed number of electrons, the one-norm of first quantization scales as \({\mathcal{O}}({D}^{3})\) compared to \({\mathcal{O}}({D}^{4})\) for the equivalent second quantization method, where D is the number of orbitals. b Similarly, for H4 the one-norm scales better in first than in second quantization. c Hydrogen chains with varying number of atoms, NA, with the number of orbitals per atom, NA/D, being constant. The number of electrons, N, is proportional to the number of atoms and orbitals, N = O(NA) = O(D). Therefore, the \({\mathcal{O}}({N}^{2})\) dependence of the one-norm in first quantization leads to worse scaling. d For a dense, real Hamiltonian, the number of unique, non-zero terms is the same for first and second quantization. e For the H4 molecule, the number of unique non-zero terms is greater for first quantization. f The number of unique, non-zero terms has better scaling in second quantization than first for the Hydrogen chain.
