Reducing the molecular electronic Hamiltonian encoding costs on quantum computers by symmetry shifts

Computational cost of energy estimation for molecular electronic Hamiltonians via quantum phase estimation (QPE) grows with the spectral norm of the Hamiltonian. In this work we propose a preprocessing procedure that reduces the norm of the Hamiltonian without changing its eigen-spectrum for the target states of a particular symmetry. The new procedure, block-invariant symmetry shift (BLISS), builds an operator T such that the cost of implementing H-T is reduced compared to that of H, yet H-T acts on the symmetric subspaces of interest the same way as H does. BLISS performance is demonstrated for linear combination of unitaries (LCU)-based QPE approaches on a set of small molecules. Using the number of electrons as the symmetry specifying the target set of states, BLISS provided a factor of 2-3 reduction of 1-norm compared to that in a single Pauli product LCU decomposition.