  Computational cost of energy estimation for molecular electronic Hamiltonians
via Quantum Phase Estimation (QPE) grows with the difference between the
largest and smallest eigenvalues of the Hamiltonian. In this work we propose a
preprocessing procedure that reduces the norm of the Hamiltonian without
changing its eigenspectrum 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 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 reduction of
1-norm for several LCU decompositions compared to their unshifted versions.
