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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. 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