  The computational cost of quantum algorithms for physics and chemistry is
closely linked to the spectrum of the Hamiltonian, a property that manifests in
the necessary rescaling of its eigenvalues. The typical approach of using the
1-norm as an upper bound to the spectral norm to rescale the Hamiltonian suits
the most general case of bounded Hermitian operators but neglects the influence
of symmetries commonly found in chemical systems. In this work, we introduce a
hierarchy of symmetry-aware spectral bounds that provide a unified
understanding of the performance of quantum phase estimation algorithms using
block-encoded electronic structure Hamiltonians. We present a variational and
numerically tractable method for computing these bounds, based on orbital
optimization, to demonstrate that the computed bounds are smaller than
conventional spectral bounds for a variety of molecular benchmark systems. We
also highlight the unique analytical and numerical scaling behavior of these
bounds in the thermodynamic and complete basis set limits. Our work shows that
there is room for improvement in reducing the 1-norm, not yet achieved through
methods like double factorization and tensor hypercontraction, but highlights
potential challenges in improving the performance of current quantum algorithms
beyond small constant factors through 1-norm reduction techniques alone.
