  Over the past three decades significant reductions have been made to the cost
of estimating ground-state energies of molecular Hamiltonians with quantum
computers. However, comparatively little attention has been paid to estimating
the expectation values of other observables with respect to said ground states,
which is important for many industrial applications. In this work we present a
novel expectation value estimation (EVE) quantum algorithm which can be applied
to estimate the expectation values of arbitrary observables with respect to any
of the system's eigenstates. In particular, we consider two variants of EVE:
std-EVE, based on standard quantum phase estimation, and QSP-EVE, which
utilizes quantum signal processing (QSP) techniques. We provide rigorous error
analysis for both both variants and minimize the number of individual phase
factors for QSPEVE. These error analyses enable us to produce constant-factor
quantum resource estimates for both std-EVE and QSP-EVE across a variety of
molecular systems and observables. For the systems considered, we show that
QSP-EVE reduces (Toffoli) gate counts by up to three orders of magnitude and
reduces qubit width by up to 25% compared to std-EVE. While estimated resource
counts remain far too high for the first generations of fault-tolerant quantum
computers, our estimates mark a first of their kind for both the application of
expectation value estimation and modern QSP-based techniques.
