  Many quantum algorithms, including recently proposed hybrid classical/quantum
algorithms, make use of restricted tomography of the quantum state that
measures the reduced density matrices, or marginals, of the full state. The
most straightforward approach to this algorithmic step estimates each component
of the marginal independently without making use of the algebraic and geometric
structure of the marginals. Within the field of quantum chemistry, this
structure is termed the fermionic $n$-representability conditions, and is
supported by a vast amount of literature on both theoretical and practical
results related to their approximations. In this work, we introduce these
conditions in the language of quantum computation, and utilize them to develop
several techniques to accelerate and improve practical applications for quantum
chemistry on quantum computers. We show that one can use fermionic
$n$-representability conditions to reduce the total number of measurements
required by more than an order of magnitude for medium sized systems in
chemistry. We also demonstrate an efficient restoration of the physicality of
energy curves for the dilation of a four qubit diatomic hydrogen system in the
presence of three distinct one qubit error channels, providing evidence these
techniques are useful for pre-fault tolerant quantum chemistry experiments.
