  Quantum computers potentially have an exponential advantage over classical
computers for the quantum simulation of many-fermion quantum systems.
Nonetheless, fermions are more expensive to simulate than bosons due to the
fermionic encoding -- a mapping by which the qubits are encoded with fermion
statistics. Here we generalize the contracted quantum eigensolver (CQE) to
avoid fermionic encoding of the wave function. In contrast to the variational
quantum eigensolver, the CQE solves for a many-fermion stationary state by
minimizing the contraction (projection) of the Schr\"odinger equation onto two
fermions. We avoid fermionic encoding of the wave function by contracting the
Schr\"odinger equation onto an unencoded pair of particles. Solution of the
resulting contracted equation by a series of unencoded two-body exponential
transformations generates an unencoded wave function from which the energy and
two-fermion reduced density matrix (2-RDM) can be computed. We apply the
unencoded and the encoded CQE algorithms to the hydrogen fluoride molecule, the
dissociation of oxygen O$_{2}$, and a series of hydrogen chains. Both
algorithms show comparable convergence towards the exact ground-state energies
and 2-RDMs, but the unencoded algorithm has computational advantages in terms
of state preparation and tomography.
