  Simulating electronic structure on a quantum computer requires encoding of
fermionic systems onto qubits. Common encoding methods transform a fermionic
system of $N$ spin-orbitals into an $N$-qubit system, but many of the fermionic
configurations do not respect the required conditions and symmetries of the
system so the qubit Hilbert space in this case may have unphysical states and
thus can not be fully utilized. We propose a generalized qubit-efficient
encoding (QEE) scheme that requires the qubit number to be only logarithmic in
the number of configurations that satisfy the required conditions and
symmetries. For the case of considering only the particle-conserving and
singlet configurations, we reduce the qubit count to an upper bound of
$\mathcal O(m\log_2N)$, where $m$ is the number of particles. This QEE scheme
is demonstrated on an H$_2$ molecule in the 6-31G basis set and a LiH molecule
in the STO-3G basis set using fewer qubits than the common encoding methods. We
calculate the ground-state energy surfaces using a variational quantum
eigensolver algorithm with a hardware-efficient ansatz circuit. We choose to
use a hardware-efficient ansatz since most of the Hilbert space in our scheme
is spanned by desired configurations so a heuristic search for an eigenstate is
sensible. The simulations are performed on IBM Quantum machines and the Qiskit
simulator with a noise model implemented from a IBM Quantum machine. Using the
methods of measurement error mitigation and error-free linear extrapolation, we
demonstrate that most of the distributions of the extrapolated energies using
our QEE scheme agree with the exact results obtained by Hamiltonian
diagonalization in the given basis sets within chemical accuracy. Our proposed
scheme and results show the feasibility of quantum simulations for larger
molecular systems in the noisy intermediate-scale quantum (NISQ) era.
