  We describe the contextual subspace variational quantum eigensolver (CS-VQE),
a hybrid quantum-classical algorithm for approximating the ground state energy
of a Hamiltonian. The approximation to the ground state energy is obtained as
the sum of two contributions. The first contribution comes from a noncontextual
approximation to the Hamiltonian, and is computed classically. The second
contribution is obtained by using the variational quantum eigensolver (VQE)
technique to compute a contextual correction on a quantum processor. In general
the VQE computation of the contextual correction uses fewer qubits and
measurements than the VQE computation of the original problem. Varying the
number of qubits used for the contextual correction adjusts the quality of the
approximation. We simulate CS-VQE on tapered Hamiltonians for small molecules,
and find that the number of qubits required to reach chemical accuracy can be
reduced by more than a factor of two. The number of terms required to compute
the contextual correction can be reduced by more than a factor of ten, without
the use of other measurement reduction schemes. This indicates that CS-VQE is a
promising approach for eigenvalue computations on noisy intermediate-scale
quantum devices.
