  Harnessing the full power of nascent quantum processors requires the
efficient management of a limited number of quantum bits with finite lifetime.
Hybrid algorithms leveraging classical resources have demonstrated promising
initial results in the efficient calculation of Hamiltonian ground states--an
important eigenvalue problem in the physical sciences that is often classically
intractable. In these protocols, a Hamiltonian is parsed and evaluated
term-wise with a shallow quantum circuit, and the resulting energy minimized
using classical resources. This reduces the number of consecutive logical
operations that must be performed on the quantum hardware before the onset of
decoherence. We demonstrate a complete implementation of the Variational
Quantum Eigensolver (VQE), augmented with a novel Quantum Subspace Expansion,
to calculate the complete energy spectrum of the H2 molecule with near chemical
accuracy. The QSE also enables the mitigation of incoherent errors, potentially
allowing the implementation of larger-scale algorithms without complex quantum
error correction techniques.
