Implementation of the Projective Quantum Eigensolver on a Quantum
  Computer
We study the performance of our previously proposed Projective Quantum
Eigensolver (PQE) on IBM's quantum hardware in conjunction with error
mitigation techniques. For a single qubit model of H$_2$, we find that we are
able to obtain energies within 4 millihartree (2.5 kcal/mol) of the exact
energy along the entire potential energy curve, with the accuracy limited by
both stochastic error and inconsistent performance of the IBM devices. We find
that an optimization algorithm using direct inversion of the iterative subspace
can converge swiftly, even to excited states, but stochastic noise can cause
large parameter updates. For the four-site transverse-field Ising model at the
critical point, PQE with an appropriate application of qubit tapering can
recover 99% of the correlation energy, even discarding several parameters. The
large number of CNOT gates needed for the additional parameters introduces a
concomitant error that, on the IBM devices, results in loss of accuracy,
despite the increased expressivity of the trial state. Error extrapolation
techniques and tapering or postselection are recommended to mitigate errors in
PQE hardware experiments.
Jonathon P. Misiewicz Francesco A. Evangelista