Implementation of the Projective Quantum Eigensolver on a Quantum Computer

Abstract

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 H2, 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 the stochastic error and the 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 prompt large parameter updates. For the 4-site transverse-field Ising model at its critical point, PQE with an appropriate application of qubit tapering can recover 99% of the correlation energy, even after 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 a 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.

Fighting Noise with Noise: A Stochastic Projective Quantum Eigensolver

“…We find (see middle panel of Figure ) that the same number of sampled Hamiltonian terms can be used per step as in standard ground state calculations to generate generally good quality excited state energies, although once again, where states are close in energy this may lead to convergence onto different states than the one targeted. This behavior has been observed for conventional PQE as well, in the presence of noise . This significantly expands on the range of systems for which folded spectrum methods are tractable.…”

“…This behavior has been observed for conventional PQE as well, in the presence of noise. 64 This significantly expands on the range of systems for which folded spectrum methods are tractable.…”

“…If the HF determinant is used, then Δ i = ϵ i + ϵ j + ... – ϵ a – ϵ b – ..., where ϵ i are Hartree–Fock spin-orbital energies. The residuals r i l can be computed using expectation values of equivalent cost as the overall energy, as either r boldi normall = ⟨ Ω i false( π / 4 false) | Û † Ĥ Û | Ω i false( π / 4 false) ⟩ − 1 2 false⟨ ϕ 0 false| Û † Ĥ Û | ϕ 0 ⟩ − 1 2 false⟨ ϕ boldi false| Û † Ĥ Û | ϕ i ⟩ or equivalently r boldi normall = 1 2 false[ false⟨ normalΩ boldi ( π / 4 ) false| Û † Ĥ Û | Ω i fals...…”

“…If the HF determinant is used, then Δ i = ϵ i + ϵ j + ... – ϵ a – ϵ b – ..., where ϵ i are Hartree–Fock spin-orbital energies. The residuals r i l can be computed using expectation values of equivalent cost as the overall energy, as either or equivalently where |Ω i (θ)⟩ = e θτ̂ i |ϕ 0 ⟩ = e θ( â i – â i † ) |ϕ 0 ⟩ = cos θ|ϕ 0 ⟩ ± sin θ|ϕ i ⟩. The cost of PQE residuals is therefore comparable to that of VQE gradients computed using the parameter-shift rule, which also employs additional energy measurements to obtain the gradient elements. − …”