Quantum-classical hybrid algorithm using an error-mitigating $N$-representability condition to compute the Mott metal-insulator transition

Quantum algorithms for molecular electronic structure have been developed with lower computational scaling than their classical counterparts, but emerging quantum hardware is far from being capable of the coherence, connectivity, and gate errors required for their experimental realization. Here we propose a class of quantum-classical hybrid algorithms that computes the energy from a two-electron reduced density matrix (2-RDM). The 2-RDM is constrained by $N$-representability conditions, constraints for representing an $N$-electron wave function, which mitigate noise from the quantum circuit. We compute the strongly correlated dissociation of doublet ${\mathrm{H}}_3$ into three hydrogen atoms. The hybrid quantum-classical computer matches the energies from full configuration interaction to 0.1 kcal/mol, one-tenth of “chemical accuracy,” even in the strongly correlated limit of dissociation. Furthermore, the spatial locality of the computed one-electron RDM reveals that the quantum computer accurately predicts the Mott metal-insulator transition.

Figure 1

Dissociation curve for the doublet ${\mathrm{H}}_3$ with respect to the bond distance from the center H to the two exterior H atoms in a linear geometry. The crosses were calculated with a variational quantum algorithm on the quantum computer, while the line was generated with a full configuration interaction (FCI) calculation on a classical computer. Energies are listed in hartree. The inset plot depicts the error from the full-CI method as a function of the separated distance, reported in millihartree. The dashed line is at 1.6 millihartree, which corresponds to 1.0 kcal/mol, a number that is generally used as a guide for chemical accuracy. The effect of errors is discussed in the text and in Appendix pp1. Variability from a single run is on the order of 2–0.2 millihartree, and in our optimization we purposefully over sample the target region.

Figure 2

Shows the sum of squares of the off-diagonal elements, $\tau$, of the 1-RDM of ${\mathrm{H}}_3$ in the local Löwdin atomic orbital basis along the dissociation curve of ${\mathrm{H}}_3$. Here, $\tau ={\sum }_{i\ne j}{\parallel {}^1{D}_j^i\parallel }^2$ where $i,j$ are orbital indices in the Löwdin atomic orbital basis. The Hartree-Fock result is shown as a dashed-dotted line, the FCI result is solid, and our variational quantum computation is shown as crosses. The bottom dashed line shows the dissociated limit where $\tau =0$, and the natural orbitals approach the atomic ones. That ${\mathrm{H}}_3$ serves as a Mott insulator can be seen between these distances, as $\tau \to 0$ with increasing distance, highlighting the mean field and 2-electron approaches.

Figure 3

Energy errors of the (a) first and (b) second macro-iterations of the Nelder-Mead simplex optimization of ${\mathrm{H}}_3$ are shown as a function of the number of energy evaluations. The variational design of the algorithm allows for lower errors in the final energy than present in the sampling. The energy errors in millihartree are measured relative to the energies from full configuration interaction. The blue dashed line shows the energy of the best simplex point along the optimization, while the green dotted line shows the “chemical accuracy” threshold of 1.6 millihartree. Error bars correspond to a 90% confidence interval based largely on sampling errors in the quantum computer. The edge hydrogens in ${\mathrm{H}}_3$ have a separation of 1.34 Å.