Table 1 Basis states of the VQE optimized statevector of 4 NEB images along the reaction path, in which the CASSCF optimizations were constrained to produce the \(\langle S^2 \rangle = 0\) singlet. The orbital basis is arranged such that all basis states have the form \(|\dots , (n^{\uparrow } \ n^{\downarrow })_i, (n^{\uparrow } \ n^{\downarrow })_{i+1}, \dots \rangle\), where \(i\) labels the molecular orbitals (shown in Fig. 1 for image 1), and \(n^{\sigma }\) is the \(\sigma\) spin orbital occupation which corresponds to a qubit state (with qubit index \(2i\) (\(2i+1\)) for \(\sigma =\ \uparrow \ (\downarrow )\)) after JW transformation. The basis states were initially chosen from the largest weight Slater determinants in CASSCF and their coefficients were then optimized in VQE to produce (ideal, noiseless) energies that match the CASSCF energies to less than 10\(^{-6}\) Hartrees. State circuits prepared using an adaption of a methodology involving controlled Givens rotations³³, and circuits compiled to the Quantinuum H1-1 device. Rightmost columns report the number of one-qubit PhasedX and \(R_Z\) gates, and two-qubit ZZPhase gates (see Appendix C of the Supplementary Information for more details).

From: Measuring correlation and entanglement between molecular orbitals on a trapped-ion quantum computer