Spin-state energetics of heme-related models with the variational quantum eigensolver

TL;DR

This work investigates spin-state energetics of a simplified deoxy-myoglobin model Fe(CH3N2)2-(OH2) using the variational quantum eigensolver (VQE). It combines state-averaged orbital-optimized VQE with unitary CC-based ansätze (notably $k$-UpCCGSD, $k=4$) and AVAS-constructed active spaces up to 10 orbitals, benchmarked against classical CASSCF and CCSD(T) references. The study finds that VQE can reproduce spin-state energetics with chemical accuracy for smaller active spaces and shows convergence toward CCSD(T) as the active space enlarges; the triplet state benefits from a multi-reference initial state, while the quintet remains near single-reference. These results provide a practical workflow and benchmarks for applying VQE to strongly correlated transition-metal systems, highlighting both the promise and the current hardware- and algorithmic- limitations that guide future hardware-aware developments.

Abstract

We present numerical calculations of the energetic separation between different spin states (singlet, triplet and quintet) for a simplified model of a deoxy-myoglobin protein using the variational quantum eigensolver (VQE) algorithm. The goal is to gain insight into the workflow and challenges of VQE simulations for transition metal complexes, with emphasis on methodology over hardware-specific implementation. The numerical calculations are performed using an in-house statevector simulator with single- and multi-reference trial wavefunctions based on the k-unitary pair coupled-cluster generalized singles and doubles or k-UpCCGSD ansatz. The spin-state energetics for active spaces of increasing size up to 10 spatial orbitals (20 spin orbitals or qubits) are computed with VQE and were found to agree with the classical complete active self-consistent field or CASSCF method to within 1-4 kcal/mol. We evaluate relevant multi-reference diagnostics and show that the spin states computed with VQE possess a sufficient degree of multi-reference character to highlight the presence of strong electron correlation effects. Our numerical simulations show that in the ideal case, the VQE algorithm is capable of reproducing spin-state energetics of strongly correlated systems such as transition metal complexes for both single- and multi-reference trial wavefunctions, asymptotically achieving good agreement with results from classical methods as the number of active orbitals increases.

Figures (10)

• Figure 1: Molecular structures of (a) Oxy-myoglobin in blackfin tuna (PDB https://www.rcsb.org/structure/3qm5), (b)FeP(Im), (c)Fe(C3H5N2)2(OH2) and (d)Fe(CH3N2)2-(OH2), which is the transition metal complex studied in this work. Here, (b) is the most realistic model of the active site of deoxy-myoglobin (oxy-myoglobin without dioxygen) where the imidazole group from the side chain histidine is bound to the heme group, while (c) and (d) are model systems of (b), with (d) being the most similar to (b). Different colors correspond to different atoms. Here, red corresponds to oxygen (O), blue to nitrogen (N), gray to carbon (C), and orange to iron (Fe). The ribbons in (a) represent the surrounding polymers of the protein. The visualizations were made with PyMol and Jmol. See Appendix \ref{['sec:molecular_geometries']} for more details about the structures.
• Figure 2: Spin-state energetics of Fe(CH3N2)2-(OH2) as a function of the number of active space orbitals for the (a) T0 and (d) T1 initial states (dashed lines show reference CCSD(T) values from Ref. Strickland2007). Energy differences between the VQE and CASSCF spin-state energetics of Fe(CH3N2)2-(OH2) as a function of the number of active space orbitals for the (b) T0 and (e) T1 initial states. Energy differences between VQE and CASSCF for the individual spin states of Fe(CH3N2)2-(OH2) as a function of the number of active space orbitals for the (c) T0 and (f) T1 initial states. See Appendix \ref{['sec:energy_traces']} for energy traces during the optimization and Appendix \ref{['sec:active_space_total_energies']} for the absolute/total energies for each active space.
• Figure B.1: The (8e,10o) initial active space for Fe(CH3N2)2-(OH2) as computed with AVAS. This active space is based on the ROHF wave function of the quintet ground state Fe(CH3N2)2-(OH2) and is generated with a threshold value of 0.70. From top to bottom, the active space consists of 4 unoccupied orbitals, 4 singly occupied orbitals, and 2 doubly occupied orbitals. Unlike the doubly and unoccupied orbitals, when the open-shell option is set to 3, the singly occupied orbitals are included regardless of the threshold value. The unoccupied orbitals are predominantly of Fe $4d$ character; 72%, 81%, 89% and 89%, respectively. The first doubly occupied orbital has 13% Fe $3d$, 25% Fe $4d$ and 0% O $2p_{z}$, while the second one has 99 % Fe $3d$, 36 % Fe 4d and 95% O $2p_{z}$. The surfaces depict isosurfaces at an isovalue of 0.025, with positive (red) and negative (blue) phases of the wavefunction. These isosurfaces were rendered using Jmol.
• Figure C.1: Schematic representation of the (a) T0 and (b) T1 initial states for the singlet $(S = 0)$, triplet $(S = 1)$ and quintet $(S = 2)$ spin states, respectively. Spatial orbitals, represented by the horizontal bars, are doubly occupied with one $\alpha$-electron (blue) and one $\beta$-electron (red) as far as possible, then the orbitals are singly occupied by $\alpha$-electrons (blue).
• Figure D.1: Energy traces for the singlet $(S = 0)$, triplet $(S = 1)$ and quintet $(S = 2)$ spin states for the T0 initial states, respectively. The plots show relative energies (with respective to CASSCF energies) during VQE optimization for the active spaces (a) (6e,5o), (b) (8e,6o), (c) (8e,7o), (d) (8e,8o), (e) (8e,9o) and (f) (8e,10o).