# Ground-state energy of the [4Fe-4S] iron–sulfur cluster

Iron–sulfur clusters carry out electron transfer and redox chemistry across biology, and their dense low-lying electronic states, which cannot be observed directly, expose the limits of simple models [1]. The Quantum Advantage Tracker hosts a 36-orbital, 54-electron model of a [4Fe-4S] cluster, and the task is to find a wavefunction with the lowest energy, a score anyone can check. Published DMRG energies are still lower than every entry on the tracker.

## Task

The Hamiltonian is given by its integrals in the file `4fe_4s_54e_36o.fcidump` (36 spatial orbitals, 54 electrons, $M_S=0$):

$$H=\sum_{pq,\sigma}h_{pq}\,a^\dagger_{p\sigma}a_{q\sigma}+\frac12\sum_{pqrs,\sigma\tau}(pq|rs)\,a^\dagger_{p\sigma}a^\dagger_{r\tau}a_{s\tau}a_{q\sigma}.$$

Find a normalized state $|\psi\rangle$ with 27 spin-up and 27 spin-down electrons and report $E=\langle\psi|H|\psi\rangle$ in hartree ($E_h$). By the variational principle $E$ is an upper bound on the ground-state energy, so lower is better. The determinant space has about $8.86\times10^{15}$ elements, well beyond exact diagonalization.

## Background

Sharma et al. (2014) computed the low-energy spectra of [2Fe-2S] and [4Fe-4S] clusters with the density matrix renormalization group (DMRG) [1]. The active space used here, Fe 3d and S 3p orbitals from a localized density-functional calculation, follows Li and Chan (2017) [2]. IBM and RIKEN put the Hamiltonian on the tracker in November 2025 with energies from sample-based quantum diagonalization (SQD), a workflow that ran an IBM Heron processor together with the Fugaku supercomputer [3]. The tracker files it as a baseline benchmark, its category for reference instances on which quantum and classical methods are compared, including ones where the state of the art is classical.

## Data and evaluation

- Input: the FCIDUMP file in the tracker folder; its README describes the active space.
- Score: the energy in $E_h$. The tracker's variational pathway asks for upper bounds and for evidence that a method respects the variational principle, so extrapolated energies are not scores.
- Checking: a submitted wavefunction, such as a list of determinants with coefficients or a matrix product state, can be re-evaluated independently against the integrals.
- Baseline: TrimCI with $6.2\times10^7$ determinants, $-327.175\,E_h$ (tracker entry `TrimCI_202511_6e7`), the best energy on the tracker. It is not the best known: see the DMRG values below.

## What is known

Tracker entries, in $E_h$, all from November 2025:

- Restricted Hartree–Fock $-326.547$ and CISD $-326.742$ (IBM, RIKEN).
- SQD with circuit and orbital optimization on IBM Heron r2 and 76,032 Fugaku nodes, with 77 qubits and about 3,500 two-qubit gates: $-326.913$ (IBM, RIKEN; workflow in [3]).
- TrimCI (Zhang and Otten, University of Wisconsin–Madison) [4]: $-326.810$ with $10^3$ determinants; $-326.913$ with $1.3\times10^4$ determinants in 15 minutes on 20 cores; $-327.175$ with $6.2\times10^7$ determinants in 14 hours on 1,536 cores.

The literature has lower values than any tracker entry. The TrimCI paper quotes a DMRG energy of $-327.240\,E_h$ at bond dimension 4000 [4], and the SQD paper takes $-327.239\,E_h$ as its DMRG reference [3]. A DMRG energy at fixed bond dimension is itself variational, so the best known upper bound is about $0.065\,E_h$ below the tracker's best; no DMRG result has been submitted to the tracker. Extrapolations give about $-327.263\,E_h$ for TrimCI [4] and $-327.118\pm0.02\,E_h$ for SQD extrapolated to zero energy variance [3]; these are estimates, not upper bounds.

## What would count as progress

- A variational energy below the DMRG values near $-327.24\,E_h$, the lowest computed (not extrapolated) energies quoted above. This would be a new best for the model.
- A checkable wavefunction below $-327.175\,E_h$ submitted to the tracker, for example a reproduced DMRG calculation with its matrix product state, so that the tracker reflects the known state of the art.
- A quantum or hybrid method that reaches the classical numbers at lower cost; the tracker records runtime and hardware next to each energy.

Results go to the tracker through its variational-problems submission form.

## References

1. S. Sharma, K. Sivalingam, F. Neese and G. K.-L. Chan, "Low-energy spectrum of iron–sulfur clusters directly from many-particle quantum mechanics", Nature Chemistry 6, 927–933 (2014). [doi:10.1038/nchem.2041](https://doi.org/10.1038/nchem.2041)
2. Z. Li and G. K.-L. Chan, "Spin-projected matrix product states: versatile tool for strongly correlated systems", J. Chem. Theory Comput. 13, 2681–2695 (2017). [doi:10.1021/acs.jctc.7b00270](https://doi.org/10.1021/acs.jctc.7b00270)
3. T. Shirakawa et al., "Closed-loop calculations of electronic structure on a quantum processor and a classical supercomputer at full scale", preprint (2025). [arXiv:2511.00224](https://arxiv.org/abs/2511.00224)
4. H. Zhang and M. Otten, "From Random Determinants to the Ground State", preprint (2025). [arXiv:2511.14734](https://arxiv.org/abs/2511.14734)

## Source

Imported from [Quantum Advantage Tracker 4fe_4s_54e_36o](https://github.com/quantum-advantage-tracker/quantum-advantage-tracker.github.io/tree/main/data/variational-problems/hamiltonians/4fe_4s), curated by Quantum Advantage Tracker reviewers (IBM, Flatiron Institute, BlueQubit, Algorithmiq and others). Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.