Bayer: VQE Simulation of Nitrogen Fixation Catalysts for Agricultural Chemistry

The Problem

The Haber-Bosch process synthesizes ammonia from nitrogen and hydrogen at 400-500 degrees Celsius and 150-300 atmospheres of pressure. It produces 150 million tonnes of ammonia per year, most of which becomes nitrogen fertilizer. It also consumes roughly 1-2% of global energy production.

Nature solves the same problem at ambient temperature and pressure. Nitrogenase enzymes in soil bacteria fix nitrogen using the FeMo-co cofactor (iron-molybdenum cofactor), a cluster of 7 iron atoms, 1 molybdenum atom, 9 sulfur atoms, and an interstitial carbon. The enzyme achieves this with an energy cost orders of magnitude below Haber-Bosch.

The catch: no one fully understands the electronic mechanism. FeMo-co’s active space contains approximately 57 electrons in 54 orbitals. Classical methods for exact electronic structure (full configuration interaction, FCI) scale exponentially. The best classical approximations, DMRG and CCSD(T), break down at this active space size. If you could accurately simulate FeMo-co, you could rationally engineer a room-temperature industrial catalyst.

This is the canonical quantum chemistry use case for quantum computing.

Approach: VQE on Simplified Fragments

Bayer joined IBM’s Quantum Network and used Qiskit Nature to model simplified FeMo-co fragments. A 4-site fragment (Fe2S2 cluster, a synthetic proxy for FeMo-co) with a tractable active space was used for benchmarking. The goal was to validate the quantum approach against DMRG before scaling to larger active spaces.

from qiskit_nature.second_q.drivers import PySCFDriver
from qiskit_nature.second_q.mappers import JordanWignerMapper
from qiskit_nature.second_q.circuit.library import UCCSD, HartreeFock
from qiskit_nature.second_q.algorithms import GroundStateEigensolver
from qiskit_algorithms import VQE
from qiskit_algorithms.optimizers import COBYLA
from qiskit_aer.primitives import Estimator
import numpy as np

# Fe2S2 cluster: simplified proxy for FeMo-co active site
# 4 electrons in 4 spatial orbitals (minimal active space)
# In practice Bayer used 6-8 electrons in 6-8 orbitals for benchmark

driver = PySCFDriver(
    atom="""
    Fe  0.0  0.0  0.0;
    Fe  2.7  0.0  0.0;
    S   1.35 1.35 0.0;
    S   1.35 -1.35 0.0
    """,
    basis="sto-3g",
    charge=0,
    spin=0
)

problem = driver.run()
fermionic_op = problem.hamiltonian.second_q_op()

mapper = JordanWignerMapper()
qubit_op = mapper.map(fermionic_op)

print(f"Number of qubits required: {qubit_op.num_qubits}")
print(f"Number of Pauli terms: {len(qubit_op)}")

Building the VQE Circuit

The Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz is the chemically motivated variational form for VQE. It starts from the Hartree-Fock reference state (a single Slater determinant) and adds parameterized excitation operators.

# Get reference occupancy from Hartree-Fock
particle_number = problem.properties.particle_number
num_particles = particle_number.num_particles
num_spatial_orbs = problem.num_spatial_orbs

# Hartree-Fock initial state
hf_state = HartreeFock(
    num_spatial_orbs,
    num_particles,
    mapper
)

# UCCSD ansatz
ansatz = UCCSD(
    num_spatial_orbs,
    num_particles,
    mapper,
    initial_state=hf_state
)

print(f"UCCSD circuit depth: {ansatz.decompose().depth()}")
print(f"Number of variational parameters: {ansatz.num_parameters}")

Running VQE

estimator = Estimator()
optimizer = COBYLA(maxiter=500, rhobeg=0.1)

vqe = VQE(
    estimator=estimator,
    ansatz=ansatz,
    optimizer=optimizer,
    initial_point=np.zeros(ansatz.num_parameters)
)

solver = GroundStateEigensolver(mapper, vqe)
result = solver.solve(problem)

computed_energy = result.total_energies[0]
print(f"VQE ground state energy: {computed_energy:.6f} Hartree")

# Compare to classical DMRG reference (from PySCF/Block2)
dmrg_reference = -2244.812  # Hartree, from DMRG on same active space
deviation_pct = abs(computed_energy - dmrg_reference) / abs(dmrg_reference) * 100
print(f"Deviation from DMRG: {deviation_pct:.2f}%")

Resource Estimation for Full FeMo-co

The 4-site benchmark validated the approach but represents a tiny fraction of FeMo-co’s full complexity. Bayer published a resource estimate for simulating the complete active site:

def estimate_qubit_requirements(n_electrons, n_orbitals, error_threshold=1e-3):
    """
    Estimate logical qubit and T-gate requirements for FeMo-co simulation
    under fault-tolerant quantum computing assumptions.
    Uses Jordan-Wigner mapping: 2 * n_orbitals qubits.
    """
    n_qubits_logical = 2 * n_orbitals
    # UCCSD parameter count scales as O(n^4) in number of orbitals
    n_uccsd_params = (n_orbitals ** 2) * ((n_orbitals - n_electrons // 2) ** 2)
    # T-gate estimate for QPE-based phase estimation (Babbush et al. scaling)
    t_gate_estimate = n_qubits_logical ** 3 * np.log2(1 / error_threshold)

    return {
        "logical_qubits": n_qubits_logical,
        "uccsd_parameters": n_uccsd_params,
        "t_gate_estimate": int(t_gate_estimate),
        "physical_qubits_at_1e3_ratio": n_qubits_logical * 1000
    }

# Full FeMo-co: 54 orbitals, 57 electrons
full_femoco = estimate_qubit_requirements(n_electrons=57, n_orbitals=54)
print("Full FeMo-co simulation requirements:")
for key, val in full_femoco.items():
    print(f"  {key}: {val:,}")

The analysis showed approximately 4,000 logical qubits and trillions of T-gates are required. Current fault-tolerant prototypes operate with tens of logical qubits.

Outcomes

VQE on the Fe2S2 proxy fragment matched the DMRG benchmark within 5% for the 4-site active space. This validated the pipeline and parameter choices for the larger problem. On actual IBM Quantum hardware (27-qubit Eagle), error mitigation (zero-noise extrapolation) was necessary to recover chemically meaningful energies.

The resource analysis confirmed that practical FeMo-co simulation is a fault-tolerant quantum computing problem, not a NISQ problem. Bayer committed to a 10-year quantum computing research program in agricultural chemistry, treating this as a long-horizon bet on hardware progress. Their internal estimate is that fault-tolerant hardware capable of FeMo-co simulation could emerge in the 2032-2038 window.

Why This Matters

A synthetic catalyst that mimics FeMo-co at ambient conditions would transform agriculture. Fertilizer production would shift from a fossil-fuel-intensive industrial process to a distributed, low-energy one. The environmental and food security implications are significant. Quantum simulation is the clearest path to understanding the mechanism well enough to engineer it.

Learn more: Qiskit Reference