• Materials

Oak Ridge: Quantum Simulation of Strongly Correlated Materials

Oak Ridge National Laboratory

ORNL's quantum computing group used VQE and Trotterized time evolution on IBM Quantum hardware to simulate the Hubbard model, a benchmark for strongly correlated materials like high-temperature superconductors, comparing results against classical tensor network methods.

Key Outcome
VQE reproduced ground state energies within 5% of exact diagonalization for the 4-site Hubbard model on noisy hardware. Trotter simulation showed qualitatively correct dynamics. Scalability analysis identified approximately 200 logical qubits as the threshold for classically intractable Hubbard simulations. Results published in Physical Review Research.

The Problem

Strongly correlated electron systems are among the most consequential unsolved problems in condensed matter physics. Materials like high-temperature cuprate superconductors, Mott insulators, and frustrated quantum magnets exhibit collective behavior that emerges from strong electron-electron interactions. Density functional theory (DFT), the workhorse of classical materials simulation, fails for these systems because it approximates electron correlation in ways that break down when interactions are strong.

Understanding the ground states and dynamics of these materials could unlock room-temperature superconductors, enabling lossless power transmission and transforming energy infrastructure. The catch: accurate classical simulation of strongly correlated systems scales exponentially with system size.

Oak Ridge National Laboratory’s quantum computing group, operating under the DOE National Quantum Information Science Research Centers (NQISRC), focuses on the Hubbard model as a tractable benchmark for this class of problems.

Hubbard Model Simulation with VQE

The Hubbard model captures the essential physics of strongly correlated materials with two parameters: hopping amplitude t (electron mobility) and on-site repulsion U (interaction strength). Exact diagonalization is limited to about 20 sites; DMRG handles quasi-1D systems but struggles in 2D.

from qiskit_nature.second_q.mappers import JordanWignerMapper
from qiskit_nature.second_q.hamiltonians import FermiHubbardModel
from qiskit_nature.second_q.hamiltonians.lattices import LineLattice
from qiskit.primitives import Estimator
from qiskit_algorithms import VQE
from qiskit_algorithms.optimizers import SLSQP
from qiskit.circuit.library import EfficientSU2
import numpy as np

# 4-site 1D Hubbard chain (8 spin-orbitals -> 8 qubits after mapping)
num_sites = 4
t = 1.0   # hopping amplitude
U = 4.0   # on-site Coulomb repulsion (U/t = 4 is strongly correlated regime)

lattice = LineLattice(num_nodes=num_sites, boundary_condition="open")
hubbard = FermiHubbardModel(lattice, onsite_interaction=U, hopping_parameter=-t)

hamiltonian = hubbard.second_q_op()
mapper = JordanWignerMapper()
qubit_op = mapper.map(hamiltonian)

print(f"Hamiltonian: {qubit_op.num_qubits} qubits, "
      f"{len(qubit_op)} Pauli terms")

# Hardware-efficient ansatz
ansatz = EfficientSU2(qubit_op.num_qubits, reps=3,
                      entanglement="linear")

# VQE with noise-aware setup
estimator = Estimator()
optimizer = SLSQP(maxiter=500)

vqe = VQE(estimator=estimator,
          ansatz=ansatz,
          optimizer=optimizer)

result = vqe.compute_minimum_eigenvalue(qubit_op)
ground_energy = result.eigenvalue.real

print(f"VQE ground state energy: {ground_energy:.4f}")
print(f"Number of VQE iterations: {result.cost_function_evals}")

Trotter Time Evolution

Beyond ground state energies, ORNL also implemented Trotterized real-time evolution to study quantum dynamics, which is relevant for understanding non-equilibrium phenomena in superconductors.

The first-order Trotter approximation decomposes the time evolution operator into a sequence of two-qubit gates. For small timesteps, this captures the correct dynamics. ORNL compared Trotter results for magnetization dynamics against exact classical propagation, finding qualitative agreement even on noisy hardware with error mitigation applied.

Results

ORNL benchmarked the quantum approach against three classical methods: exact diagonalization (ED), density matrix renormalization group (DMRG), and quantum Monte Carlo (QMC).

  • 4-site Hubbard (8 qubits): VQE ground energy within 5% of ED, using error mitigation (zero-noise extrapolation)
  • 6-site Hubbard (12 qubits): VQE energy within 12% of ED; additional noise from deeper circuit degraded accuracy
  • Trotter dynamics: Qualitatively correct spin correlation dynamics for 4-site chain; quantitative agreement required more Trotter steps than current hardware depth allows
  • Scaling analysis: Extrapolating circuit depth and qubit requirements, roughly 200 error-corrected logical qubits are needed to simulate a 2D Hubbard cluster beyond classical reach

The results were published in Physical Review Research and form part of the DOE’s quantum advantage roadmap for materials science.

What’s Next

ORNL’s roadmap targets three milestones for quantum materials simulation:

  1. Noise reduction: Demonstrating that error mitigation plus improved hardware can close the gap on 8-12 site Hubbard models
  2. Fermi-Hubbard in 2D: The 2D Hubbard model on a square lattice is believed to capture cuprate superconductor physics; classical methods struggle here and a 100-qubit demonstration is planned
  3. Real materials: Moving from model Hamiltonians to ab initio Hamiltonians for specific candidate materials (nickelate superconductors)

Learn more: Qiskit Reference