VQE Solver — Variational Quantum Eigensolver

What is VQE?

VQE (Peruzzo et al. 2014) minimizes the expectation value ⟨ψ(θ)|H|ψ(θ)⟩ of a Hamiltonian H with respect to variational circuit parameters θ. By the variational principle, this expectation value is an upper bound on the ground state energy — the optimization finds the tightest bound achievable by the chosen ansatz circuit.

VQE is primarily used for quantum chemistry (molecular electronic structure) and condensed matter physics (Ising models, Heisenberg models). It is not designed for combinatorial optimization — use QAOA or GPU Annealing for QUBO problems.

Usage

python

import nerox

client = nerox.Client()

# Hydrogen molecule ground state (Jordan-Wigner mapped to 4 qubits)
H2_hamiltonian = {
    "terms": [
        {"pauli": "II", "coeff": -1.0523732},
        {"pauli": "IZ", "coeff":  0.3979374},
        {"pauli": "ZI", "coeff": -0.3979374},
        {"pauli": "ZZ", "coeff": -0.0112801},
        {"pauli": "XX", "coeff":  0.1809312},
    ],
    "n_qubits": 4,
}

job = client.optimize.vqe(
    hamiltonian=H2_hamiltonian,
    ansatz="uccsd",             # uccsd | hardware_efficient | hva
    optimizer="l_bfgs_b",       # l_bfgs_b | adam | cobyla
    max_iter=500,
)

result = job.wait(timeout=7200)
print(f"Ground state energy: {result.objective:.6f} Hartree")
print(f"Circuit depth: {result.circuit_depth}")
print(f"Gate count: {result.n_gates}")

Supported ansätze

UCCSD

Unitary Coupled Cluster Singles and Doubles. Chemically motivated, good accuracy on small molecules, expensive in gate count.

Hardware Efficient

Alternating layers of parameterized single-qubit rotations and CNOT entanglers. Low gate count, suitable for noise-resilient simulation.

HVA

Hamiltonian Variational Ansatz. Uses the problem Hamiltonian structure directly. Well-suited for physics models (Heisenberg, Hubbard).

Limitations

VQE is limited to ~50 qubits (GPU statevector simulation). Runtime is 10 minutes to several hours depending on Hamiltonian size, ansatz depth, and optimizer convergence. It is research-grade, not production-grade. For production combinatorial optimization, use GPU Annealing.