VQE for H2 Molecule Ground State

Overview

The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm for finding the ground state energy of molecules. This circuit implements VQE for the hydrogen molecule (H₂), the simplest molecular system.

The Algorithm

VQE uses a parameterized quantum circuit (ansatz) to prepare trial states, measures the energy, and uses classical optimization to minimize it.

CODE
┌─────────────────────────────────────────────────┐
│                   VQE Loop                       │
│                                                  │
│  1. Prepare |ψ(θ)⟩ with quantum circuit         │
│  2. Measure ⟨ψ(θ)|H|ψ(θ)⟩                       │
│  3. Classical optimizer updates θ               │
│  4. Repeat until converged                      │
│                                                  │
└─────────────────────────────────────────────────┘

The Ansatz

Hardware-efficient ansatz for 2 qubits:

CODE
┌──────────┐     ┌──────────┐
q_0: ┤ RY(θ₀)  ├──■──┤ RY(θ₂)  ├
     ├──────────┤┌─┴─┐├──────────┤
q_1: ┤ RY(θ₁)  ├┤ X ├┤ RY(θ₃)  ├
     └──────────┘└───┘└──────────┘

The H2 Hamiltonian

In the STO-3G basis, the H₂ Hamiltonian can be written as:

CODE
H = c₀I + c₁Z₀ + c₂Z₁ + c₃Z₀Z₁ + c₄(X₀X₁ + Y₀Y₁)

At equilibrium bond length (~0.735 Å), the exact ground state energy is approximately -1.137 Hartree.

Running the Circuit

PYTHON
from circuit import run_circuit, optimize_vqe

# Single evaluation with default parameters
result = run_circuit()
print(f"Energy: {result['energy']:.4f} Ha")

# Run full VQE optimization
opt_result = optimize_vqe(max_iterations=50)
print(f"Optimized energy: {opt_result['optimal_energy']:.4f} Ha")

Expected Output

Metric	Value
Exact ground state	-1.137 Ha
VQE estimate (optimized)	-1.13 to -1.14 Ha
Chemical accuracy	±1.6 mHa (0.001 Ha)

Key Concepts

Variational Principle

The variational principle guarantees that any trial wavefunction gives an energy ≥ the true ground state energy:

⟨ψ|H|ψ⟩ ≥ E₀

Measurement

To compute ⟨H⟩, we measure each Pauli term separately:

ZZ basis: Standard measurement
XX basis: Apply H gates before measurement
YY basis: Apply S†H gates before measurement

Applications

Quantum Chemistry: Molecular ground state energies
Drug Discovery: Protein folding, molecular interactions
Materials Science: Electronic structure calculations

Limitations

Barren Plateaus: Gradients vanish for deep circuits
Noise: Real hardware introduces errors
Local Minima: Classical optimizer may get stuck