VQE H2 Ground State - Peruzzo et al. 2014

Overview

Reproduction of the original VQE paper - the first experimental demonstration of the Variational Quantum Eigensolver on a photonic quantum processor.

Property	Value
Category	Research
Difficulty	Advanced
Framework	Qiskit
Qubits	2
Paper	Nature Communications 5, 4213 (2014)

The Original Experiment

Peruzzo et al. demonstrated VQE on a photonic chip to find the ground state energy of molecular hydrogen (H2). This was the first practical quantum chemistry calculation on quantum hardware.

Key Innovation

Classical-quantum hybrid optimization
Shallow circuit amenable to NISQ devices
Proof-of-concept for quantum chemistry

The H2 Hamiltonian

In the minimal (STO-3G) basis, H2 maps to 2 qubits:

CODE
H = g₀I + g₁Z₀ + g₂Z₁ + g₃Z₀Z₁ + g₄(X₀X₁ + Y₀Y₁)

At equilibrium (R = 0.74 Å):

g₀ = -1.0523 (constant)
g₁ = 0.3979 (Z on qubit 0)
g₂ = -0.3979 (Z on qubit 1)
g₃ = -0.0112 (ZZ correlation)
g₄ = 0.1809 (XX + YY exchange)

Circuit Ansatz

The minimal UCCSD ansatz for H2:

CODE
┌───┐
q_0: ┤ X ├──────────●───
     └───┘┌───────┐┌─┴─┐
q_1: ─────┤ RY(θ) ├┤ X ├
          └───────┘└───┘

Only one parameter θ is needed for H2 at any bond length!

Running the Circuit

PYTHON
from circuit import run_circuit

# Reproduce dissociation curve (Figure 3 from paper)
result = run_circuit(
    bond_lengths=[0.3, 0.5, 0.74, 1.0, 1.5, 2.0, 2.5, 3.0],
    n_steps=30
)

for r, data in result['bond_length_scan'].items():
    print(f"R = {r:.2f} Å: E = {data['energy']:.4f} Ha")

print(f"Chemical accuracy: {result['chemical_accuracy']}")

Expected Results

Dissociation Curve

Bond Length (Å)	VQE Energy (Ha)	Exact (Ha)
0.30	~-0.2	-0.23
0.50	~-0.9	-0.95
0.74	~-1.13	-1.137
1.00	~-1.07	-1.101
1.50	~-0.98	-1.015
2.00	~-0.95	-0.978

Key Metrics

Equilibrium energy: -1.1373 Ha (exact)
Chemical accuracy: < 1.6 mHa error
Optimal θ: ~3.14 radians at equilibrium

Historical Significance

This paper:

Introduced VQE as a practical algorithm
First quantum chemistry on real quantum hardware
Hybrid approach for NISQ devices
Sparked the field of variational quantum algorithms

Paper Citation

BIBTEX
@article{peruzzo2014variational,
  title={A variational eigenvalue solver on a photonic quantum processor},
  author={Peruzzo, Alberto and McClean, Jarrod and others},
  journal={Nature Communications},
  volume={5},
  pages={4213},
  year={2014}
}

VQE H2 (Peruzzo 2014)