VQE for HeH+ Molecule Ground State

What You'll Learn:

• How VQE handles heteronuclear molecules with asymmetric Hamiltonians
• How UCCSD-inspired ansatze differ from hardware-efficient ones
• Why HeH+ has both XX and YY Pauli terms (unlike parity-reduced H2)
• The cosmological significance of the first molecule in the universe

Level: Intermediate | Time: 20 minutes | Qubits: 2 | Framework: Qiskit

Prerequisites

• H2 Ground State — VQE fundamentals, Pauli measurements
• Bell State — CX entanglement basics

The Idea

Imagine the universe 100,000 years after the Big Bang. The cosmic plasma is cooling. Helium nuclei (Z=2) and protons (Z=1) are the most abundant species. As the temperature drops below ~4000 K, a helium atom can capture a proton to form HeH+ — the first molecular bond in the history of the universe.

HeH+ is a 2-electron system like H2, but with a twist: the nuclear charges are asymmetric (He has charge +2, H has charge +1). This breaks the homonuclear symmetry that simplified H2's Hamiltonian. The electrons are pulled more toward helium, creating a polar ionic bond instead of the symmetric covalent bond of H2.

For VQE, this asymmetry means:

• The Z₀ and Z₁ coefficients have different magnitudes (not just opposite signs)
• Both XX and YY Pauli terms appear in the Hamiltonian
• The optimization landscape is less symmetric, requiring a more flexible ansatz

How It Works

The UCCSD-Inspired Ansatz

Instead of the hardware-efficient RY-CX-RY structure used for H2, HeH+ uses a UCCSD-inspired ansatz with a CX-RY-CX sandwich:

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

The CX-RY(θ₂)-CX pattern implements a controlled rotation: when qubit 0 is |1⟩, qubit 1 gets an extra RY rotation. This is a building block of the Unitary Coupled Cluster with Singles and Doubles (UCCSD) ansatz, which is chemically motivated rather than purely hardware-motivated.

With only 3 parameters, this compact ansatz can reach the HeH+ ground state.

The Hamiltonian: 6 Pauli Terms

Key difference from H2: HeH+ keeps the Y₀Y₁ term because the heteronuclear symmetry doesn't allow it to be eliminated by parity reduction.

Step-by-Step: Measuring the Energy

Three measurement circuits (one more than H2):

1. ZZ basis (computational): Gives ⟨Z₀⟩, ⟨Z₁⟩, ⟨Z₀Z₁⟩
2. XX basis (H on both qubits): Gives ⟨X₀X₁⟩
3. YY basis (S†H on both qubits): Gives ⟨Y₀Y₁⟩

PYTHONCopy
from circuit import run_circuit, optimize_vqe

result = run_circuit()
print(f"Energy: {result['energy']:.4f} Ha")

opt = optimize_vqe(max_iterations=100, shots=2048, seed=42)
print(f"Optimized: {opt['optimal_energy']:.4f} Ha")

The Math

Homonuclear vs. Heteronuclear

Asymmetric Hamiltonian

The Z₀ and Z₁ coefficients have equal magnitude but opposite sign (+0.5449 vs -0.5449) in this simplified model. In the full Hamiltonian, they would differ in magnitude too. The sign asymmetry already captures the key physics: the bonding and antibonding orbitals have different energies due to the unequal nuclear charges.

Expected Output

Running the Circuit

PYTHONCopy
from circuit import run_circuit, optimize_vqe, verify_vqe

# Quick evaluation
result = run_circuit()
print(f"Energy: {result['energy']:.4f} Ha (error: {result['error']:.4f})")

# Optimization
opt = optimize_vqe(max_iterations=100, shots=2048, seed=42)
print(f"Optimized: {opt['optimal_energy']:.4f} Ha")

# Verification
v = verify_vqe()
for check in v["checks"]:
    status = "PASS" if check["passed"] else "FAIL"
    print(f"[{status}] {check['name']}: {check['detail']}")

Try It Yourself

1. Compare with H2: Run both H2 and HeH+ VQE circuits. Notice HeH+ needs 3 measurement circuits (ZZ, XX, YY) while H2 needs only 2. Why?

2. Swap to hardware-efficient ansatz: Replace the UCCSD ansatz with RY-CX-RY (like H2). Does it still converge? Is it faster or slower?

3. Stretch the bond: Modify the Z coefficients to simulate a stretched bond (increase the asymmetry). How does the energy change?

4. Remove the YY term: Set Y₀Y₁ = 0 and re-optimize. How much does the ground state energy change? This quantifies the YY contribution.

Cosmological Significance

HeH+ was the first molecule to form in the universe:

• ~100,000 years after Big Bang: Temperature drops below ~4000 K
• Formation: He + H⁺ → HeH⁺ (radiative association)
• Role: Initiated molecular chemistry in the primordial gas
• 2019: First astrophysical detection in planetary nebula NGC 7027 by SOFIA telescope

The detection was published in Nature (Gusten et al., 2019) and confirmed a 94-year-old prediction from quantum mechanics.

What's Next

• LiH Ground State — Scale to 4 qubits with lithium hydride
• H2 Ground State — Review the simpler 2-qubit case
• QAOA MaxCut — Another variational algorithm for optimization

References

• Peruzzo, A. et al. (2014). "A variational eigenvalue solver on a photonic quantum processor." Nature Communications 5, 4213. DOI: 10.1038/ncomms5213
• Gusten, R. et al. (2019). "Astrophysical detection of the helium hydride ion HeH+." Nature 568, 357-359. DOI: 10.1038/s41586-019-1090-x
• Hempel, C. et al. (2018). "Quantum chemistry calculations on a trapped-ion quantum simulator." Physical Review X 8, 031022. DOI: 10.1103/PhysRevX.8.031022