VQE for LiH Molecule Ground State

What You'll Learn:

• How to scale VQE from 2 qubits (H2) to 4 qubits (LiH)
• How hardware-efficient ansatze use layered RY-RZ-CX structures
• How Pauli string measurements work for multi-qubit Hamiltonians
• Why larger molecules are harder: barren plateaus, parameter space, and noise

Level: Intermediate | Time: 30 minutes | Qubits: 4 | Framework: Qiskit

Prerequisites

• H2 Ground State — VQE fundamentals on 2 qubits
• Bell State — CX entanglement basics

The Idea

Hydrogen (H2) was a warm-up: 2 qubits, 4 parameters, a tiny Hamiltonian you could diagonalize by hand. Now we scale up. Lithium hydride (LiH) has 4 electrons — 3 from lithium and 1 from hydrogen — requiring 4 qubits in the minimal basis. The Hamiltonian grows from 5 terms to dozens (we use a simplified 9-term version), and the ansatz needs 16 parameters across 2 layers.

This is where VQE starts to earn its keep. At 4 qubits, classical diagonalization is still easy (16x16 matrix). But the same approach applied to larger molecules (caffeine: ~160 electrons) would require matrices far too large for any classical computer. VQE's hybrid quantum-classical loop scales more gracefully.

The catch: more parameters mean a larger optimization landscape, more circuit depth, and increased sensitivity to noise. Welcome to the real challenges of quantum chemistry.

How It Works

The Ansatz: 2-Layer Hardware-Efficient

Each layer has: (1) RY+RZ rotations on every qubit, then (2) a linear CX chain.

CODECopy
Layer 1                                  Layer 2
┌────────┐┌────────┐                     ┌─────────┐┌─────────┐
q0: ┤ RY(θ₀)├┤ RZ(θ₁)├──■──────────     ┤ RY(θ₈) ├┤ RZ(θ₉) ├──■──
    ├────────┤├────────┤┌─┴─┐             ├─────────┤├─────────┤┌─┴─┐
q1: ┤ RY(θ₂)├┤ RZ(θ₃)├┤ X ├──■──       ┤ RY(θ₁₀)├┤ RZ(θ₁₁)├┤ X ├──■──
    ├────────┤├────────┤└───┘┌─┴─┐        ├─────────┤├─────────┤└───┘┌─┴─┐
q2: ┤ RY(θ₄)├┤ RZ(θ₅)├─────┤ X ├──■──  ┤ RY(θ₁₂)├┤ RZ(θ₁₃)├─────┤ X ├──■──
    ├────────┤├────────┤     └───┘┌─┴─┐   ├─────────┤├─────────┤     └───┘┌─┴─┐
q3: ┤ RY(θ₆)├┤ RZ(θ₇)├──────────┤ X ├  ┤ RY(θ₁₄)├┤ RZ(θ₁₅)├──────────┤ X ├
    └────────┘└────────┘          └───┘   └─────────┘└─────────┘          └───┘

• RY: Controls the polar angle on the Bloch sphere (amplitude mixing)
• RZ: Controls the azimuthal angle (phase control)
• CX chain: Creates nearest-neighbor entanglement (0→1→2→3)
• 16 parameters: 4 qubits × 2 rotations × 2 layers

The linear CX connectivity matches most superconducting hardware (IBM, Google), making this ansatz directly executable without transpilation overhead.

The Hamiltonian: 9 Dominant Pauli Terms

The full LiH Hamiltonian in STO-3G has 100+ Pauli terms. We use a simplified version with the 9 largest contributions:

Measuring Multi-Qubit Pauli Strings

Each 4-character Pauli string requires its own measurement circuit. Before measuring, rotate each qubit into the appropriate basis:

• I or Z: No rotation (computational basis)
• X: Apply H (Hadamard)
• Y: Apply S†H

For example, to measure XXII, apply H on qubits 0 and 1 (the X positions), leave qubits 2 and 3 unchanged, then measure all in the computational basis.

PYTHONCopy
from circuit import run_circuit, optimize_vqe

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

# Full optimization (takes longer — 16 parameters)
opt = optimize_vqe(max_iterations=200, shots=2048, seed=42)
print(f"Optimized: {opt['optimal_energy']:.4f} Ha")

The Math

Scaling from H2 to LiH

The parameter count grows as O(qubits × layers), and measurement circuits grow with the number of unique Pauli bases. This is the fundamental scaling challenge of VQE.

Barren Plateaus

With 16 parameters and 2 layers, the optimization landscape becomes flatter:

∂E/∂θ_i → 0  as  circuit depth → ∞

The gradient of the cost function vanishes exponentially with circuit depth, making optimization harder. This is the barren plateau problem. Strategies:

• Use few layers (we use 2)
• Initialize near identity (small angles)
• Use problem-informed ansatze instead of hardware-efficient ones

Expected Output

With a simplified 9-term Hamiltonian, VQE may not reach chemical accuracy. The full 100+ term Hamiltonian is needed for production-quality results.

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})")

# Full optimization
opt = optimize_vqe(max_iterations=200, shots=2048, seed=42)
print(f"Optimized: {opt['optimal_energy']:.4f} Ha")
print(f"Chemical accuracy: {opt['chemical_accuracy']}")

# 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. Add a 3rd layer: Change layers=3 and use 24 parameters. Does the extra depth improve the energy, or does it hit a barren plateau?

2. Reduce to 1 layer: Use only 8 parameters. How much accuracy is lost? This quantifies the expressibility vs. trainability trade-off.

3. Try full connectivity: Replace the linear CX chain with all-to-all CX (0→1, 0→2, 0→3, 1→2, 1→3, 2→3). Does this improve convergence?

4. Compare optimizers: Try method="Nelder-Mead" or method="Powell". Which handles the 16-dimensional landscape best?

5. Plot the potential energy surface: Run VQE at Li-H bond lengths from 1.0 to 3.0 A (adjust coefficients). Does VQE reproduce the Morse potential shape?

What's Next

• H2O Ground State — 4 qubits with active space reduction for water
• BeH2 Ground State — 6 qubits, the largest in the VQE progression
• QAOA MaxCut — Variational algorithms for optimization (not chemistry)

Applications

References

• Kandala, A. et al. (2017). "Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets." Nature 549, 242-246. DOI: 10.1038/nature23879
• Peruzzo, A. et al. (2014). "A variational eigenvalue solver on a photonic quantum processor." Nature Communications 5, 4213. DOI: 10.1038/ncomms5213
• McClean, J.R. et al. (2018). "Barren plateaus in quantum neural network training landscapes." Nature Communications 9, 4812. DOI: 10.1038/s41467-018-07090-4