VQE for BeH2 Molecule Ground State

Overview

Beryllium hydride (BeH₂) is a linear 6-electron molecule that provides a step up in complexity from smaller molecules like H₂ and LiH. It's a common benchmark for testing VQE scalability.

The Molecule

BeH₂ structure:

CODE
H—Be—H

Properties:

Linear geometry
6 electrons (Be: 4, H: 1 each)
Symmetric stretching modes
Bond length: ~1.33 Å

The Ansatz

Hardware-efficient ansatz with 6 qubits:

CODE
┌────────┐     ┌────────┐     ┌────────┐     ┌────────┐
q_0: ┤ RY(θ₀)├──■──┤ RZ(θ₆)├─────┤ RY(θ₁₂)├──■──┤ RZ(θ₁₈)├
     ├────────┤┌─┴─┐├────────┤     ├────────┤┌─┴─┐├────────┤
q_1: ┤ RY(θ₁)├┤ X ├┤ RZ(θ₇)├──■──┤ RY(θ₁₃)├┤ X ├┤ RZ(θ₁₉)├──...
     ├────────┤└───┘├────────┤┌─┴─┐├────────┤└───┘├────────┤
q_2: ┤ RY(θ₂)├─────┤ RZ(θ₈)├┤ X ├┤ RY(θ₁₄)├─────┤ RZ(θ₂₀)├
     ...

Parameters

Property	Value
Qubits	6
Layers	2
Parameters	24
Hamiltonian terms	~50+ (simplified)

Running the Circuit

PYTHON
from circuit import run_circuit

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

Expected Output

Metric	Value
Exact ground state	-15.835 Ha
Qubits required	6
Chemical accuracy target	±1.6 mHa

Comparison Table

Molecule	Electrons	Qubits	Parameters
H₂	2	2	4
HeH⁺	2	2	3
LiH	4	4	16
BeH₂	6	6	24
H₂O	10	4*	20

*Active space reduction

Applications

Scalability testing: Bridge between small and large molecules
Linear molecules: Understanding symmetric stretching
Algorithm development: Testing ansatz designs

Challenges at This Scale

More parameters: 24 vs 4 for H₂
Barren plateaus: Gradients may vanish
Noise sensitivity: More qubits = more errors
Optimization landscape: More local minima