VQE for LiH Excited States — 4-Qubit VQD

What You'll Learn:

• How VQD scales from 2-qubit H2 to 4-qubit LiH with a richer electronic structure
• How ionic-covalent character mixing affects excited state structure
• How to build multi-qubit measurement circuits for a 14-term Hamiltonian
• Why LiH is a standard benchmark for quantum chemistry on NISQ devices

Level: Advanced | Time: 35 minutes | Qubits: 4 | Framework: Qiskit

Prerequisites

• H2 Excited States — VQD algorithm, overlap penalties, β selection
• H2 Ground State — VQE fundamentals, Pauli measurement
• LiH Ground State — 4-qubit ansatz, multi-qubit measurement

The Idea

Hydrogen (H2) is quantum chemistry's "Hello World" — 2 electrons, 1 bond, 2 qubits. Lithium hydride (LiH) is the next step: 4 electrons, a polar bond, and a 4-qubit active space. The electronic structure is richer because LiH has both ionic character (Li⁺H⁻, electrons transferred to hydrogen) and covalent character (shared electrons).

At equilibrium (~1.6 A), the ground state is predominantly ionic. But as you stretch the bond, the ground state smoothly transitions to covalent character — while the excited state does the opposite. At ~3 A, the two potential energy curves nearly touch in an avoided crossing. This is exactly the kind of problem where quantum computers can shine: capturing multi-reference correlations that classical single-reference methods miss.

How It Works

The LiH Hamiltonian

The 4-qubit active space (2 electrons in 4 spin-orbitals) gives a 14-term Hamiltonian:

The XX/YY terms couple spin-up and spin-down within each molecular orbital (1σ and 2σ), encoding the electron exchange symmetry required by the Pauli exclusion principle.

The 4-Qubit Ansatz

A UCCSD-inspired ansatz with CNOT-ladder entanglement:

CODECopy
Layer 1:              CNOT ladder         RZ rotations      Reverse CNOT
     ┌────────┐                          ┌────────┐
q_0: ┤ RY(θ₀) ├──■──────────────────────┤ RZ(θ₄) ├────────────────────
     ├────────┤┌─┴─┐                     ├────────┤
q_1: ┤ RY(θ₁) ├┤ X ├──■─────────────────┤ RZ(θ₅) ├──────────■────────
     ├────────┤└───┘┌─┴─┐               ├────────┤        ┌─┴─┐
q_2: ┤ RY(θ₂) ├─────┤ X ├──■────────────┤ RZ(θ₆) ├──■────┤ X ├───────
     ├────────┤     └───┘┌─┴─┐          ├────────┤┌─┴─┐  └───┘
q_3: ┤ RY(θ₃) ├──────────┤ X ├──────────┤ RZ(θ₇) ├┤ X ├──────────────
     └────────┘          └───┘          └────────┘└───┘

The CNOT ladder (0→1→2→3) creates a chain of entanglement. The reverse partial CNOT (2→3, 1→2) adds longer-range correlations, matching the structure of molecular orbital interactions.

VQD: Sequential State Finding

Same as H2 VQD, but with 16 parameters and more measurement circuits:

1. Ground state: Standard VQE with 14-term energy evaluation
2. Excited state: Minimize E + β|⟨ψ₁|ψ₀⟩|² with β = 3.0

Each energy evaluation requires 5 measurement circuits: 1 ZZ (computational), 2 XX (pairs 0-1 and 2-3), 2 YY (pairs 0-1 and 2-3).

The Math

Active Space

LiH has 4 electrons total, but the 1s² core of lithium is frozen. The active space contains 2 electrons in 4 spin-orbitals:

CODECopy
Spin-orbital mapping:
  q₀ → 1σ α (bonding, spin-up)
  q₁ → 1σ β (bonding, spin-down)
  q₂ → 2σ α (antibonding, spin-up)
  q₃ → 2σ β (antibonding, spin-down)

Ionic vs Covalent Character

At equilibrium, the ground state is:

CODECopy
|ψ_ground⟩ ≈ α|1100⟩ + β|1010⟩ + γ|0110⟩ + ...
                ionic      covalent

The dominant |1100⟩ component has both electrons in the bonding orbital — this is the ionic Li⁺H⁻ configuration. The |1010⟩ and |0110⟩ components have one electron in each orbital — covalent character.

Avoided Crossing

As the bond stretches, the ionic and covalent energies approach each other. Quantum mechanics forbids the curves from actually crossing (for states of the same symmetry), creating an avoided crossing. The ground and excited states exchange character smoothly through this region.

Expected Output

At equilibrium (R = 1.6 A):

Running the Circuit

PYTHONCopy
from circuit import run_circuit, optimize_vqd, verify_lih_vqd

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

# Full VQD: ground + excited state
opt = optimize_vqd(n_states=2, max_iterations=100, shots=2048, seed=42)
print(f"Ground:  {opt['states'][0]['energy']:.4f} Ha")
print(f"Excited: {opt['states'][1]['energy']:.4f} Ha")
print(f"Gap:     {opt['gaps'][0]:.4f} Ha")

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

Try It Yourself

1. Stretch the bond: Change the Hamiltonian coefficients for R = 2.5 A and R = 3.5 A. Watch the energy gap shrink toward the avoided crossing. (Hint: look up LiH STO-3G coefficients in Kandala et al. supplementary materials.)

2. Potential energy surface: Compute the ground and excited state energies at 5 bond lengths from 1.0 A to 4.0 A. Plot both curves — you should see the avoided crossing near 3 A.

3. Add a third state: Run optimize_vqd(n_states=3) to find the second excited state. Is the second gap larger or smaller than the first?

4. Compare with H2: The H2 VQD needs 2 qubits and 2 measurement circuits per energy evaluation. LiH needs 4 qubits and 5 circuits. How does the circuit count scale with the number of Pauli terms?

What's Next

• H2 Excited States — Simpler 2-qubit version of VQD
• SSVQE — Simultaneous multi-state optimization (alternative to sequential VQD)
• Multistate VQE — Multi-reference approach for strongly correlated systems

Applications

References

• Kandala, A. et al. (2017). "Hardware-efficient variational quantum eigensolver for small molecules." Nature 549, 242-246. DOI: 10.1038/nature23879
• 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
• Higgott, O. et al. (2019). "Variational Quantum Computation of Excited States." Quantum 3, 156. DOI: 10.22331/q-2019-07-01-156
• O'Malley, P.J.J. et al. (2016). "Scalable quantum simulation of molecular energies." Physical Review X 6, 031007. DOI: 10.1103/PhysRevX.6.031007