Variational Quantum Deflation (VQD) — General Implementation

What You'll Learn:

• How VQD scales to larger systems (4 qubits, arbitrary Hamiltonian)
• How the deflation process systematically removes found states from the search space
• Why β must increase as more states are found (cumulative penalty landscape)
• How to implement measurement-based energy computation for a spin-chain Hamiltonian

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

Prerequisites

• H2 Excited States — VQD on 2 qubits (molecular Hamiltonian)
• H2 Ground State — VQE fundamentals

The Idea

The H2 VQD tutorial shows the core algorithm on 2 qubits with a molecular Hamiltonian. This circuit generalizes VQD to arbitrary Hamiltonians and demonstrates it on the 4-qubit Transverse Field Ising Model (TFIM) — a fundamental model in condensed matter physics.

VQD works by "deflating" the Hilbert space: after finding each eigenstate, it adds a penalty that makes that region of the space expensive to visit. The optimizer is forced to explore new regions, discovering higher-energy states.

CODECopy
State 0 (VQE):      L₀ = ⟨H⟩
State 1 (VQD):      L₁ = ⟨H⟩ + β₀|⟨ψ₁|ψ₀⟩|²
State 2 (VQD):      L₂ = ⟨H⟩ + β₀|⟨ψ₂|ψ₀⟩|² + β₁|⟨ψ₂|ψ₁⟩|²

Each additional state adds one more penalty term. The penalty "walls" accumulate, creating an increasingly constrained landscape.

How It Works

The Transverse Field Ising Model

H = -Σᵢ ZᵢZᵢ₊₁ - h Σᵢ Xᵢ    (h = 0.5)

At h = 0.5 (below critical point h = 1.0), the system is in the ordered phase: spins prefer to align, and the ground state is near |0000⟩ or |1111⟩. The spectrum has clear gaps.

Measurement Strategy

The TFIM has two types of Pauli terms:

1. ZZ terms (3 for 4 qubits): Measured in the computational basis. One circuit gives all ZZ expectations simultaneously.

2. X terms (4 for 4 qubits): Each requires a Hadamard rotation before measurement. We can measure all X terms from individual single-qubit rotations on the same circuit.

Total: 5 measurement circuits per energy evaluation.

Hardware-Efficient Ansatz

CODECopy
     ┌────────┐          ┌────────┐┌────────┐          ┌────────┐┌────────┐
q_0: ┤ RY(θ₀) ├──■───────┤ RY(θ₄) ├┤ RZ(θ₅) ├──■───────┤ RY(θ₁₂)├┤ RZ(θ₁₃)├
     ├────────┤┌─┴─┐     ├────────┤├────────┤┌─┴─┐     ├────────┤├────────┤
q_1: ┤ RY(θ₁) ├┤ X ├──■──┤ RY(θ₆) ├┤ RZ(θ₇) ├┤ X ├──■──┤ RY(θ₁₄)├┤ RZ(θ₁₅)├
     ├────────┤└───┘┌─┴─┐├────────┤├────────┤└───┘┌─┴─┐├────────┤├────────┤
q_2: ┤ RY(θ₂) ├─────┤ X ├┤ RY(θ₈) ├┤ RZ(θ₉) ├─────┤ X ├┤ RY(θ₁₆)├┤ RZ(θ₁₇)├
     ├────────┤     └───┘├────────┤├────────┤     └───┘├────────┤├────────┤
q_3: ┤ RY(θ₃) ├──────────┤ RY(θ₁₀)├┤ RZ(θ₁₁)├──────────┤ RY(θ₁₈)├┤ RZ(θ₁₉)├
     └────────┘          └────────┘└────────┘          └────────┘└────────┘

20 parameters per state × 3 states = 60 total (but optimized sequentially, 20 at a time).

β Selection for Multiple States

Each penalty β must exceed the corresponding energy gap. For the TFIM at h = 0.5:

Using a uniform β = 3.0 provides safety margin for all gaps.

The Math

Deflation Guarantee

VQD converges to the k-th eigenstate if:

1. βᵢ > Eᵢ₊₁ - Eᵢ for all i < k (penalties exceed gaps)
2. The ansatz can represent the k-th eigenstate
3. The optimizer finds the global minimum

When all conditions hold, the cost function landscape has its global minimum exactly at the k-th eigenstate:

CODECopy
L_k(ψ_k) = E_k + Σᵢ βᵢ × 0 = E_k
L_k(ψ_j, j<k) = E_j + βⱼ × 1 > E_j + (E_{j+1} - E_j) = E_{j+1} ≥ E_k

So the ground states of lower cost functions are more expensive than the target state.

Error Propagation

VQD's main weakness: errors compound. If state k has error εₖ, then state k+1 sees a distorted penalty landscape. The overlap |⟨ψ_{k+1}|ψ̃_k⟩|² (where ψ̃_k is the approximate state k) differs from |⟨ψ_{k+1}|ψ_k⟩|² by O(εₖ). This shifts the optimal state k+1 by O(εₖ/gap).

Expected Output

For TFIM (4 qubits, h = 0.5):

Error increases with state index (VQD error propagation).

Running the Circuit

PYTHONCopy
from circuit import run_circuit, verify_vqd, compute_exact_eigenvalues

# Exact reference
exact = compute_exact_eigenvalues(n_states=3)
print(f"Exact: {[f'{e:.4f}' for e in exact]}")

# VQD: find 3 eigenstates sequentially
result = run_circuit(n_states=3, beta=3.0, max_iterations=50, seed=42)
for state in result['states']:
    print(f"State {state['index']}: E = {state['energy']:.4f}")

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

Try It Yourself

1. Break β: Set beta=0.3 (below the first gap ~1.0). The second state should collapse back to the ground state.

2. Find 5 states: Run run_circuit(n_states=5). Do the higher states maintain correct ordering? Does the error grow linearly or faster?

3. Critical point: Set TRANSVERSE_FIELD=1.0 (quantum phase transition). The gap closes — VQD should struggle. How large does β need to be?

4. Compare with SSVQE: Run SSVQE for the same system. Which method gets better excited states? (Hint: SSVQE finds them simultaneously, so no error propagation.)

5. Increase shots: Run with 256, 1024, 4096, 16384 shots. Does the energy variance decrease as 1/√shots?

What's Next

• SSVQE — Simultaneous optimization (no error propagation)
• Multistate VQE — Contracted Hamiltonian (few parameters)
• H2 Excited States — 2-qubit molecular VQD

Applications

References

• Higgott, O., Wang, D., Brierley, S. (2019). "Variational Quantum Computation of Excited States." Quantum 3, 156. DOI: 10.22331/q-2019-07-01-156
• McClean, J.R. et al. (2016). "The theory of variational hybrid quantum-classical algorithms." New Journal of Physics 18, 023023. DOI: 10.1088/1367-2630/18/2/023023
• Nakanishi, K.M., Mitarai, K., Fujii, K. (2019). "Subspace-search variational quantum eigensolver for excited states." Physical Review Research 1, 033062. DOI: 10.1103/PhysRevResearch.1.033062