Multistate Contracted VQE (MC-VQE)

What You'll Learn:

• How to find multiple eigenstates simultaneously using a contracted Hamiltonian approach
• How orbital rotations (Givens rotations) transform reference states into better trial states
• Why multi-reference methods are essential for strongly correlated systems
• How to solve generalized eigenvalue problems with regularized overlap matrices

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

Prerequisites

• H2 Excited States — VQD algorithm, overlap between states
• H2 Ground State — VQE fundamentals

The Idea

VQD finds excited states one at a time — ground state first, then first excited, then second. This works, but it compounds errors: each state's optimization depends on the accuracy of all previous ones. If the ground state is slightly wrong, every subsequent state is affected.

MC-VQE takes a different approach: instead of finding states one by one, it builds a small subspace of reference states and diagonalizes the Hamiltonian within that subspace. All eigenstates emerge simultaneously from a single matrix diagonalization. The quantum computer's job is to prepare good reference states; the classical computer handles the small eigenvalue problem.

Think of it like this: VQD searches the full landscape one valley at a time, while MC-VQE builds a topographic map of a small region and reads off all the valleys at once.

How It Works

Step 1: Choose Reference States

Select initial configurations that span the physically relevant subspace:

CODECopy
|Φ₁⟩ = |1100⟩    Hartree-Fock reference (2 occupied orbitals)
|Φ₂⟩ = |1010⟩    Single excitation (orbital 1 → 2)
|Φ₃⟩ = |0110⟩    Different single excitation
|Φ₄⟩ = entangled  Multi-reference superposition

Good reference sets should be linearly independent and cover the target subspace — the subspace where the lowest eigenstates live.

Step 2: Apply Orbital Rotation

A shared rotation U(θ) transforms all references simultaneously:

|Ψ_I⟩ = U(θ)|Φ_I⟩

The rotation uses Givens rotations between orbital pairs — the same operation used in classical quantum chemistry to rotate molecular orbitals:

CODECopy
Givens rotation between qubits i and j:

q_i: ──RY(θ)──■──────────────
               │
q_j: ─────────X──RY(-θ/2)──■──
                             │
q_i: ───────────────────────X──

For 4 qubits, there are 6 orbital pairs → 6 rotation parameters.

Step 3: Build Contracted Matrices

Compute the Hamiltonian and overlap in the reference subspace:

CODECopy
H_IJ = ⟨Ψ_I|H|Ψ_J⟩    (energy matrix)
S_IJ = ⟨Ψ_I|Ψ_J⟩      (overlap matrix)

These are small matrices (4×4 for 4 references) that can be classically diagonalized.

Step 4: Solve the Generalized Eigenvalue Problem

H·C = S·C·E

The eigenvalues E give the energies, and the eigenvectors C give each eigenstate as a linear combination of reference states.

Step 5: Optimize θ

Minimize the ground state eigenvalue by adjusting the orbital rotation parameters. After optimization, all eigenstates are obtained "for free" from the same diagonalization.

The Math

Contracted Hamiltonian

The full Hamiltonian H acts on a 2^n-dimensional space. The contracted Hamiltonian H_c acts on an m-dimensional subspace (m << 2^n):

H_c = P† H P

where P = [|Ψ₁⟩, |Ψ₂⟩, ..., |Ψₘ⟩] is the matrix of reference states.

The eigenvalues of H_c are upper bounds to the true eigenvalues (by the variational principle applied to each subspace direction).

Generalized Eigenvalue Problem

When references are non-orthogonal (S ≠ I), we solve HC = SCE. This is handled by:

1. SVD of S: S = UΣV†
2. Regularize: discard singular values below threshold
3. Transform: H' = S^{-1/2} H S^{-1/2}
4. Standard diagonalization of H'

Transverse Field Ising Model

H = -Σᵢ ZᵢZᵢ₊₁ - h Σᵢ Xᵢ

At h = 0.5 (ordered phase), the ground state is near the ferromagnetic configuration. The first excited state involves a domain wall.

Expected Output

Accuracy depends on how well the reference states span the target subspace. Ground state is typically most accurate.

Running the Circuit

PYTHONCopy
from circuit import run_circuit, verify_mcvqe, compute_exact_eigenvalues

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

# MC-VQE
result = run_circuit(max_iterations=100, seed=42)
for state in result['states']:
    print(f"E_{state['index']}: {state['energy']:.4f} "
          f"(exact: {state['exact']:.4f}, error: {state['error']:.4f})")

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

Try It Yourself

1. Add more references: Create a 5th reference state (e.g., |0011⟩) and observe how the accuracy of higher excited states improves.

2. Vary the transverse field: Change h from 0.1 to 2.0. At h = 1.0 (critical point), the system is hardest to solve — do you need more references?

3. Compare with VQD: Run VQD for the same Hamiltonian. Which method gets closer to the exact excited state energies? Which converges faster?

4. Reference quality: Replace the entangled reference with |0011⟩. Does the subspace quality (ground state accuracy) improve or degrade?

What's Next

• SSVQE — Simultaneous optimization without contracted Hamiltonian
• VQD Algorithm — Sequential deflation approach
• H2 Excited States — Simplest excited state example

Comparison: MC-VQE vs Other Methods

Applications

References

• Parrish, R.M. et al. (2019). "Quantum Computation of Electronic Transitions Using a Variational Quantum Eigensolver." Physical Review Letters 122, 230401. DOI: 10.1103/PhysRevLett.122.230401
• Huggins, W.J. et al. (2020). "A non-orthogonal variational quantum eigensolver." New Journal of Physics 22, 073009. DOI: 10.1088/1367-2630/ab867b
• 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