Multistate Contracted VQE (MC-VQE)

Overview

MC-VQE is a multi-reference approach for strongly correlated systems. Instead of optimizing a single wavefunction, it builds a contracted Hamiltonian in a subspace of reference states, then diagonalizes to get all eigenstates simultaneously.

Property	Value
Category	Chemistry
Difficulty	Advanced
Framework	Qiskit
Qubits	4
Depth	Variable
Gates	RY, RZ, CX, H

The Algorithm

1. Choose Reference States

Select initial configurations: |Φ₁⟩, |Φ₂⟩, ..., |Φₙ⟩

2. Apply Orbital Rotation

Transform references: |Ψᵢ⟩ = U(θ)|Φᵢ⟩

3. Build Contracted Matrices

CODE
H_IJ = ⟨Ψ_I|H|Ψ_J⟩
S_IJ = ⟨Ψ_I|Ψ_J⟩

4. Solve Generalized Eigenvalue

CODE
H·C = S·C·E

5. Optimize θ

Minimize E₀ (ground state) or weighted sum.

Reference States

CODE
|Φ₁⟩ = |1100⟩  (HF reference)
|Φ₂⟩ = |1010⟩  (single excitation)
|Φ₃⟩ = |0110⟩  (single excitation)
|Φ₄⟩ = superposition (multi-reference)

Orbital Rotation Circuit

Givens rotations between orbital pairs:

CODE
┌────────┐     ┌──────────┐
q_i: ┤ RY(θ)  ├──■──┤ RY(-θ/2) ├
     └────────┘┌─┴─┐└──────────┘
q_j: ──────────┤ X ├────────────
               └───┘

Running the Circuit

PYTHON
from circuit import run_circuit

result = run_circuit(n_qubits=4)

print(f"References: {result['n_references']}")
for state in result['states']:
    print(f"E_{state['index']}: {state['energy']:.4f}")
    print(f"  Coefficients: {state['coefficients']}")

Expected Output

State	MC-VQE	Exact	Error
E₀	~-4.50	-4.50	<1%
E₁	~-3.50	-3.50	<1%
E₂	~-2.50	-2.50	<2%
E₃	~-1.50	-1.50	<2%

Advantages Over Single-Reference

Aspect	Single-Ref VQE	MC-VQE
Strong correlation	Poor	Good
Excited states	Sequential	Simultaneous
State character	Single config	Mixed
Near-degeneracy	Unstable	Stable

Applications

Bond breaking: H₂ dissociation, transition states
Transition metals: d-orbital degeneracy
Excited states: Conical intersections
Photochemistry: Multi-state dynamics

Multistate VQE