Subspace-Search VQE (SSVQE)

Overview

SSVQE finds multiple eigenstates simultaneously in a single optimization, unlike VQD which finds them sequentially. It optimizes a weighted sum of energies while enforcing orthogonality between states.

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

The Algorithm

Cost Function

CODE
L(θ) = Σₖ wₖ ⟨ψₖ(θₖ)|H|ψₖ(θₖ)⟩ + λ Σᵢ<ⱼ |⟨ψᵢ|ψⱼ⟩|²

Where:

wₖ: Weights (larger for ground state)
λ: Orthogonality penalty strength

Weight Assignment

To ensure proper ordering:

w₀ > w₁ > w₂ > ...
Example: w = [3, 2, 1] for 3 states

Circuit Structure

Each state k has its own parameters θₖ:

CODE
State 0: |ψ₀(θ₀)⟩
     ┌────────┐     ┌────────┐┌────────┐
q_0: ┤ RY(θ₀₀)├──■──┤ RY(θ₀₄)├┤ RZ(θ₀₅)├
     └────────┘  │  └────────┘└────────┘
                 ...

State 1: |ψ₁(θ₁)⟩
     ┌────────┐     ┌────────┐┌────────┐
q_0: ┤ RY(θ₁₀)├──■──┤ RY(θ₁₄)├┤ RZ(θ₁₅)├
     └────────┘  │  └────────┘└────────┘
                 ...

Running the Circuit

PYTHON
from circuit import run_circuit

# Find 3 lowest eigenstates
result = run_circuit(n_qubits=4, n_states=3)

for state in result['states']:
    print(f"E_{state['rank']}: {state['energy']:.4f}")

print(f"Max overlap: {result['max_overlap']:.4f}")
print(f"Orthogonality: {result['orthogonality_achieved']}")

Expected Output

State	SSVQE	Exact	Error
E₀	~-4.50	-4.50	<1%
E₁	~-3.50	-3.50	<2%
E₂	~-2.50	-2.50	<3%

Comparison: SSVQE vs VQD

Aspect	SSVQE	VQD
Optimization	Simultaneous	Sequential
Parameters	O(K × P)	O(K × P)
Measurements	K times more	Standard
Parallelism	High	Low
Overlap control	Explicit penalty	Deflation

Key Concepts

Weight Selection

Weights encode target ordering
Larger weight → tends to lower energy
Common: wₖ = K - k (linear)

Orthogonality Penalty

Essential for state separation
λ should be > largest gap expected
Too large: optimization difficult

Applications

Full spectrum: Find all low-lying states at once
Degenerate systems: Handle near-degenerate states
Quantum dynamics: Construct propagators

SSVQE Algorithm