Expressibility Analysis

Overview

Expressibility measures how uniformly a variational ansatz can cover the Hilbert space. More expressive circuits can represent a wider variety of quantum states.

Property	Value
Category	Ansatz
Difficulty	Advanced
Framework	Cirq
Qubits	4
Purpose	Analysis

Methodology

Fidelity Sampling

For pairs of random parameters (θ₁, θ₂):

CODE
F = |⟨ψ(θ₁)|ψ(θ₂)⟩|²

Comparison to Haar Random

Haar random states follow Beta(1, 2ⁿ-1) distribution.

KL Divergence

CODE
Expressibility = D_KL(P_ansatz || P_Haar)

Lower value → more expressive.

Analyzed Ansatze

Simple (Product States)

CODE
|ψ⟩ = ⊗ RY(θᵢ)|0⟩

Least expressive - no entanglement.

Hardware-Efficient

CODE
[RY-RZ per qubit] → [CZ ladder] → repeat

Moderate expressibility.

Strongly Entangling

CODE
[RX-RY-RZ per qubit] → [All-to-all CNOT] → repeat

Most expressive.

Running the Circuit

PYTHON
from circuit import run_circuit

result = run_circuit(n_qubits=4, n_layers=2, n_samples=200)

for name, data in result['ansatze'].items():
    print(f"{name}: {data['expressibility']:.4f}")

print(f"Ranking: {result['ranking']}")

Expected Results

Ansatz	Expressibility	Entanglement
Simple	~0.8-1.0	None
HEA	~0.2-0.5	Linear
Strong	~0.05-0.2	Full

Trade-offs

More Expressive	Less Expressive
Harder to train	Easier to train
More gates	Fewer gates
Barren plateaus	Gradient signal
Higher accuracy potential	Limited states

Applications

Ansatz design: Choose appropriate expressibility
Resource estimation: Gates vs. power
Benchmarking: Compare circuit families