Quantum Simulation

from openai import OpenAI

client = OpenAI(
    api_key=SYNAPTIQ_API_KEY,
    base_url="https://api.synaptiq.contact/v1",
)

prompt = """
Simulate the following quantum experiment: Bell state preparation.

Initialize two qubits in the |00⟩ state.
Apply a Hadamard gate to the first qubit.
Apply a CNOT gate with the first qubit as control and the second as target.
Show the mathematical derivation, quantum circuit, and measurement statistics.
"""

completion = client.chat.completions.create(
    model="synaptiq-2-quantum",
    messages=[
        {"role": "user", "content": prompt}
    ]
)

print(completion.choices[0].message.content)

Bell State Preparation and Analysis

Mathematical Formalism

We begin with a 2-qubit system in the ground state:

|ψ₀⟩ = |00⟩

First, we apply a Hadamard gate to the first qubit:

H|0⟩ = (1/√2)(|0⟩ + |1⟩)

So our state becomes:

|ψ₁⟩ = (1/√2)(|0⟩ + |1⟩) ⊗ |0⟩ = (1/√2)(|00⟩ + |10⟩)

Next, we apply a CNOT gate with the first qubit as control and the second as target:

CNOT|ψ₁⟩ = (1/√2)(|00⟩ + |11⟩)

This is the Bell state |Φ⁺⟩, one of the maximally entangled two-qubit states.

Quantum Circuit

q_0: ──[H]──●──
             │
q_1: ────────X──

Measurement Statistics

When we measure this Bell state, quantum theory predicts we should observe either both qubits in state |0⟩ or both in state |1⟩, with equal probability (50% each).

| Measurement | Count | Probability |
|-------------|-------|------------|
| 00          | 512   | 0.5        |
| 11          | 512   | 0.5        |
| 01          | 0     | 0.0        |
| 10          | 0     | 0.0        |

Insights

This experiment demonstrates quantum entanglement, one of the most profound phenomena in quantum mechanics. The key observations:

1. The qubits are in a superposition of states until measured.
2. After entanglement, the measurement outcomes of the two qubits are perfectly correlated - if one is measured as |0⟩, the other will always be |0⟩, and similarly for |1⟩.
3. This correlation persists regardless of the spatial separation between the qubits, which Einstein famously referred to as "spooky action at a distance."
4. The Bell state created here (|Φ⁺⟩) is maximally entangled, meaning it cannot be factored into a product of individual qubit states.