Bell State Preparation via Variational Quantum Circuit

Preparing the Bell state |Φ+⟩ = (|00⟩ + |11⟩)/√2 using a single-parameter variational quantum circuit, optimised with a hand-implemented parameter-shift rule.

Built with PennyLane.

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What this does

A variational circuit consisting of an RY rotation followed by a CNOT gate is trained to produce equal-weight superposition across |00⟩ and |11⟩. The cost function minimises the expectation values ⟨Z₀⟩² + (⟨Z₀Z₁⟩ − 1)², driving the system toward the target entangled state.

Optimisation uses gradient descent with the parameter-shift rule — a hardware-compatible method that estimates gradients using only circuit evaluations at shifted parameters (p ± π/2), without accessing internal quantum state. This makes the approach directly transferable to real hardware.

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Circuit

RY is drawn from a universal gate set {RX, RY, CNOT} adopted for this project — sufficient for arbitrary single- and two-qubit operations.

RY(p) ──●──
        │
   I ───⊕──

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Results

Optimization Performance

The model was trained using Gradient Descent with a tolerance ($\epsilon$) of $10^{-4}$.

• Convergence: The algorithm reached the stopping criterion at iteration 1410.
• Final Parameter ($p$): $\approx 1.5956$ (converging toward the theoretical value of $\pi/2 \approx 1.5708$).
• Final Error: $0.0001$.

At the optimal parameter, the circuit produces $|00\rangle$ and $|11\rangle$ with equal probability ($\approx 0.5$ each), confirming successful Bell state preparation.

Observable	Initial ($p = 3.0$)	Optimised
$P(|00\rangle)$	~0.99	~0.50
$P(|11\rangle)$	~0.01	~0.50

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Project structure

bell-state-vqc/
├── bell_state_vqc.ipynb   # main notebook
├── figures/               # plots
│   ├── circuit-diagram.png
│   ├── states-probabilities.png
│   └── parameter-evolution.png
├── requirements.txt
└── README.md

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Run it

pip install -r requirements.txt
jupyter notebook bell_state_vqc.ipynb

Requirements

pennylane
numpy
matplotlib
seaborn

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Notes

Code written independently as part of self-directed study into variational quantum algorithms. Documentation refined iteratively.