Variational Circuit

A variational circuit (or parameterized quantum circuit, PQC) is a quantum circuit containing rotation gates with tunable parameters θ = (θ₁, θ₂, ..., θₙ). The circuit prepares a trial state |ψ(θ)⟩ whose properties depend on the parameters. A classical optimizer adjusts the parameters to minimize a cost function C(θ) computed from quantum measurements. The parameter-shift rule provides an analytical gradient of quantum circuits: ∂C/∂θᵢ = [C(θᵢ + π/2) − C(θᵢ − π/2)] / 2. This enables gradient-based optimization. Variational circuits underlie VQE, QAOA, QML (quantum neural networks), and quantum kernel methods. The expressibility of a variational circuit (how many states it can represent) and its trainability (susceptibility to barren plateaus) are key research questions. PennyLane and HLQuantum both support differentiable variational circuits with PyTorch and JAX integration.