QML Capstone

QML Capstone: Variational Quantum Optimizer

Every variational quantum algorithm, VQE for chemistry, QAOA for optimization, quantum classifiers for machine learning, is the same engine under the hood: a parameterized circuit, a cost measured from its output, gradients of that cost, and a classical loop that walks downhill. In this capstone you build that engine in miniature and verify every moving part: the cost function from real circuit counts, exact gradients via the parameter shift rule, gradient descent to the minimum, and a final check that you actually arrived.

The Ansatz and the Objective

The circuit is two parameterized rotations and one entangler: $R_y({\theta }_1)$ on qubit 0, $R_y({\theta }_2)$ on qubit 1, then CNOT(0, 1). The objective is to maximize the probability of measuring $|11⟩$, written as a cost to minimize:

Work out the landscape by hand. The outcome 11 requires qubit 0 to read 1, which happens with probability ${\sin }^2({\theta }_1/2)$. And because the CNOT flips qubit 1 exactly when qubit 0 is 1, reading 11 requires qubit 1 to have been 0 before the CNOT, with probability ${\cos }^2({\theta }_2/2)$. So

with minimum $C=-1$ at ${\theta }_1=\pi$, ${\theta }_2=0$. Your optimizer should find that point without being told.

The Parameter Shift Rule

For gates of the form $R_y(\theta )$, the gradient of any measured expectation is given exactly by two extra circuit evaluations per parameter: