QCQI - Chapter 4: Quantum Circuits & Universality

Learning Objectives¶

• Circuit model, controls, and measurements.

• ZYZ decomposition; synthesize 1-qubit unitaries.

• Universal sets: Clifford+T, $\{H,T,\mathrm{CNOT}\}$; SK idea.

Circuit Model¶

Controlled operations: apply $U$ if control is $|1\rangle$.

Eureka! Control gives conditional evolution without breaking unitarity.

ZYZ Decomposition¶

$U=R_z(\alpha)R_y(\beta)R_z(\gamma)$ (up to phase).

Use it to compile arbitrary 1-qubit gates.

Universality¶

Clifford+T is universal; any entangling 2-qubit + all 1-qubit gates is universal.

Eureka! CNOTs are the “nonlocal currency” to be spent sparingly.

Schematics¶

Hands-on Notebook¶

Companion notebook (multi-backend): L04_QCQI_Ch04_Quantum_Circuits_and_Universality

Set backend in the first cell to cirq, pennylane, braket, pyquil, or qiskit.

Mini-Lab¶

• Extract ZYZ angles; verify fidelity.

• Coarse Clifford+T approximation study vs. error.

• Controlled-$R_z$: create/measure entanglement.