QCQI - Chapter 2: Qubits, States & the Bloch Sphere

Learning Objectives¶

• Parametrize pure states $(\theta,\phi)$ and read off Bloch components.

• Connect $\mathrm{SU}(2)$ unitaries and $\mathrm{SO}(3)$ rotations.

• Measure along $X,Y,Z$ via basis change and estimate expectations.

• Perform single-qubit tomography and interpret $\rho=\frac{1}{2}(I+\vec r\cdot\vec\sigma)$.

State Parametrization¶

$|\psi(\theta,\phi)\rangle=\cos\frac{\theta}{2}|0\rangle+e^{i\phi}\sin\frac{\theta}{2}|1\rangle$, $\vec r=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$.

Eureka! Two angles $\Rightarrow$ two degrees of freedom (global phase drops out).

Unitaries as Rotations¶

$U=R_z(\alpha)R_y(\beta)R_z(\gamma)$; adjoint action rotates $\vec r$.

Eureka! $\mathrm{SU}(2)\to \mathrm{SO}(3)$: double-cover relation.

Measuring $X,Y,Z$¶

Use $H$ (for $X$) and $S^\dagger H$ (for $Y$) before $Z$-readout.

Estimate $\langle \sigma_k\rangle=(N_0-N_1)/(N_0+N_1)$.

Tomography¶

Collect $\langle X\rangle,\langle Y\rangle,\langle Z\rangle$.

Set $\rho=\frac{1}{2}(I+\vec r\cdot\vec\sigma)$; check purity $\mathrm{Tr}(\rho^2)$.

Eureka! Noise shrinks $\|\vec r\|$: Bloch-ball radius encodes mixedness.

Schematics¶

Hands-on Notebook¶

Companion notebook (multi-backend): L02_QCQI_Ch02_Qubits_States_and_Bloch_Sphere

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

Mini-Lab¶

• Plot $(\langle X\rangle,\langle Y\rangle,\langle Z\rangle)$ across $\theta$ for fixed $\phi$.

• Add noise; watch the Bloch vector shrink.

• Compare tomography across two backends.