QCQI – Chapter 2 Detailed Summary: Qubits, States & the Bloch Sphere

Learning Objectives¶

• Parametrize a single-qubit pure state and map to the Bloch sphere.

• Interpret single-qubit unitaries as rotations; use ZYZ decomposition.

• Measure along $X,Y,Z$ (basis change) and estimate expectation values.

• Perform linear-inversion tomography: $\rho=\frac12(I+\vec r\cdot \vec\sigma)$.

States and Bloch Vectors¶

$|\psi(\theta,\phi)\rangle=\cos\frac{\theta}{2}|0\rangle+e^{i\phi}\sin\frac{\theta}{2}|1\rangle$, $\vec r=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$, and $\rho=\frac12(I+\vec r\cdot \vec\sigma)$.

Eureka! Pure states lie on the Bloch surface ($\|\vec r\|=1$); noise shrinks $\|\vec r\|<1$.

Unitaries as Rotations¶

$U=R_z(\alpha)R_y(\beta)R_z(\gamma)$, with $R_{\vec n}(\vartheta)=e^{-i(\vartheta/2)\vec n\cdot\vec\sigma}$.

Eureka! $\mathrm{SU}(2)$ double-covers $\mathrm{SO}(3)$: global phases vanish at the density level.

Measurements via Basis Change¶

$\langle \sigma_x\rangle = \langle Z\rangle$ after $H$;   $\langle \sigma_y\rangle = \langle Z\rangle$ after $S^\dagger H$;   $\langle \sigma_z\rangle = \langle Z\rangle$ directly.

Tomography¶

Estimate $\langle X\rangle,\langle Y\rangle,\langle Z\rangle$ from repeated shots, then set $\hat\rho=\frac12(I+\hat m_x X+\hat m_y Y+\hat m_z Z)$.

Purity: $\operatorname{Tr}(\rho^2)=\tfrac{1+\|\vec r\|^2}{2}$.

Worked Examples¶

State preparation. Apply $R_y(\theta)$ then $R_z(\phi)$ to $|0\rangle$.

Expectations from counts. $\langle Z\rangle=(N_0-N_1)/(N_0+N_1)$; for $X(Y)$ prepend $H(S^\dagger H)$.

Global vs relative phase. $e^{i\gamma}|\psi\rangle$ unchanged; relative phases impact interference after basis rotations.

Pseudo-code Aids¶

Pseudocode — Prepare $|\psi(\theta,\phi)\rangle$ from $|0\rangle$

Input: angles $(\theta,\phi)$ Steps: apply $R_y(\theta)$; then $R_z(\phi)$ Output: state $|\psi(\theta,\phi)\rangle$

Pseudocode — Measure $\langle \sigma_k\rangle$

Input: axis $k\in\{x,y,z\}$, shots $S$ if $k=x$: prepend $H$; if $k=y$: prepend $S^\dagger$ then $H$ measure $Z$ for $S$ shots; return $(N_0-N_1)/(N_0+N_1)$

Pseudocode — Single-qubit tomography (linear inversion)

Input: unknown state, shots per basis $S$ estimate $\hat m_x,\hat m_y,\hat m_z$ as above set $\hat\rho=\frac12(I+\hat m_x X+\hat m_y Y+\hat m_z Z)$

Schematics¶

Bloch Sphere (axes only)¶

Basis-change circuits¶

Hands-on Notebook (Multi-Backend)¶

Run: L02_QCQI_Ch02_Qubits_States_and_Bloch_Sphere

Set backend to cirq, pennylane, braket, pyquil, or qiskit.