Quantum Computing I Bloch Sphere

The Bloch sphere is a geometric representation of qubit states as points on the surface of a unit sphere. Many operations on single qubits that are commonly used in quantum information processing can be neatly described within the Bloch sphere picture.

Bloch Sphere

Bloch Sphere

A pure state $\ket{\psi}$ is a representative of a equivalence class

$$\{ \mathrm{e}^{\mathrm{i}\chi} \ket{\psi}\mid\alpha\in\mathbb{R}\}$$

One can map the state space of a two-level system $\mathbb{C}^2$ to the unit 3D sphere $S^2$ by^[1]

$$\ket{\psi} = \cos\frac{\theta}{2} \ket{1}+\mathrm{e}^{\mathrm{i}\phi }\sin\frac{\theta}{2} \ket{0}$$

$$\boldsymbol{n}(\theta,\phi)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$$

where $\theta \in [0,\pi]$ and $\phi \in [0,2\pi)$.

Bloch vector

We set the basis of the state space to $\{\ket{1},\ket{0}\}$, which means $\ket{1}=\begin{pmatrix}1\\ 0\end{pmatrix}$ and $\ket{0}=\begin{pmatrix}0 \\ 1\end{pmatrix}$.

/* Note: This definition is different from the order of kets that are usually chosen. The advantage of this kind of choice is that the Hamiltonian will have the form $H = \frac{1}{2} \hbar \omega_0 \sigma_z$ instead of $-\sigma_z$, since we set $\ket{0} \equiv \ket{g}$ and $\ket{1} \equiv \ket{e}$. */

State	Coordinates on $S^2$	$(\theta,\phi)$	Ket	Vector
$\ket{+}_z \equiv\ket{1}$	$(0,0,1)$	$(0,0)$	$\ket{1}$	$\begin{pmatrix}1\\ 0\end{pmatrix}$
$\ket{-}_z=\ket{0}$	$(0,0,-1)$	$(\pi,0)$	$\ket{0}$	$\begin{pmatrix}0 \\ 1\end{pmatrix}$
$\ket{+}_x$	$(1,0,0)$	$(\frac{\pi}{2},0)$	$\frac{\ket{1}+\ket{0}}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 1\end{pmatrix}$
$\ket{-}_x$	$(-1,0,0)$	$(\frac{\pi}{2},\pi)$	$\frac{\ket{1}-\ket{0}}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -1\end{pmatrix}$
$\ket{+}_y$	$(0,1,0)$	$(\frac{\pi}{2},\frac{\pi}{2})$	$\frac{\ket{1}+\mathrm{i}\ket{0}}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ \mathrm{i}\end{pmatrix}$
$\ket{-}_y$	$(0,-1,0)$	$(\frac{\pi}{2},\frac{3\pi}{2})$	$\frac{\ket{1}-\mathrm{i}\ket{0}}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -\mathrm{i}\end{pmatrix}$

Theorem Antipodal points on the Bloch sphere corresponds to a pair of mutually orthogonal state vectors.

Rotation

A anticlockwise rotation around $\hat{n}$ by $\theta$ on the Bloch sphere is described by rotation operator

$$R(\theta) = \exp\left[-\frac{\mathrm{i}}{2} \theta (\hat{n}\cdot{\boldsymbol{\sigma}})\right]$$

Rotation around the x,y,z axis are

$$R_x= \exp\left(-\frac{\mathrm{i}}{2} \theta \sigma_x\right)= \begin{pmatrix} \cos\frac{\theta}{2} & -\mathrm{i}\sin\frac{\theta}{2} \\ -\mathrm{i}\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}$$

$$R_y= \exp\left(-\frac{\mathrm{i}}{2} \theta \sigma_y\right)= \begin{pmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\ \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}$$

$$R_z= \exp\left(-\frac{\mathrm{i}}{2} \theta \sigma_z\right)= \begin{pmatrix} \mathrm{e}^{-\mathrm{i}\theta/2} & 0 \\ 0 & \mathrm{e}^{\mathrm{i}\theta/2} \end{pmatrix}$$

Rotation around $(\cos\phi,\sin\phi,0)$ is defined as $R_\phi$, with

$$R_\phi(\theta)= \exp\left(-\frac{\mathrm{i}}{2} \theta \sigma_\phi\right)= \begin{pmatrix} \cos\frac{\theta}{2} & -\mathrm{i}\mathrm{e}^{-\mathrm{i}\phi}\sin\frac{\theta}{2} \\ -\mathrm{i}\mathrm{e}^{\mathrm{i}\phi}\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}$$

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1. Bloch sphere by Cmglee is licensed under CC BY-SA 4.0. Available at https://commons.wikimedia.org/wiki/File:Bloch_sphere.svg. ↩︎