# Bloch sphere
**Bloch sphere** is a geometrical representation of a single qubit state as a point on the surface of a unit sphere in three-dimensional space. It provides an intuitive way to visualize qubit states and the effect of quantum gates as rotations. The Bloch sphere is named after physicist Felix Bloch.

A single qubit $\lvert\psi\rangle = a\lvert 0\rangle + b\lvert 1\rangle$ has two complex probability amplitudes $a, b \in \mathbb{C}$, subject to the normalization constraint $|a|^2 + |b|^2 = 1$. Because global phase is physically unobservable, the qubit is fully described by just two real parameters: a polar angle $\theta \in [0, \pi]$ and an azimuthal angle $\varphi \in [0, 2\pi)$. We can therefore write any pure qubit state in the following standard form.

$$\lvert\psi\rangle = \cos\frac{\theta}{2}\lvert 0\rangle + e^{i\varphi}\sin\frac{\theta}{2}\lvert 1\rangle$$

## Computational basis states
The two computational basis states $\lvert 0\rangle$ and $\lvert 1\rangle$ sit at the north and south poles of the Bloch sphere respectively. The state $\lvert 0\rangle$ corresponds to $\theta = 0$ (north pole, coordinates $(0,0,1)$), while $\lvert 1\rangle$ corresponds to $\theta = \pi$ (south pole, coordinates $(0,0,-1)$). The six cardinal states — $\lvert 0\rangle$, $\lvert 1\rangle$, $\lvert +\rangle$, $\lvert -\rangle$, $\lvert i\rangle$, $\lvert -i\rangle$ — sit at the six poles of the three coordinate axes.

## Bloch vector
Any qubit state $\lvert\psi\rangle$ maps to a point on the surface of the unit sphere. The Cartesian coordinates $(x, y, z)$ of the corresponding Bloch vector are obtained from the probability amplitudes via the [[hopf-fibration]].

$$x = 2\,\mathfrak{Re}(a^* b), \qquad y = 2\,\mathfrak{Im}(a^* b), \qquad z = |a|^2 - |b|^2$$

These coordinates satisfy $x^2 + y^2 + z^2 = 1$, confirming that every pure state lies on the surface of the unit sphere. Mixed states occupy the interior of the sphere, where $x^2 + y^2 + z^2 < 1$.

## Quantum gates as rotations
Single-qubit gates act on the Bloch sphere as rotations. The rotation gates $R_x(\theta)$, $R_y(\theta)$, $R_z(\theta)$ rotate the Bloch vector by angle $\theta$ around the $x$-, $y$-, and $z$-axes respectively. The Pauli gates $X$, $Y$, $Z$ are $\pi$ rotations about the corresponding axes. The Hadamard gate $H$ corresponds to a $\pi$ rotation about the axis halfway between $x$ and $z$, mapping $\lvert 0\rangle \leftrightarrow \lvert +\rangle$ and $\lvert 1\rangle \leftrightarrow \lvert -\rangle$.