Qubits

The fundamental unit of information for classical computers is the bit - which can be one of two numbers - zero or one. In contrast, the fundamental unit of information for quantum computers are complex vectors.

Definition: Qubit

A quantum bit (qubit) is a two-dimensional complex vector whose Euclidean norm is $1$.

Bra-Ket Notation

Quantum States are pretty much always expressed via bra-ket notation.

Note: Euclidean Norm

The condition that the Euclidean norm must be equal to $1$ reflects a fundamental truth about the physical nature of quantum systems.

Notation: Common States

Four common qubit states have reserved names and notations:

Name	Notation	State
Zero	$|0⟩$	$\left[\begin{array}{c}1\\ 0\end{array}\right]$
One	$|1⟩$	$\left[\begin{array}{c}0\\ 1\end{array}\right]$
Plus	$|+⟩$	$\left[\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right]$
Minus	$|-⟩$	$\left[\begin{array}{c}\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}\end{array}\right]$
i	$|\mathrm{i}⟩$	$\left[\begin{array}{c}\frac{1}{\sqrt{2}}\\ \mathrm{i}\frac{1}{\sqrt{2}}\end{array}\right]$
Minus i	$|-\mathrm{i}⟩$	$\left[\begin{array}{c}\frac{1}{\sqrt{2}}\\ -\mathrm{i}\frac{1}{\sqrt{2}}\end{array}\right]$

Bloch Sphere

A single qubit can be visualized intuitively as an arrow pointing from the origin to a point on the so-called Bloch sphere.

Measurement

Measurement is one of the fundamental operations we can perform on qubits. Given a qubit $|q⟩\in {\mathbb{C}}^2$ or $⟨q|\in {\mathbb{C}}^{1\times 2}$, we first choose an orthonormal basis $\{|b_1⟩,|b_2⟩\}$ or $\{⟨b_1|,⟨b_2|\}$ for ${\mathbb{C}}^2$ or ${\mathbb{C}}^{1\times 2}$, respectively. We then write $|q⟩$ or $⟨q|$ as a linear combination of the corresponding basis:

$$|q⟩=\alpha |b_1⟩+\beta |b_2⟩\text{or}⟨q|=\alpha ⟨b_1|+\beta ⟨b_2|$$

Definition: Superposition

If $\alpha \ne 0$ and $\beta \ne 0$, we say that $|q⟩$ is a superposition of $|b_1⟩$ and $|b_2⟩$ (or analogously for $⟨q|$, $⟨b_1|$ and $⟨b_2|$).

Definition: Pure State

If $\alpha =0$ or $\beta =0$ but $\alpha \ne \beta$, we say that $|q⟩$ is a pure state (or analogously for $⟨q|$).

Since the basis is orthonormal, the coefficients $\alpha ,\beta \in \mathbb{C}$ are subject to the following condition in order for the norm of the qubit to be equal to $1$:

$$|\alpha {|}^2+|\beta {|}^2=1$$

When we measure $|q⟩$ with respect to $\{|b_1⟩,|b_2⟩\}$, it randomly changes to either $|b_1⟩$ or $|b_2⟩$:

• $|q⟩$ can become $|b_1⟩$ with probability $|\alpha {|}^2$.
• $|q⟩$ can become $|b_1⟩$ with probability $|\beta {|}^2$.

We also say that $|q⟩$ collapses into $|b_1⟩$ or $|b_2⟩$. The situation is analogous for $⟨q|$ and $\{⟨b_1|,⟨b_2|\}$.