In quantum information, a qubit is the fundamental unit of information, similar to how we use bits in our classical computers. I will talk about how we can realize a qubit in lab in some other article, for now let's jump into the mathematics of qubits. The state of a two-state quantum system can be defined by a qubit, which is represented as a unit-length complex vector in a 2D Hilbert space.

As we learned in linear algebra, an arbitrary vector can be written as a linear combination of orthonormal basis vectors in that space. By defining basis vectors, we can expand any arbitrary qubit in terms of these bases. Which is useful because now we only need to think about the basis vectors while doing gate operations.

There are two bases used very frequently: the Z-basis and the X-basis. Let’s not go into detail on why they’re called Z or X just yet—that’ll be covered later (hopefully :p).

• Z-basis: Represented by $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$, which we call the 0 and 1 states of a qubit

• X-basis: Represented by $\begin{bmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{bmatrix}$ and $\begin{bmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \end{bmatrix}$, which we call the + and - states of a qubit:

Since these are vectors and not just numbers, we use bra-ket notation, which is ubiquitous in quantum computing.
$$|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad |1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$

One thing you’ll want to remember is that the counterpart to a vertical "ket" vector is a horizontal row vector called a "bra" (like $\langle1|$). They are the transpose complex conjugates of each other. This conjugate transpose is typically denoted by a dagger ($\dagger$). So $(|1\rangle)^\dagger = \langle1|$, and the ket with nx1 dimensions changes to 1xn (n is 2 for qubit):
$$\begin{bmatrix} 0 \\ 1 \end{bmatrix}^\dagger = \begin{bmatrix} 0 & 1 \end{bmatrix}$$

Using the above knowledge, an arbitrary qubit $|\psi\rangle$ can be defined as:

$$|\psi\rangle = \alpha \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \beta \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} \alpha \ \beta \end{bmatrix}$$

where $\alpha,\beta \in \mathbb{C}$, and $|\alpha|^2 + |\beta|^2 = 1$ (since the qubit is always of unit length).

The last thing I’d like to cover is that we can redefine our state in a different basis. For example, we know that:

$$\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{bmatrix} + \frac{1}{\sqrt{2}}\begin{bmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \end{bmatrix}$$
$$\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{bmatrix} - \frac{1}{\sqrt{2}}\begin{bmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \end{bmatrix}$$

Substituting this into our definition of $|\psi\rangle$, we get:

$$|\psi\rangle = \frac{\alpha + \beta}{\sqrt{2}}\begin{bmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{bmatrix} + \frac{\alpha - \beta}{\sqrt{2}}\begin{bmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \end{bmatrix}$$

where $\alpha,\beta \in \mathbb{C}$, and $|\alpha|^2 + |\beta|^2 = 1$ for normalisation

I agree that this article was more abstract and mathematical than you probably had expected, but you can trust me when I say that it'll be really useful while learning about gates in quantum computing.