Section 1.6 Linear Algebra in Quantum Computation
As mentioned in the section on Classical Computing Section 1.4, qubits can be in the state \(\ket{0}\) or \(\ket{1}\text{.}\) What’s more, these states can be represented with vectors.
\begin{equation*}
\ket{0} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}
\end{equation*}
and
\begin{equation*}
\ket{1} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}
\end{equation*}
This notation was first introduced by mathematician Paul Dirac and is known as "Dirac Notation" or "Bra-Ket Notation." In this notation, the \(\ket{}\) symbol represents a qubit state and is referred to as a "ket." At a glance, it can be seen that the vectors \(\ket{0}\) and \(\ket{1}\) are each normal and are orthogonal to each other. Additionally, any point in two dimensional space could be described with a linear combination of these two vectors, meaning they form a basis (we will discuss exactly which space they form a basis for in the following section). Put together, this means \(\ket{0}\) and \(\ket{1}\) form an orthonormal basis. All qubits must be normalized in order to be expressed properly.
Systems with multiple qubits are described by vectors in higher dimensions, which will be discussed later.
