Here you will learn the mathematics of representing quantum information and quantum computing operations.
Qubit
A qubit is represented by a two-dimensional complex vector usually denoted by \(|\psi\rangle\). Historically \(\psi\) is used as the wavefunction of a quantum particle like an electron or any other object that behaves quantum mechanically.
Ket and Bra
The \(|\rangle\) is called a ket introduced by Paul Dirac. It’s a way of distinguishing a column vector \(|v\rangle\) (ket) from its dual row vector \(\langle v|\) (called bra). The state of the qubit, \(|\psi\rangle\), can be explicitly written as a two component column vector
\[|\psi\rangle = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}\]where \(c_1\) and \(c_2\) are complex numbers. The qubit can also be represented by its dual row vector
\[\langle\psi| = \begin{pmatrix} c_1^* & c_2^* \end{pmatrix}\]These two representations are considered equal.
Inner product and normalization
Now, the elements of the state vector \(c_1\) and \(c_2\) cannot be any numbers as they are limited by the normalization condition. It is a postulate of quantum mechanics that the state of a qubit is normalized which means the magnitude of the state vector is unity. By definition the magnitude (or norm) of a complex vector \(\|\psi\|\) is equal to the square root of its inner product with its dual row vector.
\[\|\psi\| = \sqrt{\langle \psi | \cdot | \psi \rangle}\]The inner product \(\langle \psi | \cdot | \psi \rangle\) is often abbreviated as \(\langle \psi | \psi \rangle\). It is worth mentioning that the inner product is always defined between a bra and a ket. Hence, called bracket.
We can write the inner product explicitly in terms of \(c_1\) and \(c_2\)
Again, the normalization dictates that \(\sqrt{\langle\psi|\psi\rangle}=1\).
In other words
Rays
Let’s assume we have a qubit at a specific state \(\)
Single Qubit Gates
Gates are operations that change the state of qubits. That is what quantum computing means. You compute by applying gates on your qubits.
Multiple Qubits
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