# Qubit

A Qubit (or QBit) is a unit of measure used in quantum computing.

Like a bit in normal (non-quantum) computing, a Qubit has two distinct states, 0 state and the 1 state. However, unlike the normal bit, a qubit can have a state that is somewhere in-between, called a "superposition."

If you try measure a qubit that is in a superposition, the qubit will change, and become one of two states. The state the qubit changes to depends on how it is measured. For simplicity, let's assume we are measuring in a way that will make the qubit change to either a 0 state or a 1 state.

A qubit can be represented as a 2-element column vector.

A qubit in the 0 state looks like ${\begin{bmatrix}1\\0\end{bmatrix}}$ .

A qubit in the 1 state looks like ${\begin{bmatrix}0\\1\end{bmatrix}}$ .

In general, a qubit state will look like ${\begin{bmatrix}\alpha \\\beta \end{bmatrix}}$ , where $|\alpha |^{2}+|\beta |^{2}=1$ .

α and β are called amplitudes. They can be complex numbers. Each state has an amplitude.

By squaring a state's amplitude, you can get the probability of measuring that state.

Each state can also have a phase. The phase is part of the amplitude and is what can make the amplitude a complex number.

A state's phase is like how much that state has rotated. The angle of phase is usually represented as either Φ or φ. Let's use φ.

φ can go from 0 to $2\pi$ radians. The angle sort of goes into an Euler identity, where instead of $e^{i\pi }$ , the $\pi$ gets substituted with the angle φ. The state's phase becomes $e^{i\varphi }$ .

This expression $e^{i\varphi }$ is a phase factor that becomes part of a state's amplitude. It gets multiplied with the amplitude.

A phase angle of 0 makes the amplitudes positive real numbers, since $e^{i0}=1$ .

A phase angle of $\pi$ makes the amplitudes negative real numbers, since $e^{i\pi }=-1$ . (This is Euler's identity)

A phase angle of ${\frac {\pi }{2}}$ makes the amplitudes positive imaginary numbers, since $e^{i\pi /2}=i$ .

A phase angle of ${\frac {3\pi }{2}}$ makes the amplitudes negative imaginary numbers, since $e^{i3\pi /2}=-i$ .

Beyond 0 and $2\pi$ , the phase angle just wraps back around again, since it is just a rotation.

An example qubit may look like ${\frac {1}{\sqrt {2}}}{\begin{bmatrix}1\\-1\end{bmatrix}}$ . There is a 50% chance of measuring a 0 or a 1. There is a phase of 1 on the 0 state's amplitude. There is a phase of -1 on the 1 state's amplitude.

Qubits are generally written as kets, which look like $|\psi \rangle$ . Kets are part of Bra-Ket notation, also known as Dirac notation. Kets are a way of saying column vector.

The 0 and 1 state are written as $|0\rangle$ and $|1\rangle$ respectively.

A general qubit in ket notation will be written as $|\psi \rangle =\alpha |0\rangle +\beta |1\rangle$ .

This equation is exactly the same as $|\psi \rangle ={\begin{bmatrix}\alpha \\\beta \end{bmatrix}}$ , since $\alpha |0\rangle +\beta |1\rangle =\alpha {\begin{bmatrix}1\\0\end{bmatrix}}+\beta {\begin{bmatrix}0\\1\end{bmatrix}}={\begin{bmatrix}\alpha \\0\end{bmatrix}}+{\begin{bmatrix}0\\\beta \end{bmatrix}}={\begin{bmatrix}\alpha \\\beta \end{bmatrix}}$ 