Qubit

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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 .

A qubit in the 1 state looks like .

In general, a qubit state will look like , where .

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 radians. The angle sort of goes into an Euler identity, where instead of , the gets substituted with the angle φ. The state's phase becomes .

This expression 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 .

A phase angle of makes the amplitudes negative real numbers, since . (This is Euler's identity)

A phase angle of makes the amplitudes positive imaginary numbers, since .

A phase angle of makes the amplitudes negative imaginary numbers, since .

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

An example qubit may look like . 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 . 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 and respectively.

A general qubit in ket notation will be written as .

This equation is exactly the same as , since