# The QuBit

## The QuBit

Computers use switching states of transistors that can assume the classical state 0 or 1 to store information or perform computing operations. In a quantum computer, the smallest unit of information is called a QuBit. Unlike the bit, which assumes either one or the other state (0 or 1), a QuBit Q consists of an arbitrary superposition of these two states:

\Ket{Q}=\alpha\Ket{0}+\beta\Ket{1}\qquad(1)

α and β are (in general) complex numbers indicating the fraction for 0 and 1, respectively. The state 0 resp. 1 and the state Q are represented as vectors on the surface of a Bloch sphere [1]:

\Ket{0}

\Ket{1}

\Ket{Q}=\alpha\Ket{0}+\beta\Ket{1}

## Excursus: Vector representation and braket notation

\Ket{0}=\begin{bmatrix}

1 \\

0 \\

\end{bmatrix}

;

\Ket{1}=\begin{bmatrix}

0 \\

1 \\

\end{bmatrix}\qquad(2)

\Bra{0}=\begin{bmatrix}

1 & 0

\end{bmatrix}

;

\Bra{1}=\begin{bmatrix}

0 & 1

\end{bmatrix}\qquad(3)

The first row in the column vectors or first column in the row vectors refers to the value |0〉 and the second row or column refers to the value |1〉(see also representation on the Bloch sphere).

The values for α and β are normalized so that their square results in the value 1:

|\alpha|^{2} + |\beta|^{2}=1\qquad(4)

In this case, |α|^{2} and |β|^{2} indicate the probabilities of which state the QuBit is currently in. A permissible state for a QuBit is shown in the next figure, where the probability of encountering 0 or 1 is 50% to 50% [1].:

\Ket{Q}=\frac{1}{\sqrt{2}}\Ket{0} + \frac{1}{\sqrt{2}}\Ket{1}\qquad (5)

\Ket{+}

=

=

\frac{1}{\sqrt{2}}

\frac{1}{\sqrt{2}}\Ket{0}

+

+

\frac{1}{\sqrt{2}}

\frac{1}{\sqrt{2}}\Ket{1}

This state is abbreviated as |+〉.

*Question: When does the QuBit “decide” for one of the two states?*

### Measurement of a QuBit state

Once the state of a QuBit is determined by a measurement, it transitions to one of the two classical states 0 or 1 with probability |α|^{2} or |β|^{2}, respectively.

## Sources

[1]: M. Ellerhoff, Mit Quanten Rechnen. Wiesbaden: Springer Spektrum , ISBN 978-3-658-31221-3