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}

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

\Ket{0}

\Ket{1}

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

Here the so-called Bra-Ket notation is used. This notation is a simplified representation for column vectors | 〉 (Bra) and ⟨ | for row vectors (Ket), respectively.

Excursus: Vector representation and braket notation.

\Ket{0}=\begin{bmatrix} 1 \\ 0 \end{bmatrix}; \Ket{1}=\begin{bmatrix} 0 \\ 1 \end{bmatrix}

\Bra{0}=\begin{bmatrix} 1 & 0 \end{bmatrix}; \Bra{1}=\begin{bmatrix} 0 & 1 \end{bmatrix}

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

|\alpha|^{2} + |\beta|^{2}=1

|α|² and |β|² indicate the probabilities of which state the QuBit is currently in. A possible state for a QuBit where the probability for |0> and |1> is 50% is |+>. This state is shown in the next figure [1]:

\Ket{Q}=\Ket{+}=\frac{1}{\sqrt{2}}\Ket{0} + \frac{1}{\sqrt{2}}\Ket{1}

After a measurement the QuBit takes the state of the measurement result, i.e. if one measures 0 with probability |α|², afterwards the QuBit has state |0>.

Sources

[1] M. Ellerhoff. Mit Quanten Rechnen. ISBN 978-3-658-31221-3

Version: 17.06.2024