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:

<br /> \Ket{Q}=\alpha\Ket{0}+\beta\Ket{1}\qquad(1)<br />

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]:

<br /> \Ket{0}<br />

<br /> \Ket{1}<br />

<br /> \Ket{Q}=\alpha\Ket{0}+\beta\Ket{1}<br />

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

Excursus: Vector representation and braket notation

<br /> \Ket{0}=\begin{bmatrix}<br /> 1 \\<br /> 0 \\<br /> \end{bmatrix}<br /> ;<br /> \Ket{1}=\begin{bmatrix}<br /> 0 \\<br /> 1 \\<br /> \end{bmatrix}\qquad(2)<br />
<br /> \Bra{0}=\begin{bmatrix}<br /> 1 & 0<br /> \end{bmatrix}<br /> ;<br /> \Bra{1}=\begin{bmatrix}<br /> 0 & 1<br /> \end{bmatrix}\qquad(3)<br /> 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:

<br /> |\alpha|^{2} + |\beta|^{2}=1\qquad(4)<br />

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].:

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

<br /> \Ket{+}<br />
<br /> =<br />

<br /> =<br />

<br /> \frac{1}{\sqrt{2}}<br />

<br /> \frac{1}{\sqrt{2}}\Ket{0}<br />
<br /> +<br />
<br /> +<br />
<br /> \frac{1}{\sqrt{2}}<br />

<br /> \frac{1}{\sqrt{2}}\Ket{1}<br />

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