In today's computers, the basic unit of information is a bit, which can be either 0 or 1.
Similarily, in Quantum computing the basic unit of information is a quantum bit or qubit.
In contrast to a classical bit, a qubit can assume states of the form
where and are called amplitude with
In contrast to a classical bit, which only has either the value or , a qubit can be in a so-called superposition state in between.
Yet, it is not possible to read this state directly from a qubit. To obtain information about a qubit it has to be measured, which destroys the superposition.
When measuring a qubit, the state is observed with probability and the state with probability .
The state of a qubit is considered to be a two-dimensional vector with complex entries.
The so-called state vector is:
This can be specified as a linear combination of the two-dimensional standard basis vectors:
The notation or is called Dirac- or Bra-Ket- notation (from ``bracket'', bra: , ket: ).
If , then , i.e., the complex conjugate row vector.
Graphical Representation of a Qubit
To represent a qubit graphically, one would naively need four dimensions, since and are two complex numbers, each with a real and imaginary part.
However, if we assume that and are real numbers, we can represent a qubit as follows^{.css-6phhhi{transition-property:var(--chakra-transition-property-common);transition-duration:var(--chakra-transition-duration-fast);transition-timing-function:var(--chakra-transition-easing-ease-out);cursor:pointer;outline:2px solid transparent;outline-offset:2px;position:relative;color:var(--chakra-colors-brand-600);-webkit-text-decoration:none;text-decoration:none;width:-webkit-fit-content;width:-moz-fit-content;width:fit-content;}.css-6phhhi:focus-visible,.css-6phhhi[data-focus-visible]{box-shadow:var(--chakra-shadows-outline);}.css-6phhhi:hover,.css-6phhhi[data-hover]{color:var(--chakra-colors-brand-600);}.css-6phhhi:hover::before,.css-6phhhi[data-hover]::before{-webkit-transform:scaleX(1);-moz-transform:scaleX(1);-ms-transform:scaleX(1);transform:scaleX(1);}.css-6phhhi::before{content:"";position:absolute;display:block;width:100%;height:2px;bottom:calc(var(--chakra-space-0-5) * -1);left:0px;background-color:var(--chakra-colors-brand-600);-webkit-transform:scaleX(0);-moz-transform:scaleX(0);-ms-transform:scaleX(0);transform:scaleX(0);transform-origin:top left;-webkit-transition:-webkit-transform 0.3s ease;transition:transform 0.3s ease;}1}
We use the value of on the x-axis and the value of on the y-axis.
As , qubits always lie exactly on the dotted circle.
For a qubit with complex amplitudes, i.e. , we can still represent it with only three dimensions using the so-called Bloch sphere:
Through transformations, we can convert the qubit formula from the previous section into the following form
where and .
and are sufficient to completely describe a rotation on the Bloch sphere, where is also denoted as the (relative) phase of a qubit. The possible states a qubit can take are precisely represented by the surface of the Bloch sphere^{2}