Quantum Computing and Quantum Information

The Quantum Bit

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 follows¹

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 sphere²

Footnotes

Footnotes

1. Aaronson, Scott. "Introduction to quantum information science II lecture notes." (2022). Figure 3.1 ↩

2. You can look at the following to illustrate this:https://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/blochsphere/blochsphere.html. ↩