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

Placeholder Image Alt

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:

Placeholder Image Alt

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



  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.

Quantum Computing and Quantum Information
The Quantum Gates

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