# Quantum Computing and Quantum Information

# Quantum Gates

Similar to classical computer science, we can describe operations on qubits with gates. Some of the most important ones are the three so-called **Pauli gates** (also known as *X-,Y- and Z-gates*) and the Hadamard gate.
Mathematically, the state transition is described with a **unitary matrix**, e.g:

This applies to simple gates on a single qubit as well as in more complex systems with several qubits.

The application of a gate

To calculate the new state

Geometrically, the application of a gate on a single qubit can be seen as a rotation of the state vector around some axis in the Bloch sphere.

# Pauli Gates

## X Gate

The X gate, or NOT gate, behaves similarly to a *NOT* gate on a
classical bit. It describes a rotation with angle

Diagram

`OPENQASM 2.0; include "qelib1.inc"; qreg q[1]; x q[0];`

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The representation as a unitary matrix is

Thus, when applied to

In Dirac notation, we write:

We can also derive this matrix by correlating the inputs and outputs in
Braket notation (

## Y Gate

In a Bloch sphere, the Y gate describes the rotation by the angle

In a circuit, it is represented as follows:

Diagram

`OPENQASM 2.0; include "qelib1.inc"; qreg q[1]; y q[0];`

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The representation as a unitary matrix is:

Applied to

In Dirac notation, we write:

## Z gate

In a Bloch sphere, the Z gate describes the rotation by the angle

In a circuit, it is represented as follows:

Diagram

`OPENQASM 2.0; include "qelib1.inc"; qreg q[1]; z q[0];`

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The representation as a unitary matrix is:

Applied to

In Dirac notation, we write:*4-dimensional complex vector space* of all complex

## Characteristics of Pauli Gates

Together with the identity matrix, the Pauli gates form a basis of the
*4-dimensional complex vector space* of all complex

# Hadamard Gate

The Hadamard gate, named after the French mathematician Jacques
Hadamard, plays an important role in quantum computing because it can
put a qubit from a "classical" basis state to a *superposition* state.
For example, an equal superposition is obtained by applying the Hadamard
gate to the state

In a circuit, it is represented as follows:

Diagram

`OPENQASM 2.0; include "qelib1.inc"; qreg q[1]; h q[0];`

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The unitary matrix is given as:

Applied to

In Dirac notation, we write:

# Arbitrary Rotation Gates

A qubit can also be rotated by any angle and axis in order to "reach"
all possible states on the Bloch surface. There are different sets of
so-called universal gate sets in the literature which are sufficient for
this. We use the rotation gates

This results in the following rotation matrices:

In the circuit language, these gates look as follows:

Diagram

`OPENQASM 2.0; include "qelib1.inc"; qreg q[1]; rx(pi/2) q[0]; ry(pi/2) q[0]; rz(pi/2) q[0];`

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# Measurement

When measuring a qubit, the respective (binary) result

Diagram

`OPENQASM 2.0; include "qelib1.inc"; qreg q[1]; creg c[1]; measure q[0] c[0];`

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There is no matrix representation -- because measuring is not an operation like any other: it reads a bit of classical information from a qubit. Furthermore, in contrast to the previous gates, measurements are not reversible. In circuits, we draw a double line to distinguish classical bits from qubits.

More on measurements is described in QuBit Measurements