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 on a qubit is described with

To calculate the new state , the matrix corresponding to the gate is multiplied from the left by the initial state of the qubit:

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 around the x-axis of the Bloch sphere. In a circuit, it is represented as follows:

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

The representation as a unitary matrix is

Thus, when applied to and this results in:

In Dirac notation, we write:

) as in the previous section. For this, we sum the matrices of all possible outputs for the basis vectors:

We can also derive this matrix by correlating the inputs and outputs in Braket notation () as in the previous section. For this, we sum the matrices of all possible outputs for the basis vectors:

Y Gate

In a Bloch sphere, the Y gate describes the rotation by the angle around the y-axis.

In a circuit, it is represented as follows:

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

The representation as a unitary matrix is: and this results in:

Applied to and this results in:

In Dirac notation, we write:

around the z-axis.

Z gate

In a Bloch sphere, the Z gate describes the rotation by the angle around the z-axis.

In a circuit, it is represented as follows:

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

The representation as a unitary matrix is:

Applied to and this results in

In Dirac notation, we write:4-dimensional complex vector space of all complex matrices. This means that every complex matrix can be represented by a linear combination of these four matrices. Further, the three Pauli gates are self-inverse, i.e.:

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 matrices. This means that every complex matrix can be represented by a linear combination of these four matrices. Further, the three Pauli gates are self-inverse, i.e.:

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 , because then the amplitudes of and are equal; a measurement would result in observing the qubit in state or with 50% probability respectively. In the Bloch sphere, the gate describes a rotation of around the z-axis, followed by a rotation of around the y-axis.

In a circuit, it is represented as follows: and we get

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

The unitary matrix is given as:

Applied to and we get

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 , and , defined as follows:

or is obtained with probabilities depending on the state of the qubit. This process can also be represented with a gate:

This results in the following rotation matrices:

In the circuit language, these gates look as follows:

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

Measurement

When measuring a qubit, the respective (binary) result or is obtained with probabilities depending on the state of the qubit. This process can also be represented with a gate:

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

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