# Quantum Computing and Quantum Information

# Multi-Qubit Systems

In order to create circuits with several qubits, we first need to
describe the mathematical state of more than one qubit. As with the
representation of classical bits, it should be possible to represent
*register* or *quantum register*.
Theoretically quantum registers can represent an infinite number of
states, but when measured in a certain basis, there are only

## Quantum Register

The combination of two qubits into a register

Here, for the amplitudes

To simplify this notation, the amplitudes

As notation convention, the tensor product of qubits

If a quantum register is measured, the state

For the formal definition of registers with

**Definition**
A quantum register with

The measurement of a quantum register therefore results in the state

For clarity, the binary representation is preferred in the following, so the basis states of a 2-qubit register are:

One can see that in the vector representation there is always exactly one 1 at the position (starting from 0) that corresponds to the binary number.

## Simple Multi-Qubit Gates

To transform a quantum register to a new state, gates can be applied to
multiple qubits. In the following circuit, the state

For example, we can combine the single qubit gates presented in the previous section as QASM code:

Diagram

`OPENQASM 2.0; include "qelib1.inc"; qreg q[2]; u(pi,pi/2,pi/4) q[0]; u(pi,pi/2,pi/8) q[1];`

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The tensor product of the two 1-qubit matrices then results in the 2-qubit gate, i.e:

For example, for the

It is also possible to apply multi-qubit gates only to parts of a register:

If no operation is performed on qubit

Diagram

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

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The result of the tensor product is:

Using the example of the

And the corresponding matrix is calculated in the same way:

If this matrix is applied to a state vector (e.g.,

In the QASM language, this looks as follows:

Diagram

`OPENQASM 2.0; include "qelib1.inc"; qreg q[3]; x q[1]; x q[2];`

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## Controlled Multi-Qubit Gates

There are special multi-qubit gates that require several qubits as input
and cannot be represented as a tensor product of individual gates. This
allows that the state of one qubit can influence the state of another.
An example of this is the *Controlled-NOT* gate, or short *CNOT* gate:

Diagram

`OPENQASM 2.0; include "qelib1.inc"; qreg q[2]; cx q[0],q[1];`

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The so-called *control qubit* *target qubit*

If the control qubit is in a basis state

If we apply this matrix to the state vector

The principle of controlling gates is not limited to the X gate, it can be combined with all other gates, e.g:

Diagram

`OPENQASM 2.0; include "qelib1.inc"; qreg q[2]; cy q[0],q[1]; cz q[0],q[1]; ch q[0],q[1]; cu(pi,pi/2,pi/4,pi/8) q[0],q[1];`

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In general, the matrix of a controlled gate

Similarly, in Braket notation a controlled gate

The principle of controlled gates is not limited to two qubits, but can
also be used with more qubits, e.g. in the form of the so-called
*controlled CNOT* or *CCNOT* gate, which is also known as the *Toffoli
gate* named after its inventor.

Diagram

`OPENQASM 2.0; include "qelib1.inc"; qreg q[3]; ccx q[0],q[1],q[2];`

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The NOT gate is only applied to the target qubit if both control qubits
are

Similarly in Braket notation, we can define multi-controlled gates as