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 classical states with qubits. This combination is called (as in classical computer science) register or quantum register. Theoretically quantum registers can represent an infinite number of states, but when measured in a certain basis, there are only possible measurement results.

Quantum Register

The combination of two qubits into a register is the tensor product of the state vectors, :

Here, for the amplitudes and it still applies that:

To simplify this notation, the amplitudes and are combined to :

As notation convention, the tensor product of qubits can be combined to :

If a quantum register is measured, the state is obtained with probability .

For the formal definition of registers with qubits, it makes sense to switch from binary to decimal notation:

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 [^1] is transformed to using the multi-qubit operation :

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

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];

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

For example, for the gate, this results in

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

If no operation is performed on qubit , this corresponds to multiplication by the identity matrix (or ). The identity matrix can also be represented as a gate:

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

The result of the tensor product is:

Using the example of the gate, this works as follows:

And the corresponding matrix is calculated in the same way:

If this matrix is applied to a state vector (e.g., ), the desired result is obtained:

In the QASM language, this looks as follows:

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

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:

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

The so-called control qubit controls the application of the X gate to the target qubit .

If the control qubit is in a basis state , an X gate is applied to the target qubit only if the control qubit is . The state of the control qubit remains unchanged. For a control qubit in superposition, only the amplitudes of the basis states in which the control qubit is are affected. The matrix of the gate is:

If we apply this matrix to the state vector , for example, we obtain the result:

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

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];

In general, the matrix of a controlled gate on a state with as the control qubit and as the target qubit looks like this:

Similarly, in Braket notation a controlled gate is defined as:

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.

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

The NOT gate is only applied to the target qubit if both control qubits are . As with the CNOT, the state of the two control qubits remains unchanged. For the CCNOT gate we get the matrix

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