CNOT Gate

CNOT GATE

Principle

CNOT gate means controlled NOT gate. It is a quantum logic gate which plays a vital role in the designing of quantum computers.

The CNOT gate will have two‒qubit operation, wherein the first qubit is referred as the control qubit and the second qubit is referred as the target qubit.

Symbolic Representation

The Symbolic representation of CNOT gate is as shown in Fig. 4.5.

Concept

A CNOT gate basically implements a reversible Ex‒OR. It can be used to generate entanglement. The CNOT gate can be logically represented as shown in Fig. 4.6.

From Fig. 4.6 we can see that the control (x) and the target (y) are shown as two horizontal lines. Here, we can also notice that the output 'y' depends on the input source 'a' and is shown by an interconnecting vertical line from 'a' to 'y' and to one of the inputs of the EX‒OR gate i.e., the target input 'b' as shown in Fig. 4.5.

Logical Operation

Inputs

The input 'a' is typically called the source, and input 'b' is known as the target input.

Here, the output 'x' depends on input 'a' i.e.,

(i) When, the source a = 0, then control x= 0, i.e, x takes the value of source.

Similarly, (ii) when the source a = 1, then the control x = 1 as shown in truth table.

Thus, the source is also called the control input and controls the application of the NOT operation on the target input.

Outputs

The Output of CNOT gate is 

Here, the output 'y' depends on the source 'a' and target 'b', i.e.,

(i) When the source 'a' value is '0' then the output 'y' will be equal to the value of 'b'.

(ii) When the source 'a' value is 1, then output 'y' will have the inverse value of 'b'

Therefore, we can say that 'y' is the inverse of target 'b'. when the source 'a' is 1 otherwise the output y will be equal to b.

Truth Table

The truth table for CNOT gate is obtained based on the above phenomana and is as shown in the Table. 4.1

Table 4.1 Truth Table for CNOT gate

Case (i) When a=0 and b = 0 (or) 1

Here, When a = 0, then y will be equal to 'b'

Case (ii) When a = 1 and b = 0 (or) 1

Here, when the source 'a' is equal to '1' and when the target 'b' value is zero, then the output y will have the inverse of target 'b' i.e., zero will become 1 (or) 1 will become zero..

In other words, whether 'y' get the inverted value of the target 'b' (or not), it is controlled by source 'a'.

Hence, because of this reason, CNOT is known as a controlled NOT gate.

It can be seen from the truthtable that, in CNOT gate the inputs can be uniquely determined from the outputs, thus verfying the reversibility of the gate.

Matrix representation of CNOT gate

The action of CNOT gate can be represented by the matrix

Quantum Entanglement

From the truth table, we can see that, if the target qubit is |0> and the control qubit is either |0> and |1> then the output target 'y' takes the value of the control qubit, i.e, it, becomes a copy of the control qubit, but the control qubit itself does not change.

However, a super position in the control qubit results in the entaglement of control and target qubits.

Thus, When two or more particles link up in a certain way, no matter how far apart they are in space, their states remains linked. That means they share a common, united quantum state.

So observations of one of the particles can automatically provide information about the other entangled particles, regardless of the distance between them. And any action to one of these particles will invariably impact the other in the entangled system.

This united state is known as quantum entanglement.