What is Quantum Computing? (Part 4)

In my previous post, I introduced 2 operators, which are acting on 1 qubit. In this post, I will introduce another crucial operator CNOT, which acts on 2 qubits.

Controlled-NOT (CNOT) is a binary operator. The first qubit is the control qubit and the second is the target qubit. If the control qubit is in the state |0>, then we apply the NOT operator (X) to the target qubit.

For example, if we apply CNOT on the state |10>, it will become |11>, where |00> is the 4x1 vertical vector (1,0,0,0), |01> is (0,1,0,0), |10> is (0,0,1,0), and |11> is (0,0,0,1).

You can apply the matrix multiplication of the above CNOT matrix on the 4x1 vertical vector (0,0,1,0), and it will give you the vector (0,0,0,1).

CNOT is an important qubit operator because we can use the CNOT gate to entangle 2 qubits in QC. Entanglement is a special quantum property. By definition, 2 states are in entanglement if the measurement of one is correlated with the other in a way that is stronger than correlations in the classical world. In other words, the states are not separable.

Next, we will try to combine the Hadamard operator (H) with CNOT. Let’s start with 2 qubits q0 and q1. Both are in the state of |0>. Then we apply H on q0, that is H(q0), and then apply CNOT on q0 and q1. The first H operator will produce the superposition of states. H(q0) = (1/sqrt(2))|0>+(1/sqrt(2))|1>. The second CNOT operator entangles 2 qubits into a non-separable state: (1/sqrt(2))|00> + (1/sqrt(2))|11>

You may notice that the output 2 qubits must be the same 00 or 11, and each will have 0.5 chance to occurs. Therefore it is a perfectly random and perfectly correlated pair of qubits. It is called Bell state.