Superdense Coding-Math & Implementation

For classical communication there is a limit on how much information can be send with a given number of bits. We can send no more than n bits of information using n bits. However, there are ways that we can push the boundary which are not possible classically. Hence, we can use superdense coding using quantum computer.

Definition:

It is a procedure that allows someone to send two classical bits to another party using just a single qubit of information.

Here we will use 4 types of gates while implementing this procedure.

1. H gate -> This puts a qubit into superposition.
2. X gate -> X|0> -> |1> & X|1> -> |0>
3. CX gate -> It takes two qubits one is control and another is target. If the control bit is 0 , it does nothing to the target qubit, otherwise if the control bit is 1, X gate is applied on target qubit.
4. Z gate -> It only change the phase of a qubit when it is in |1> state i.e, Z|0> -> |0> , Z|1> -> -|1>

The procedure is implemented as follows,

For keeping track , we have put 4 track point here , namely W1,W2,W3,W4.

Now let's see how this procedure works mathematically,

Now let's see its coding part, here we will use Qiskit for the implementation,

from qiskit import QuantumCircuit,Aer,execute

qc_3rd_party = QuantumCircuit(2,2)

qc_3rd_party.h(1)

qc_3rd_party.cx(1,0)


message_list = ['00', '01', '10', '11']

qc_Alice = QuantumCircuit(2,2)

for msg in message_list:

    if msg[-2]=='1':

        qc_Alice.z(1)

    if msg[-1]=='1':

        qc_Alice.x(1)

    qc_Bob = QuantumCircuit(2,2)

    qc_Bob.cx(1,0)

    qc_Bob.h(1)

    qc_Bob.measure([0,1],[0,1])

    complete_qc = qc_3rd_party.compose(qc_Alice.compose(qc_Bob))

    backend = Aer.get_backend('qasm_simulator')

    print(f'For message = {msg}\n')

    count = backend.run(complete_qc).result().get_counts()

    print(count,"\n")

    qc_3rd_party.data=[]

    qc_Alice.data=[]

    qc_Bob.data=[]

The output of the code is ,

Comments

[Alex Roor]

I admire how you backed up your arguments with solid evidence and real-world examples. It makes your points much more compelling.