$\require{cancel} \newcommand{\Ket}[1]{\left|{#1}\right\rangle} \newcommand{\Bra}[1]{\left\langle{#1}\right|} \newcommand{\Braket}[1]{\left\langle{#1}\right\rangle} \newcommand{\Rsr}[1]{\frac{1}{\sqrt{#1}}} \newcommand{\RSR}[1]{1/\sqrt{#1}} \newcommand{\Verti}{\rvert} \newcommand{\HAT}[1]{\hat{\,#1~}} \DeclareMathOperator{\Tr}{Tr}$
Superdense Coding¶
First created in June 2018
Superdense Coding is not quantum teleportation. It is the reverse, and illustrated here as a general concept.
Alice is sending two classical bits to Bob. They pre-share a pair of entangled qubits $\Ket{\alpha}\otimes\Ket{\beta}$, expressed in this context as $\Ket{\psi}=\Ket{\alpha\beta}$.
$G=(H\otimes I)(cX)(U\otimes I)$
$= (H\otimes I) \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} u_{00} & 0 & u_{01} & 0\\ 0 & u_{00} & 0 & u_{01}\\ u_{10} & 0 & u_{11} & 0\\ 0 & u_{10} & 0 & u_{11} \end{bmatrix} =\frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 1 & 0 & -1 & 0\\ 0 & 1 & 0 & -1 \end{bmatrix} \begin{bmatrix} u_{00} & 0 & u_{01} & 0\\ 0 & u_{00} & 0 & u_{01}\\ 0 & u_{10} & 0 & u_{11}\\ u_{10} & 0 & u_{11} & 0 \end{bmatrix} =\frac{1}{\sqrt{2}} \begin{bmatrix} u_{00} & u_{10} & u_{01} & u_{11}\\ u_{10} & u_{00} & u_{11} & u_{01}\\ u_{00} & -u_{10} & u_{01} & -u_{11}\\ -u_{10} & u_{00} & -u_{11} & u_{01} \end{bmatrix} .$
$\Ket{\psi} =G\frac{1}{\sqrt{2}}(\Ket{00}+\Ket{11}) =G\frac{1}{\sqrt{2}}[1,0,0,1]^T =\frac{1}{2}[u_{00}+u_{11},~u_{10}+u_{01},~u_{00}-u_{11},~-u_{10}+u_{01}]^T .$
For $I,\qquad\qquad(u_{00}, u_{01}, u_{10}, u_{11})=(1,0,0,1),\quad\Ket{\psi}=[1,0,0,0]^T$.
For $X,\qquad\qquad(u_{00}, u_{01}, u_{10}, u_{11})=(0,1,1,0),\quad\Ket{\psi}=[0,1,0,0]^T$.
For $Z,\qquad\qquad(u_{00}, u_{01}, u_{10}, u_{11})=(1,0,0,-1),\quad\Ket{\psi}=[0,0,1,0]^T$.
For $Y'=ZX,\quad(u_{00}, u_{01}, u_{10}, u_{11})=(0,1,-1,0),\quad\Ket{\psi}=[0,0,0,1]^T$.
import time, datetime
from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister, QISKitError
from qiskit import available_backends, execute, register, get_backend
# import state tomography functions
from qiskit.tools.visualization import plot_histogram, plot_state, circuit_drawer
def run_job(qc, backend, shots=128):
# Run circuit qc on backend bknd
print('Job started at ' + time.strftime('%H:%M:%S, %d %b, %Y'))
print('Doing ' + str(shots) + ' shot(s) on backend ' + backend)
#job = execute(qc, backend=backend, shots=128, max_credits=3)
job = execute(qc, backend=backend, shots=shots, max_credits=3)
ts0 = datetime.datetime.now().replace(microsecond=0)
lapse = 0
interval = 3
while not job.done:
#print('Status @ {} seconds'.format(interval * lapse))
ts1 = datetime.datetime.now().replace(microsecond=0)
print('Lapse #' + str(lapse) + ' ' + str(ts1 - ts0) + ' so far')
print(job.status)
time.sleep(interval)
lapse += 1
print(job.status)
print('Job stopped at ' + time.strftime('%H:%M:%S, %d %b, %Y'))
return job
### Retry Point: Superdense Coding ###
q = QuantumRegister(2)
c = ClassicalRegister(2)
qc = QuantumCircuit(q, c)
# | Bob Alice >
# q[0] is Bob's; q[1] is Alice's
# Use U=Z as an example coding |10>
# Preparing the Bell State
qc.h(q[1])
qc.cx(q[1], q[0])
# 1/sqrt(2)(|00>+|11>)
# Alice codes |00>
# Alice codes |01>
#qc.x(q[1])
# Alice codes |10>
qc.z(q[1])
# Alice codes |11>
#qc.x(q[1])
#qc.z(q[1])
# Decoding | Bob Alice > = |10>
qc.cx(q[1], q[0])
qc.h(q[1])
circuit_drawer(qc)
job = execute(qc, backend='local_statevector_simulator')
job.result().get_statevector(qc)
qc.measure(q, c)
job = run_job(qc, 'local_qasm_simulator')
plot_histogram(job.result().get_counts(qc))
