Enigma 002 : The Four Hair Colours

Enigma 002 : The Four Hair Colours#

Are you ready to shift into second gear? At a carnival, Alice, Bob, Charlie, and Dahlia decide to enter a contest to win an inter galactic trip. The challenge? Each person must guess their own hair color. The team will have to work together to find the right strategy to determine the orange or indigo color of their hair. You will apply a new concept, the principle of parity.

Make sure to watch the following video before getting started with this exercise:

Run the cell below to install the necessary packages.

import sys
!{sys.executable} -m pip install qiskit==1.1.1
!{sys.executable} -m pip install qiskit_aer==0.14.2
!{sys.executable} -m pip install pylatexenc==2.10

# Import necessary modules
import numpy as np
from qiskit import QuantumCircuit

Exercise 1 - Code writing#

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The enigma video presented a quantum circuit to solve the hair color problem with 4 people. Here’s an example of the code associated with the circuit:

problem_qc = QuantumCircuit(8)

problem_qc.h(0)
problem_qc.h(1)
problem_qc.h(2)
problem_qc.h(3)
problem_qc.barrier(0, 1, 2, 3, 4, 5, 6, 7)

# You check if the number of indigo hair color in front of you is even or odd
problem_qc.cx(1,4)
problem_qc.cx(2,4)
problem_qc.cx(3,4)
problem_qc.barrier(0, 1, 2, 3, 4, 5, 6, 7)

# Everyone takes note of the answer
problem_qc.cx(4,5)
problem_qc.cx(4,6)
problem_qc.cx(4,7)
problem_qc.barrier(0, 1, 2, 3, 4, 5, 6, 7)

# Bob checks the parity of the hair color in front of him
problem_qc.cx(2,5)
problem_qc.cx(3,5)
problem_qc.barrier(0, 1, 2, 3, 4, 5, 6, 7)

# Charlie and Dahlia take note of the answer
problem_qc.cx(5,6)
problem_qc.cx(5,7)
problem_qc.barrier(0, 1, 2, 3, 4, 5, 6, 7)

# Charkie checks the parity of Dahlia's hair color
problem_qc.cx(3,6)
problem_qc.barrier(0, 1, 2, 3, 4, 5, 6, 7)

# Dahlia takes note of Charlie's hair color
problem_qc.cx(6,7)

Can you adapt the circuit for 6 people?

problem_qc = QuantumCircuit(12)

### Start writing your code here. ###


# Visualize the circuit
problem_qc.draw('mpl')

Exercise 2 - Code writing#

Simplify the code with a for loop. Can you write a circuit for any number of people using a for loop?

nb_players = 5

nb_qubits = nb_players*2

problem_qc = QuantumCircuit(nb_qubits)

for i in range(nb_players):
    problem_qc.h(i)

start_qubit = 1

### Add the rest of the code here. ###


# Visualize the circuit
problem_qc.draw('mpl')

Exercise 3 - Quick quiz#

The goal of this enigma is to determine all the hair colors with the highest probability. Thus, what is the condition for 100% of the players to correctly guess their hair color?

Exercise 4 - Quick quiz#

In all of the quantum circuits for this enigma, one qubit is never entangled. Which one is it?

Exercise 5 - Quick quiz#

Run the following code to execute the quantum circuit for 4 players on a simulator.

Note

When running quantum algorithms, simulators are often used to test the quantum circuits without monopolizing the ressources of a real quantum computer. Simulators are classical computers that mimic the behaviors of quantum computers.

# Quantum circuit for 4 players
nb_players = 4
nb_qubits = nb_players*2

problem_qc = QuantumCircuit(nb_qubits)

for i in range(nb_players):
    problem_qc.h(i)

start_qubit = 1

for j in range(nb_players, nb_qubits-start_qubit):
    problem_qc.barrier()
    for i in range(start_qubit, nb_players):
        problem_qc.cx(i, j)
    problem_qc.barrier()
    for k in range(j+1, nb_qubits):
        problem_qc.cx(j, k)
    start_qubit = start_qubit+1

# Execute the circuit and draw the histogram
measured_qc = problem_qc.measure_all(inplace=False)
simulator = AerSimulator()
result = simulator.run(transpile(measured_qc, simulator), shots=1024).result()
counts = result.get_counts(measured_qc)
plot_histogram(counts)

Can you explain the significance of the first four qubits (starting from the right) in any given measured state?
Can you explain the significance of the last four qubits (starting from the right) in any given measured state?