​Quantum Coding

The code for Shor's algorithm involves manipulating qubits to perform modular exponentiation and then applying a quantum Fourier transform to find the period of the modular exponentiation function. This period can then be used to efficiently find the prime factors of the original number. While this is a simplified explanation, the underlying quantum code for Shor's algorithm demonstrates the power of quantum computing to solve complex problems in a fraction of the time it would take a classical computer.

 

Quantum code for algorithms like Shor's demonstrates the potential of quantum computing to revolutionize fields such as cryptography and drug discovery, by solving problems that are currently intractable with classical computers.

 

The quantum Fourier transform (QFT) is a fundamental operation in quantum computing that acts on the amplitudes of a quantum state. It is the quantum analog of the classical discrete Fourier transform (DFT) and plays a central role in many quantum algorithms, including Shor's algorithm for factoring large numbers.

 

The QFT transforms a quantum state consisting of N qubits into its Fourier transform. Mathematically, it can be defined as:

 

QFT = (1/√(2^n)) * Σx Σy exp(2 * π * i * x * y / N) |y⟩⟨x|

 

Where N is the number of basis states, n is the number of qubits, x and y are integers from 0 to N-1, and |y⟩ and |x⟩ represents the quantum states.

 

The QFT is implemented using a sequence of quantum gates, such as Hadamard gates, phase shift gates, and controlled phase shift gates. The exact gate sequence depends on the number of qubits and the specific form of the QFT algorithm being used.

 

The QFT is primarily used in quantum algorithms that require the manipulation of frequency information, such as factoring large numbers or solving the discrete logarithm problem. It has also found applications in other fields, such as signal processing and simulation of physical systems.

 

It is important to note that the QFT is a complex operation and requires a good understanding of quantum computing principles to fully comprehend and implement.

​

Shor's algorithm involves manipulating qubits to perform modular exponentiation and then applying a quantum Fourier transform to find the period of the modular exponentiation function. This period can then be used to efficiently find the prime factors of the original number.

 

```python

import math

from qiskit import BasicAer, QuantumCircuit, execute

 

def shor_algorithm(N):

    # Step 1: Choose a random integer 'a' (1 < a < N)

    a = 2  # You can change this to a randomly chosen 'a'

 

    # Step 2: Determine the number of qubits needed to represent N

    n = math.ceil(math.log2(N))

 

    # Step 3: Create a quantum circuit with n qubits and an additional auxiliary qubit

    circuit = QuantumCircuit(n+1, n)

 

    # Step 4: Apply Hadamard gate to the auxiliary qubit

    circuit.h(n)

 

    # Step 5: Apply a series of controlled modular exponentiation gates

    for qubit in range(n):

        circuit.append(modular_exponentiation(N, a, 2 ** qubit), [qubit] + list(range(n, n+1)))

 

    # Step 6: Apply quantum Fourier transform to the register consisting of the n control qubits

    circuit.append(quantum_fourier_transform(n).to_gate(), list(range(n)))

 

    # Step 7: Measure the control qubits

    circuit.measure(range(n), range(n))

 

    # Step 8: Simulate the circuit using a classical simulator

    backend = BasicAer.get_backend('qasm_simulator')

    job = execute(circuit, backend, shots=1)

    result = list(map(int, job.result().get_counts(circuit).keys()))[0]

 

    # Step 9: Compute the estimated value for the period

    r = continued_fractions(result / (2 ** n))

 

    return r

 

def modular_exponentiation(N, a, power):

    # Create a quantum circuit for modular exponentiation by a^power mod N

    circuit = QuantumCircuit(2, 1)

 

    # Apply repeated squaring and modular reduction

    for _ in range(power):

        circuit.x(0)

        circuit.append(controlled_modular_multiply(N, a), [0, 1])

        circuit.x(0)

 

    # Measure the result

    circuit.measure(0, 0)

 

    return circuit

 

def controlled_modular_multiply(N, a):

    # Create a quantum circuit for controlled modular multiplication by a mod N

    circuit = QuantumCircuit(3)

 

    # Apply modular multiplication by a

    circuit.swap(0, 1)

    circuit.x(2)

    circuit.append(modular_addition(N, a), [2, 1])

    circuit.x(2)

    circuit.swap(0, 1)

 

    return circuit

 

def modular_addition(N, a):

    # Create a quantum circuit for modular addition by a mod N

    circuit = QuantumCircuit(2)

 

    # Apply modular addition by a

    circuit.x(1)

    circuit.append(controlled_increment_mod_N(N, a), [0, 1])

    circuit.x(1)

 

    return circuit

 

def controlled_increment_mod_N(N, a):

    # Create a quantum circuit for controlled increment by a mod N

    circuit = QuantumCircuit(2)

 

    # Apply controlled increment by a

    circuit.cx(0, 1)

    circuit.append(increment_mod_N(N, a), [1])

    circuit.cx(0, 1)

 

    return circuit

 

def increment_mod_N(N, a):

    # Create a quantum circuit for incrementing a mod N

    circuit = QuantumCircuit(1)

 

    # Apply increment by a

    circuit.u1(a, 0)

 

    return circuit

 

def quantum_fourier_transform(n):

    # Create a quantum circuit for the quantum Fourier transform

    circuit = QuantumCircuit(n)

 

    for qubit in range