Advanced Topics

Advanced Topics in Quantum Computing

Main Topics:

• Quantum cryptography and secure communication, including quantum key distribution

• Quantum machine learning and quantum artificial intelligence

• Quantum error correction and fault tolerance

• Quantum simulations and applications in materials science and drug discovery

Quantum cryptography and secure communication:

Quantum cryptography uses the principles of quantum mechanics to create unbreakable encryption keys and ensure secure communication.

Quantum key distribution (QKD) is a protocol for generating and sharing secret keys between two parties using quantum systems. QKD relies on the fact that any attempt to intercept or measure the quantum states will change them, making it impossible for an eavesdropper to obtain the secret key without detection.

QKD has been demonstrated in various experimental setups, and several companies are developing commercial QKD systems for secure communication.

Quantum machine learning and quantum artificial intelligence:

Quantum machine learning is an emerging field that explores how quantum computing can enhance traditional machine learning algorithms or enable new types of algorithms that are not feasible on classical computers.

Quantum computing has the potential to speed up certain tasks in machine learning, such as matrix multiplication and optimization, which are at the heart of many algorithms.

Quantum algorithms, such as the quantum support vector machine and quantum neural networks, have been proposed for classification and other tasks. However, developing practical quantum machine learning algorithms is still an active area of research.

Quantum artificial intelligence is a broader field that aims to explore how quantum computing can enhance or enable various AI applications, such as natural language processing, computer vision, and robotics. Quantum-inspired algorithms have been proposed for some of these applications, but practical quantum AI is still in its infancy.

Quantum cryptography and secure communication:

Quantum cryptography uses the principles of quantum mechanics to create unbreakable encryption keys and ensure secure communication.

Quantum key distribution (QKD) is a protocol for generating and sharing secret keys between two parties using quantum systems. QKD relies on the fact that any attempt to intercept or measure the quantum states will change them, making it impossible for an eavesdropper to obtain the secret key without detection.

QKD has been demonstrated in various experimental setups, and several companies are developing commercial QKD systems for secure communication.

Quantum machine learning and quantum artificial intelligence:

Quantum machine learning is an emerging field that explores how quantum computing can enhance traditional machine learning algorithms or enable new types of algorithms that are not feasible on classical computers.

Quantum computing has the potential to speed up certain tasks in machine learning, such as matrix multiplication and optimization, which are at the heart of many algorithms.

Quantum algorithms, such as the quantum support vector machine and quantum neural networks, have been proposed for classification and other tasks. However, developing practical quantum machine learning algorithms is still an active area of research.

Quantum artificial intelligence is a broader field that aims to explore how quantum computing can enhance or enable various AI applications, such as natural language processing, computer vision, and robotics. Quantum-inspired algorithms have been proposed for some of these applications, but practical quantum AI is still in its infancy.

Quantum error correction and fault tolerance:

Quantum systems are highly susceptible to errors caused by decoherence, noise, and other environmental factors. These errors can quickly accumulate and cause the system to fail.

Quantum error correction is a technique for detecting and correcting errors in quantum systems. It relies on encoding the information into multiple quantum states and performing measurements to detect errors without disturbing the information.

Quantum fault tolerance is the ability to protect quantum systems against errors and continue to perform computations accurately even in the presence of errors.

Developing quantum error correction and fault tolerance is critical for building practical quantum computers.

Quantum simulations and applications in materials science and drug discovery:

Quantum simulations involve using quantum computers to simulate the behavior of quantum systems, such as molecules and materials, which are difficult or impossible to simulate using classical computers.

Quantum simulations have the potential to accelerate the discovery and development of new materials and drugs, as well as improve our understanding of complex physical and chemical processes.

Quantum simulations have been demonstrated for a range of applications, including predicting molecular properties, simulating chemical reactions, and optimizing materials for various applications.

However, developing practical quantum simulations for complex systems remains a significant challenge, and further research is needed to scale up the simulations to larger systems.

Also Read:

1. Quantum Computing

2. Topics-Mechanics-Algorithms

3. Error Correction and hardware

4. Programming-Applications-Advanced

5. Questions and Answers

6. Research

7. Quantum Mechanics

8. Fundamentals

9. Software & Hardware

10.Advanced Topics

Software and Hardware in Quantum

Quantum Softwares & Hardware

Softwares:

• Programming languages and frameworks for quantum computing, including Qiskit and Microsoft Q#

• Quantum simulation and emulation tools for simulating quantum circuits and algorithms

• Hybrid classical-quantum computing, including variational quantum algorithms and quantum-classical optimization

 

1. Programming Languages and Frameworks for Quantum Computing

Quantum software is a crucial aspect of quantum computing. It provides a way for programmers to design and implement quantum algorithms and applications that run on quantum hardware. There are several programming languages and frameworks available for quantum computing, each with its unique features, advantages, and disadvantages.

Qiskit

Qiskit is an open-source framework for quantum computing developed by IBM. It provides a set of tools, libraries, and APIs that allow users to build and execute quantum circuits, design and run quantum algorithms, and simulate quantum systems. Qiskit supports different quantum hardware platforms, including superconducting qubits and trapped ions, and offers a range of simulators for testing and debugging quantum circuits.

Qiskit's core components include Terra, Aer, Aqua, and Ignis. Terra is a low-level library for building and executing quantum circuits, while Aer is a high-performance simulator for testing and debugging quantum circuits. Aqua is a high-level library for building and executing quantum algorithms, and Ignis is a library for implementing and testing quantum error correction codes.

Example: Quantum teleportation using Qiskit

Here is an example of how to implement quantum teleportation using Qiskit:

python code

from qiskit import QuantumCircuit, execute, Aer

# Create a quantum circuit with three qubits and three classical bits

qc = QuantumCircuit(3, 3)

# Prepare the state to be teleported

qc.h(0)

qc.cx(0, 1)

# Create an entangled pair of qubits

qc.h(2)

qc.cx(2, 1)

# Perform a Bell measurement on the two qubits to be teleported

qc.cx(0, 2)

qc.h(0)

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

# Apply the necessary gates to the third qubit based on the measurement results

qc.z(2).c_if(0, 1)

qc.x(2).c_if(1, 1)

# Measure the third qubit and store the result in a classical bit

qc.measure(2, 2)

# Execute the circuit on a simulator and get the results

backend = Aer.get_backend('qasm_simulator')

result = execute(qc, backend=backend).result()

# Print the counts of the measurement outcomes

print(result.get_counts(qc))

Microsoft Q#

Microsoft Q# is a domain-specific programming language for quantum computing. It provides a set of features and tools for developing and executing quantum algorithms, including support for different quantum hardware platforms, a range of quantum simulators, and a quantum development kit (QDK) that includes libraries, templates, and examples.

Q# uses a syntax similar to C# and provides a range of quantum-specific data types and operations, including qubits, quantum gates, and quantum measurements. It also includes a range of classical control flow statements, such as loops and conditional statements, for controlling the flow of quantum computations.

Example: Quantum teleportation using Q#

Here is an example of how to implement quantum teleportation using Q#:

qsharp code

namespace Teleportation {

    open Microsoft.Quantum.Primitive;

    operation TeleportMessage (message : Qubit) : Unit {

        using ((teleport, message1, message2) = (Qubit(), Qubit(), Qubit())) {

            // Create an entangled pair of qubits

            H(teleport);

            CNOT(teleport, message1);

            // Perform a Bell measurement on the two qubits to be teleported

            CNOT(message, message1);

            H(message);

            let (a, b) = (M(message), M(message1));

qsharp code

namespace Teleportation {

scss

Copy code

open Microsoft.Quantum.Primitive;

operation TeleportMessage (message : Qubit) : Unit {

    using ((teleport, message1, message2) = (Qubit(), Qubit(), Qubit())) {

        // Create an entangled pair of qubits

        H(teleport);

        CNOT(teleport, message1);

        // Perform a Bell measurement on the two qubits to be teleported

        CNOT(message, message1);

        H(message);

        let (a, b) = (M(message), M(message1));

        // Apply the necessary gates to the third qubit based on the measurement results

        if (b == One) {

            X(message2);

        }

        if (a == One) {

            Z(message2);

        }

        // Teleport the message

        CNOT(teleport, message2);

        H(teleport);

        if (M(teleport) == One) {

            Z(message2);

        }

        if (M(message1) == One) {

            X(message2);

        }

        Reset(teleport);

        Reset(message1);

    }

}

}

Quantum simulation and emulation tools for simulating quantum circuits and algorithms

Quantum simulation and emulation tools are crucial for testing and verifying quantum circuits and algorithms. These tools provide a way to simulate the behavior of quantum systems without the need for expensive and complex physical hardware. They can also help researchers explore the behavior of quantum systems and develop new algorithms.

There are several quantum simulation and emulation tools available, including software simulators and cloud-based services. Some popular tools are:

Qiskit: Qiskit is an open-source framework for working with quantum circuits and algorithms. It provides a software simulator that can simulate quantum circuits with up to 32 qubits, as well as interfaces to several cloud-based quantum computing services.

Microsoft Quantum Development Kit: Microsoft's Quantum Development Kit provides a simulator that can simulate quantum circuits with up to 30 qubits, as well as a cloud-based quantum computing service.

IBM Quantum Experience: IBM's Quantum Experience is a cloud-based service that provides access to several quantum computing devices and simulators. It includes a software simulator that can simulate quantum circuits with up to 32 qubits.

Google Cirq: Google's Cirq is an open-source framework for working with quantum circuits and algorithms. It includes a simulator that can simulate quantum circuits with up to 30 qubits.

Here is an example of how to simulate a quantum circuit using Qiskit:

python code

from qiskit import QuantumCircuit, Aer, execute

# Create a quantum circuit with two qubits

qc = QuantumCircuit(2)

# Add a Hadamard gate to the first qubit

qc.h(0)

# Add a CNOT gate between the first and second qubits

qc.cx(0, 1)

# Execute the circuit on a simulator and get the results

backend = Aer.get_backend('qasm_simulator')

result = execute(qc, backend=backend).result()

# Print the counts of the measurement outcomes

print(result.get_counts(qc))

This code creates a simple quantum circuit with two qubits, applies a Hadamard gate to the first qubit, and then applies a CNOT gate between the first and second qubits. It then executes the circuit on a software simulator and prints the counts of the measurement outcomes.

Hybrid classical-quantum computing, including variational quantum algorithms and quantum-classical optimization

Hybrid classical-quantum computing is a type of computing that combines classical computing and quantum computing to solve complex problems that cannot be solved efficiently by either type of computer alone. In this approach, classical computers and quantum computers work together to perform a computation, with classical computers controlling and optimizing the quantum computations.

One example of hybrid classical-quantum computing is variational quantum algorithms. In this approach, a classical computer is used to optimize the parameters of a quantum circuit to minimize a certain cost function. The quantum circuit is then executed on a quantum computer to obtain the result. This approach has been used in applications such as quantum chemistry and optimization problems.

Here is an example of a simple variational quantum algorithm for finding the ground state of a molecular Hamiltonian using Qiskit, a quantum computing framework for Python:

python code

from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister

from qiskit import Aer, execute

from qiskit.chemistry.components.variational_forms import UCCSD

from qiskit.aqua.components.optimizers import L_BFGS_B

# Define the quantum and classical registers

q = QuantumRegister(2)

c = ClassicalRegister(2)

# Define the quantum circuit and the variational form

qc = QuantumCircuit(q, c)

qc.x(q[1])

qc.barrier()

variational_form = UCCSD(q, depth=1)

# Define the optimizer and the backend

optimizer = L_BFGS_B()

backend = Aer.get_backend('statevector_simulator')

# Define the molecular Hamiltonian

h2 = {

    'driver': {'name': 'H2', 'h1': {'X': 0.0}, 'h2': {'II': -1.137}},

    'qubit_mapping': 'parity',

    'two_qubit_reduction': True

}

# Define the objective function

def objective_function(params):

    energy = variational_form.evaluate_energy(params, h2)

    return energy.real

# Optimize the parameters of the variational form

result = optimizer.optimize(variational_form.num_parameters, objective_function)

# Set the optimized parameters to the quantum circuit

variational_form.set_parameter_vector(result[0])

variational_form.construct_circuit(qc)

# Execute the circuit on the backend and get the result

job = execute(qc, backend)

result = job.result()

print(result.get_statevector())

In this example, we use the UCCSD variational form to find the ground state of a molecular Hamiltonian for a hydrogen molecule. The objective function is the energy of the circuit evaluated on the Hamiltonian, and the L-BFGS-B optimizer is used to optimize the parameters of the circuit. Finally, we execute the circuit on a statevector simulator backend and print the result.

This example demonstrates how hybrid classical-quantum computing can be used to solve real-world problems using a combination of classical and quantum computing.

 Quantum Hardware

• Different physical systems used for quantum computing, including superconducting qubits, ion traps, and topological qubits

• Challenges in building quantum hardware, including error correction and scalability

Quantum hardware refers to the physical devices that are used for quantum computing. There are different physical systems used for quantum computing, and each has its own advantages and disadvantages. Some of the most common physical systems used for quantum computing are:

Superconducting qubits: These are qubits that are made from superconducting materials, which are materials that have zero electrical resistance at low temperatures. Superconducting qubits are commonly used in quantum computers built by IBM and Google.

Ion traps: These are qubits that are made from ions (charged atoms) that are trapped in a magnetic field. Ion traps are commonly used in quantum computers built by companies like IonQ and Honeywell. 

Topological qubits: These are qubits that are made from exotic materials that have topological properties that make them less susceptible to noise and errors. Topological qubits are still in the experimental stage, and it is not yet clear how they will be used in practical quantum computers.

Challenges in building quantum hardware:

Building quantum hardware is a challenging task, and there are several major challenges that need to be addressed:

Error correction: Quantum hardware is highly sensitive to noise and errors, which can cause errors in the computation. Error correction is a technique that is used to protect quantum states from the effects of noise and errors. 

Scalability: Quantum hardware is still in the early stages of development, and the number of qubits that can be reliably controlled and manipulated is still relatively small. Scaling up the number of qubits is a major challenge, as it requires developing new materials and technologies.

Integration: Quantum hardware is highly complex and requires specialized equipment to operate. Integrating quantum hardware with classical computing infrastructure is a major challenge, as it requires developing new software and hardware interfaces.

Example:

One example of a quantum hardware system is the IBM Q System One, which is a superconducting qubit-based quantum computer that is designed for commercial use. The system has 20 qubits, and it is housed in a custom-designed refrigerator that is designed to maintain the qubits at extremely low temperatures. The system is designed to be used by businesses and researchers who want to experiment with quantum computing. However, the system is still relatively expensive and is not yet widely available.

Also Read:

1. Quantum Computing

2. Topics-Mechanics-Algorithms

3. Error Correction and hardware

4. Programming-Applications-Advanced

5. Questions and Answers

6. Research

7. Quantum Mechanics

8. Fundamentals

9. Software & Hardware

10.Advanced Topics

Fundamentals

 Quantum Computing Fundamentals

Quantum Fundamentals:

• Qubits and quantum gates, including the Pauli matrices and the Hadamard gate

• Quantum circuits and quantum algorithms, including the Deutsch-Jozsa algorithm and Grover's algorithm

• Quantum parallelism and quantum teleportation

Quantum computing is a type of computing that utilizes quantum mechanics principles to perform operations on data. It is based on the use of quantum bits or qubits, which can represent both 0 and 1 simultaneously due to the principle of superposition. Qubits allow for the creation of quantum circuits that use quantum gates to manipulate the state of the qubits, leading to more efficient computation.

Qubits and Quantum Gates:

Qubits are the basic units of quantum information, and they can be represented as a combination of the two states 0 and 1, denoted as |0⟩ and |1⟩. A qubit can be in a superposition of these two states, which means that it can represent a linear combination of both states. This allows for more efficient computations, as multiple operations can be performed simultaneously on a qubit.

Quantum gates are the building blocks of quantum circuits and are used to manipulate the state of qubits. There are several types of quantum gates, including the Pauli-X, Pauli-Y, and Pauli-Z gates, which are analogous to the classical NOT gate. The Hadamard gate, denoted as H, is another important quantum gate that puts a qubit into a superposition state, which is useful for performing quantum algorithms.

Example:

python code

from qiskit import QuantumCircuit, execute, Aer

# Create a quantum circuit with one qubit

qc = QuantumCircuit(1, 1)

# Apply a Hadamard gate to the qubit

qc.h(0)

# Measure the qubit and store the result in a classical bit

qc.measure(0, 0)

# Execute the circuit on a simulator and get the results

backend = Aer.get_backend('qasm_simulator')

result = execute(qc, backend).result() 

# Print the counts of the measurement outcomes

print(result.get_counts(qc))

 

Quantum Circuits and Quantum Algorithms:

Quantum circuits are a series of quantum gates that manipulate qubits to perform a particular computation. Quantum algorithms are designed to use these quantum circuits to solve problems that are not easily solvable using classical computing. Some examples of quantum algorithms include the Deutsch-Jozsa algorithm and Grover's algorithm.

The Deutsch-Jozsa algorithm is a quantum algorithm that determines if a function is constant or balanced. The classical equivalent of this algorithm would require an exponential number of function evaluations, whereas the quantum algorithm can determine the nature of the function with just one evaluation. The algorithm uses a quantum oracle, which is a black box that applies a function to a qubit, and a Hadamard gate to perform the computation.

Example:

python code

from qiskit import QuantumCircuit, execute, Aer

# Define the oracle function

def oracle(qc, n):

    for qubit in range(n):

        qc.cx(qubit, n)

# Define the Deutsch-Jozsa algorithm

def deutsch_jozsa(n):

    # Create a quantum circuit with n qubits and n-1 classical bits

    qc = QuantumCircuit(n, n-1)

    # Apply Hadamard gates to all qubits

    qc.h(range(n)) 

    # Apply the oracle function

    oracle(qc, n-1)

    # Apply Hadamard gates to all qubits again

    qc.h(range(n-1)) 

    # Measure the first n-1 qubits and store the results in the classical bits

    qc.measure(range(n-1), range(n-1))

    # Execute the circuit on a simulator and get the results

    backend = Aer.get_backend('qasm_simulator')

    result = execute(qc, backend).result() 

    # Print the counts of the measurement outcomes

 

Grover's algorithm:

Grover's algorithm is a quantum algorithm that is used to search an unsorted database with N elements in O(sqrt(N)) time, which is a quadratic speedup over classical algorithms. The algorithm works by repeatedly applying an operator that amplifies the amplitude of the marked element until it can be easily identified. The algorithm uses a quantum oracle that marks the desired element and a diffusion operator that amplifies the amplitude of all states except the marked element.

Example:

python code

from qiskit import QuantumCircuit, execute, Aer

import math

# Define the oracle function

def oracle(qc, marked):

    qc.x(marked)

    qc.h(marked)

    qc.cx(marked, n)

    qc.h(marked)

    qc.x(marked)

# Define the diffusion operator

def diffusion(qc):

    qc.h(range(n))

    qc.x(range(n))

    qc.h(n-1)

    qc.mct(list(range(n-1)), n-1)

    qc.h(n-1)

    qc.x(range(n))

    qc.h(range(n))

# Define Grover's algorithm

def grover(elements, marked):

    # Determine the number of qubits needed to represent the elements

    n = math.ceil(math.log2(len(elements))) 

    # Create a quantum circuit with n qubits and n-1 classical bits

    qc = QuantumCircuit(n, n-1)

    # Apply Hadamard gates to all qubits

    qc.h(range(n)) 

    # Apply the oracle function and diffusion operator sqrt(N) times

    for i in range(int(math.sqrt(len(elements)))):

        oracle(qc, marked)

        diffusion(qc)

    # Measure the first n-1 qubits and store the results in the classical bits

    qc.measure(range(n-1), range(n-1)) 

    # Execute the circuit on a simulator and get the results

    backend = Aer.get_backend('qasm_simulator')

    result = execute(qc, backend).result()

    # Print the counts of the measurement outcomes

    print(result.get_counts(qc))

Quantum Parallelism and Quantum Teleportation:

Quantum parallelism refers to the ability of a quantum computer to perform multiple computations simultaneously by creating a superposition of all possible inputs. This is useful for certain types of problems, such as searching and optimization. Quantum teleportation is a quantum communication protocol that allows for the transfer of quantum states from one qubit to another without physically moving the qubit. This is achieved through the use of entanglement, which allows for the transfer of information instantaneously.

Example of quantum teleportation:

python code

from qiskit import QuantumCircuit, execute, Aer

# Create a quantum circuit with three qubits and three classical bits

qc = QuantumCircuit(3, 3)

# Prepare the state to be teleported

qc.h(0)

qc.cx(0, 1) 

# Create an entangled pair of qubits

qc.h(2)

qc.cx(2, 1)

# Perform a Bell measurement on the two qubits to be teleported

qc.cx(0, 2)

qc.h(0)

qc.measure([0, 1], [0, 1]) 

# Apply the necessary gates to the third qubit based on the measurement results

qc.z(2).c_if(0, 1)

qc.x(2).c_if(1, 1)

# Measure the third qubit and store the result in a classical bit

qc.measure(2, 2) 

# Execute the circuit on a simulator and get the results

backend = Aer.get_backend('qasm_simulator')

result = execute(qc, backend=backend).result()

# Print the counts of the measurement outcomes

print(result.get_counts(qc)) 

This code uses the Qiskit library to perform quantum teleportation. The quantum circuit qc is created with three qubits and three classical bits. The first qubit is prepared in a superposition state using the Hadamard gate qc.h(0) and then entangled with the second qubit using a controlled NOT gate qc.cx(0,1). The third qubit is entangled with the second qubit using the same operations.

A Bell measurement is then performed on the first and second qubits using controlled NOT gates and Hadamard gates. The measurement results are stored in the first two classical bits. Based on the measurement results, the necessary gates are applied to the third qubit to recreate the original state. The state of the third qubit is then measured and the result is stored in the third classical bit.

The execute function is used to run the quantum circuit on a simulator, and the result method is called to get the result of the simulation. Finally, the get_counts method is used to retrieve the number of times each possible outcome was measured. This is printed to the console to show the results of the teleportation protocol.

Also Read:

1. Quantum Computing

2. Topics-Mechanics-Algorithms

3. Error Correction and hardware

4. Programming-Applications-Advanced

5. Questions and Answers

6. Research

7. Quantum Mechanics

8. Fundamentals

9. Software & Hardware

10.Advanced Topics

Quantum Mechanics

Introduction to Quantum Mechanics

Introduction 

Basics of quantum mechanics, including wave-particle duality, superposition, and entanglement, Quantum states and operators, including the wave function and measurement.

Quantum mechanics is a branch of physics that deals with the behavior of matter and energy at the quantum level, which is the smallest scale of physical phenomena. Quantum mechanics is a fundamental theory of nature that is essential to the development of modern technologies, including quantum computing. It provides a mathematical framework for describing the behavior of particles at the quantum level, including their wave-like properties, superposition, and entanglement.

Basics of Quantum Mechanics:

Quantum mechanics is based on a few fundamental principles, including wave-particle duality, superposition, and entanglement. Wave-particle duality refers to the fact that particles can exhibit both wave-like and particle-like behavior, depending on how they are observed. Superposition is the idea that particles can exist in multiple states simultaneously, and entanglement is the idea that particles can become correlated in such a way that the state of one particle is dependent on the state of the other, even when they are separated by large distances.

Quantum States and Operators:

Quantum states are described by wave functions, which are complex mathematical functions that represent the probability distribution of a particle's position and momentum. The wave function can be used to calculate the probability of a particle being in a particular state, and it can also be used to calculate the expectation value of observables such as position and momentum.

Quantum operators are mathematical objects that operate on the wave function and represent physical quantities such as position and momentum. Operators can be used to calculate the probability of a particle being in a particular state, and they can also be used to calculate the expectation value of observables such as position and momentum.

Measurement in Quantum Mechanics:

In quantum mechanics, measurement is a fundamental process that collapses the wave function of a particle into a definite state. The result of a measurement is a random variable that can take on different values with different probabilities, according to the probability distribution given by the wave function. The process of measurement is often described as a "wave function collapse", because it causes the particle to become localized in a definite position or momentum state.

 

One of the most famous examples of a quantum algorithm is Shor's algorithm, which is used to factor large numbers. It takes advantage of the fact that the period of a function modulo a composite number can be efficiently found using a quantum computer, which can then be used to factor the composite number.

Here's an example of how to implement Shor's algorithm in Python using the Qiskit framework:

python code

from qiskit import QuantumCircuit, Aer, execute

from qiskit.circuit.library import QFT

import numpy as np

def shor(N):

    # Check if N is even

    if N % 2 == 0:

        return 2

    # Choose a random integer between 1 and N-1

    a = np.random.randint(1, N)

    # Check if a is a factor of N

    gcd = np.gcd(a, N)

    if gcd > 1:

        return gcd

    # Set up the quantum circuit

    qubits = QuantumCircuit(2 * int(np.log2(N)) + 1, int(np.log2(N)))

    qubits.h(range(int(np.log2(N))))

    qubits.append(QFT(int(np.log2(N))), range(int(np.log2(N))))

    qubits.x(2 * int(np.log2(N)))

    qubits.barrier()

    # Perform the quantum part of the algorithm

    for i in range(int(np.log2(N))):

        qubits.swap(i, 2 * int(np.log2(N)) - i - 1)

        for j in range(2 * int(np.log2(N)) - 1):

            qubits.cswap(j , j+1, int(np.log2(N))+i)

less code

        qubits.barrier()

 

qubits.append(QFT(int(np.log2(N))), range(int(np.log2(N))))

# Measure the quantum circuit and obtain the result

qubits.measure(range(int(np.log2(N))), range(int(np.log2(N))))

backend = Aer.get_backend('qasm_simulator')

counts = execute(qubits, backend, shots=1).result().get_counts()

measured = int(list(counts.keys())[0], 2)

# Calculate the possible period and factor

possible_period = np.gcd(measured, 2**int(np.log2(N)) - 1)

if possible_period % 2 != 0:

    return None

factor1 = np.power(a, possible_period//2, N) + 1

factor2 = np.power(a, possible_period//2, N) - 1

gcd1 = np.gcd(factor1, N)

gcd2 = np.gcd(factor2, N)

if gcd1 == 1 or gcd1 == N:

    return gcd2

else:

    return gcd1

sql code 

# Measure the first register

qubits.measure(range(int(np.log2(N))), range(int(np.log2(N))))

# Simulate the circuit and get the results

backend = Aer.get_backend('qasm_simulator')

shots = 1024

results = execute(qubits, backend, shots=shots).result()

counts = results.get_counts()

# Find the period from the results

periods = []

for output in counts:

    decimal = int(output, 2)

    period = decimal / 2 ** int(np.log2(N))

    periods.append(period)

r = np.round(np.mean(periods))

if r % 2 != 0:

    return shor(N)

# Calculate the factors using the period

x = int(np.power(a, r / 2)) % N

factor1 = np.gcd(x - 1, N)

factor2 = np.gcd(x + 1, N)

return factor1, factor2

vbnet code

In this code, we define a function `shor` that takes an integer `N` as input and returns its prime factors using Shor's algorithm. The function first checks if `N` is even and returns 2 if it is. It then chooses a random integer `a` between 1 and `N-1` and checks if `a` is a factor of `N`. If it is, the function returns the factor. If not, it sets up the quantum circuit, performs the quantum part of the algorithm, and measures the first register. The function then simulates the circuit and calculates the period from the results. It calculates the factors using the period and returns them. If the period is odd, the function recursively calls itself with the same input `N`.

Also Read:

1. Quantum Computing

2. Topics-Mechanics-Algorithms

3. Error Correction and hardware

4. Programming-Applications-Advanced

5. Questions and Answers

6. Research

7. Quantum Mechanics

8. Fundamentals

9. Software & Hardware

10.Advanced Topics