Detailed Summary: Quantum Generative Adversarial Networks (QGANs) for Geological and Atmospheric Biodetection

Overview

The project aims to develop a Quantum Generative Adversarial Network (QGAN) to enhance anomaly detection in geological and atmospheric biodetection. Leveraging both classical and quantum computing, this model will analyze geological changes, particularly focusing on fossil particles correlated with oil deposits

In simple terms, fossil particles, also known as microfossils or fossil biomarkers, are remnants of ancient organic matter from plants, algae, and microorganisms that have turned into oil over millions of years. When these organisms died long ago, their remains were buried under sediment and transformed by heat and pressure into hydrocarbons, which are the main components of crude oil. The process involves the organic matter turning into kerogen, then further breaking down into liquid and gaseous hydrocarbons like oil and natural gas. Biomarkers in oil are specific molecules that can be linked back to the original organisms, helping identify the source and conditions under which the oil formed.

The aim of this project is to isolate and detect specific particles associated with oil discovery, and to understand their relationship with oil. We seek to determine whether these particles predate oil formation or emerged subsequently, investigate their movement and generation, and identify contributing factors. By exploring the potential to reproduce these particles, we hope to replicate or regenerate natural resources and enhance oil detection capabilities. Through environmental simulations, we aim to recreate the natural processes involved in oil formation, potentially leading to innovative methods for particle multiplication, replication, or manufacturing. The outcomes of this research could have significant implications, potentially driving groundbreaking advancements in geospatial information systems (GIS) and related fields. This endeavor could potentially lead to a significant breakthrough, akin to discovering a “chasm of crude.”

The objectives include:

  1. Detection and Correlation: Identifying the presence of specific particles in fossils that correlate with oil deposits.
  2. Simulation and Replication: Determining if these conditions can be replicated to create alternative resources.
  3. Environmental Impact Study: Analyzing how atmospheric pressure, weather, time, movement, and other factors influence particle changes around resources.
  4. Resource Creation: Using findings to develop strategies for enhancing global resource availability or creating alternatives.

Concept Map

  1. Data Collection and Preprocessing
    • Geological data on fossil particles.
    • Atmospheric and environmental data.
    • Historical data on oil deposits.
  2. Model Development
    • Autoencoder for feature extraction.
    • Adversarial Network for anomaly detection.
    • Quantum components for enhanced computation.
  3. Simulation and Analysis
    • Simulating environmental impacts on particle changes.
    • Analyzing correlations between fossil particles and oil deposits.
  4. Resource Optimization
    • Identifying potential for alternative resource creation.
    • Developing models to replicate conditions for resource generation.

Initial Concept Mindmap - mermaid

Hard to read - redo

OBJECTIVES

QGAN IMPLEMENTATION PROCESS

SIMPLE DIAGRAM

STRUCTURING

modifying

reflecting code change

hmmm…

Steps to Follow

Step 1: Define the Problem and Objectives

Objective: Develop a QGAN model to isolate and detect specific particles associated with oil discovery, understand their relationship with oil, and analyze particles related to oil formation and their geological changes.

Specific Goals: - Determine Presence and Generation: Identify the specific particles that are present in geological samples and understand their genesis in relation to oil deposits. - Analyze Movement: Study the movement patterns of these particles within geological formations. - Identify Contributing Factors: Investigate the environmental and geological factors contributing to the presence and movement of these particles. - Reproduce Particles: Develop methods to reproduce these particles, aiming to replicate or regenerate natural resources and enhance oil detection capabilities.

Key Objectives: - Detection and Correlation: Identify specific particles in fossils that correlate with oil deposits to enhance detection techniques. - Simulation and Replication: Determine if the identified conditions and processes can be replicated to create alternative resources or improve existing resource extraction methods. - Environmental Impact Study: Analyze the influence of atmospheric pressure, weather, time, movement, and other environmental factors on particle changes around oil resources. - Resource Creation: Utilize the findings to develop strategies for enhancing global resource availability or creating viable alternative resources through environmental simulations.

Step 2: Data Collection and Preprocessing

Data Collection: - Geological Data: Collect data on fossil particles. - Environmental Data: Gather atmospheric and other environmental data. - Historical Data: Compile historical data on oil deposits. - Ensure data quality and relevance to the study.

Data Types: - Particle composition - Spatial distribution - Temporal changes - Environmental conditions

Data Sources: - Geological and atmospheric data, including samples of fossil particles and oil deposits.

Preprocessing: - Data Cleaning: Remove noise and irrelevant information. Handle missing values and outliers. - Feature Extraction: Identify and extract relevant features from the data using domain knowledge and autoencoders. - Data Normalization: Normalize data to ensure consistency and improve model performance. - Quantum Data Encoding: Encode classical data into quantum states suitable for quantum processing.

Step 3: Define the QGAN Architecture

Quantum Generative Adversarial Network (QGAN): - Generator: Quantum circuit that generates synthetic data samples. - Discriminator: Classical or quantum neural network that distinguishes between real and synthetic data. - Autoencoder: Classical autoencoder network to compress and reconstruct data for anomaly detection. - Encoder: Compresses the data into a lower-dimensional representation. - Decoder: Reconstructs the original data from the compressed representation.

Architecture Design: - Design the overall architecture, defining how the generator, discriminator, and autoencoder will interact.

Step 4: Hybrid Classical-Quantum Computing Setup & Implementation of the QGAN Model

Classical Preprocessing: - Use classical computing resources to preprocess and prepare data for quantum processing.

Quantum Processing: - Quantum Circuit Design: Design the quantum circuits for the generator using a quantum computing framework (e.g., Qiskit, PennyLane). Utilize quantum processors (e.g., IBM Q, Rigetti) for running quantum algorithms. - Discriminator Network: Implement the discriminator using classical deep learning frameworks like TensorFlow or PyTorch. - Autoencoder Implementation: Implement the autoencoder for anomaly detection.

Step 5: Model Training

Training the QGAN: - Train the generator and discriminator in an adversarial manner using quantum gradient descent or other optimization algorithms suitable for quantum circuits. - Training Data: Split the data into training and testing sets. - Training Process: - Train the generator to produce realistic synthetic data. - Train the discriminator to distinguish between real and synthetic data. - Use adversarial training to iteratively improve both networks. - Train the autoencoder on normal data to learn the compressed representation and reconstruction. - Use anomaly detection by evaluating reconstruction errors on test data.

Anomaly Detection: - Train the autoencoder to detect anomalies in geological and atmospheric data, identifying patterns and particles associated with oil formation.

Hybrid Model Training: - Combine quantum and classical training processes, leveraging classical optimization techniques for parameter tuning.

Step 6: Model Evaluation and Validation

Performance Metrics: - Use metrics such as accuracy, precision, recall, F1-score, and ROC-AUC to evaluate the model.

Validation: - Validate the model using a separate validation dataset.

Anomaly Detection Validation: - Validate the model’s ability to detect and isolate specific particles. - Compare detected anomalies with known oil-related particles (comparing reconstruction errors with a threshold).

Step 7: Environmental Simulations and Replication

Simulation Setup: - Create environmental simulations to mimic natural processes of oil formation. - Use the trained model to analyze and detect particles in simulated environments.

Replication and Multiplication: - Investigate the potential to replicate particles through controlled simulations. - Develop methods for particle multiplication and regeneration.

Step 8: Interpret and Analyze Results

Particle Analysis: - Analyze the detected particles to understand their relationship with oil.

Movement and Generation: - Study the movement and generation of particles using the model’s outputs.

Simulation: - Use environmental simulations to recreate the natural processes involved in oil formation.

Step 9: Integration with Geospatial Information Systems (GIS)

GIS Integration: - Integrate the model with GIS platforms to enhance oil detection capabilities. - Use geospatial data to improve the accuracy of particle detection and mapping.

Visualization and Analysis: - Visualize detected particles and their spatial distribution. - Analyze geological changes and their correlation with oil formation.

Step 10: Optimization and Refinement

Model Refinement: - Fine-tune the QGAN model based on evaluation results.

Parameter Optimization: - Optimize hyperparameters for improved performance.

Iterative Improvements: - Iteratively improve the model by incorporating new data and insights.

Step 11: Deployment, Application & Continuous Improvement

Deployment: - Deploy the trained model in a production environment for real-time geological and atmospheric biodetection. - Set up monitoring and maintenance processes.

Application: - Apply the model to new geological sites to detect and analyze fossil particles related to oil.

Continuous Improvement: - Continuously update the model with new data and findings. - Refine the model to improve accuracy and robustness.

Step 12: Documentation and Reporting

Documentation: - Document the entire process, including model architecture, training procedures, and evaluation metrics. - Provide detailed reports on findings and implications.

Reporting: - Share results with stakeholders and the scientific community. - Publish findings in relevant journals and conferences.

Moving Forward and Building From: Continuous Monitoring and Updating

Monitoring: - Continuously monitor the model’s performance and update it with new data.

Application: - Use as a template for QGAN with different goals/data based on results.

Research and Development: - Conduct ongoing research to enhance the model and explore new applications.

Environmental Resource Renewal

Title: Identifying Particle Traces for Environmental Resource Renewal Using a Discriminative Autoencoder in GIS

Objective: Develop a GIS-based anomaly detection system using a discriminative autoencoder for identifying particle traces, which are crucial for understanding and managing environmental resources.

Approach: - Discriminative Autoencoder: This neural network architecture learns to reconstruct normal data while simultaneously distinguishing anomalous patterns. - GIS Data: Geographic Information System data, such as soil samples, water quality measurements, or satellite imagery, is used to train the model. - Particle Traces: Anomalies identified by the model represent potential particle traces, which could indicate pollution, sediment transport, or other factors influencing resource renewal.

Benefits: - Enhanced Resource Management: Identification of particle traces enables targeted intervention to optimize resource renewal processes. - Improved Environmental Monitoring: Continuous monitoring of particle movement and distribution allows for early detection of potential environmental threats. - Data-Driven Decision-Making: The system provides objective data-driven insights to inform environmental resource management decisions.

Conclusion

By leveraging the Universal Axiom framework and integrating quantum and classical computing, this project aims to uncover critical insights into geological changes and resource optimization, paving the way for innovative solutions in resource creation and environmental analysis. ___

“Discriminatory AutoEncoder for Anomaly Detection in GIS to Identify Particle Traces for Environmental Resource Renewal” well that’s a mouthfull…

Example Calculations

Exponential Growth (E_n): \[ E_n = 3E_{n-1} + 2 \]

  • Base Case: \(E_0 = 1\)
  • First Iteration: \(E_1 = 3 \times 1 + 2 = 5\)
  • Second Iteration: \(E_2 = 3 \times 5 + 2 = 17\)
  • Third Iteration: \(E_3 = 3 \times 17 + 2 = 53\)

Fibonacci Sequence (F_n): \[ F_n = F_{n-1} + F_{n-2} \]

  • Base Cases: \(F_0 = 0, F_1 = 1\)
  • First Iteration: \(F_2 = 1 + 0 = 1\)
  • Second Iteration: \(F_3 = 1 + 1 = 2\)
  • Third Iteration: \(F_4 = 2 + 1 = 3\)

Axiomatic Subjectivity Scale (X): \[ X = \frac{Y_s}{Y_o} \]

  • Example: \(Y_s = 4, Y_o = 5\)
  • Calculation: \(X = \frac{4}{5} = 0.8\)

TimeSphere (Z): \[ Z = \frac{n}{T} \]

  • Example: \(n = 5, T = 10\)
  • Calculation: \(Z = \frac{5}{10} = 0.5\)

Combined Equation: \[ Intelligence_n = E_n \times (1 + F_n) \times X \times Y \times Z \times (A \times B \times C) \]

  • Example:
    • \(E_3 = 53\)
    • \(F_4 = 3\)
    • \(X = 0.8\)
    • \(Y = 0.8\)
    • \(Z = 0.5\)
    • \(A = 0.9, B = 0.85, C = 0.8\)
    • Combined: \[ Intelligence_n = 53 \times (1 + 3) \times 0.8 \times 0.8 \times 0.5 \times (0.9 \times 0.85 \times 0.8) \]

This calculation shows how each component interacts dynamically, reflecting the comprehensive nature of the Universal Axiom framework.

Exponential Growth (E_n): \(E_n = 3E_{n-1} + 2\)

  • Base Case: \(E_0 = 1\)
  • First Iteration: \(E_1 = 3 \times 1 + 2 = 5\)
  • Second Iteration: \(E_2 = 3 \times 5 + 2 = 17\)
  • Third Iteration: \(E_3 = 3 \times 17 + 2 = 53\)

Fibonacci Sequence (F_n): \(F_n = F_{n-1} + F_{n-2}\)

  • Base Cases: \(F_0 = 0, F_1 = 1\)
  • First Iteration: \(F_2 = 1 + 0 = 1\)
  • Second Iteration: \(F_3 = 1 + 1 = 2\)
  • Third Iteration: \(F_4 = 2 + 1 = 3\)

Axiomatic Subjectivity Scale (X): \(X = \frac{Y_s}{Y_o}\)

  • Example: \(Y_s = 4, Y_o = 5\)
  • Calculation: \(X = \frac{4}{5} = 0.8\)

TimeSphere (Z): \(Z = \frac{n}{T}\)

  • Example: \(n = 5, T = 10\)
  • Calculation: \(Z = \frac{5}{10} = 0.5\)

Combined Equation: \(Intelligence_n = E_n \times (1 + F_n) \times X \times Y \times Z \times (A \times B \times C)\)

  • Example:
    • \(E_3 = 53\)
    • \(F_4 = 3\)
    • \(X = 0.8\)
    • \(Y = 0.8\)
    • \(Z = 0.5\)
    • \(A = 0.9, B = 0.85, C = 0.8\)
    • Combined: \(Intelligence_n = 53 \times (1 + 3) \times 0.8 \times 0.8 \times 0.5 \times (0.9 \times 0.85 \times 0.8)\)

Mathematical Representation of Grover’s and Shor’s Algorithms and Their Practical Implications in QGAN

1. Grover’s Algorithm

Grover’s algorithm is a quantum algorithm used for searching an unsorted database or solving the unstructured search problem. It offers a quadratic speedup over classical algorithms.

Mathematical Representation:

  • Initialization: Start with a superposition of all possible states: \[ \left| \psi \right\rangle = \frac{1}{\sqrt{N}} \sum_{x=0}^{N-1} \left| x \right\rangle \]
  • Oracle (Black-box function): Mark the correct answer by flipping the sign of the amplitude of the correct state: \[ O \left| x \right\rangle = \begin{cases} -\left| x \right\rangle & \text{if } x = x_0 \\ \left| x \right\rangle & \text{otherwise} \end{cases} \]
  • Grover Diffusion Operator (Amplification): Inverts the amplitude of the state about the mean amplitude: \[ D = 2 \left| \psi \right\rangle \left\langle \psi \right| - I \]
  • Iterative Process: Repeat the application of the Oracle and Diffusion operator approximately \(\sqrt{N}\) times: \[ G = DO \] After \(O(\sqrt{N})\) iterations, the probability of measuring the correct state is close to 1.

2. Shor’s Algorithm

Shor’s algorithm is used for integer factorization, which underpins the security of many encryption systems.

Mathematical Representation:

  • Step 1 (Quantum Part): Find the period \(r\) of the function \(f(x) = a^x \mod N\) using a quantum computer.
    • Initialize a quantum state in a superposition: \[ \left| \psi \right\rangle = \frac{1}{\sqrt{q}} \sum_{x=0}^{q-1} \left| x \right\rangle \left| 0 \right\rangle \]
    • Apply the quantum Fourier transform to find the period \(r\): \[ \text{QFT} \left| f(x) \right\rangle = \frac{1}{\sqrt{r}} \sum_{k=0}^{r-1} \left| k \frac{q}{r} \right\rangle \]
    • Measure the state to get information about the period.
  • Step 2 (Classical Part): Use the period \(r\) to factorize \(N\):
    • Compute the greatest common divisor (GCD): \[ \gcd(a^{r/2} \pm 1, N) \] This gives the non-trivial factors of \(N\).

3. Practical Implications in QGAN (Quantum Generative Adversarial Networks)

QGANs combine classical GANs with quantum computing to leverage the benefits of both worlds.

  • Speedup in Training:
    • Grover’s Algorithm: Grover’s algorithm can speed up the search within the parameter space during the optimization process, enhancing the training of the quantum generator and discriminator networks.
  • Enhanced Factorization:
    • Shor’s Algorithm: The quantum generator can be used to simulate and model quantum data distributions. Shor’s algorithm can factorize large integers efficiently, which is essential in cryptographic applications within QGANs.

QGAN Structure: - Quantum Generator: Uses quantum circuits to generate data samples. The quantum speedup in sampling allows for more complex data distributions to be modeled efficiently. - Classical Discriminator: Uses classical neural networks to distinguish between real and generated data.

In theory, using these algorithms in QGANs can achieve better performance in tasks that involve large data sets or complex distributions.

Grover & Shor in GIS for Anomaly Detection using Autoencoder

Geographic Information Systems (GIS) involve the use of spatial data to manage and analyze various phenomena across different regions. Anomaly detection in GIS can help identify unusual patterns or behaviors in spatial data, which is critical for applications such as environmental monitoring, urban planning, and disaster management.

Autoencoder is a type of neural network used for unsupervised learning of efficient codings. It consists of an encoder and a decoder part. For anomaly detection, the autoencoder is trained to reconstruct normal data, and any significant deviation during reconstruction can be flagged as an anomaly.

Steps to Implement Anomaly Detection using Autoencoder in GIS

  1. Data Collection and Preprocessing:
    • Collect GIS data relevant to the application. This can include satellite images, spatial coordinates, weather data, land use maps, etc.
    • Preprocess the data to a suitable format for training. This may involve normalization, data cleaning, and transforming spatial data into a grid or raster format.
  2. Autoencoder Architecture:
    • Design an autoencoder neural network with an appropriate architecture to handle spatial data. For example, Convolutional Autoencoders (CAE) are often used for image-like data due to their ability to capture spatial hierarchies.
    from tensorflow.keras.layers import Input, Conv2D, MaxPooling2D, UpSampling2D
from tensorflow.keras.models import Model

input_img = Input(shape=(128, 128, 1))  # Example for 128x128 grayscale images

# Encoder
x = Conv2D(32, (3, 3), activation='relu', padding='same')(input_img)
x = MaxPooling2D((2, 2), padding='same')(x)
x = Conv2D(16, (3, 3), activation='relu', padding='same')(x)
encoded = MaxPooling2D((2, 2), padding='same')(x)

# Decoder
x = Conv2D(16, (3, 3), activation='relu', padding='same')(encoded)
x = UpSampling2D((2, 2))(x)
x = Conv2D(32, (3, 3), activation='relu', padding='same')(x)
x = UpSampling2D((2, 2))(x)
decoded = Conv2D(1, (3, 3), activation='sigmoid', padding='same')(x)

autoencoder = Model(input_img, decoded)
autoencoder.compile(optimizer='adam', loss='binary_crossentropy')
  3. Training the Autoencoder:
    • Train the autoencoder on normal GIS data (without anomalies).
    autoencoder.fit(train_data, train_data, epochs=50, batch_size=256, shuffle=True, validation_data=(val_data, val_data))
  4. Anomaly Detection:
    • Use the trained autoencoder to reconstruct test data. Calculate the reconstruction error for each sample.
    • Set a threshold for the reconstruction error. Samples with errors above this threshold are considered anomalies.
    reconstructed = autoencoder.predict(test_data)
reconstruction_error = np.mean(np.abs(reconstructed - test_data), axis=(1, 2, 3))
anomaly_threshold = np.percentile(reconstruction_error, 95)  # For example, using the 95th percentile as threshold

anomalies = test_data[reconstruction_error > anomaly_threshold]
  5. Visualization and Analysis:
    • Visualize the detected anomalies on a GIS map to understand the spatial distribution of anomalies.
    • Perform further analysis to investigate the causes of the anomalies.
    import matplotlib.pyplot as plt

for i in range(len(anomalies)):
    plt.figure(figsize=(10, 4))

    # Original image
    plt.subplot(1, 2, 1)
    plt.title('Original Image')
    plt.imshow(test_data[i].reshape(128, 128), cmap='gray')

    # Reconstructed image
    plt.subplot(1, 2, 2)
    plt.title('Reconstructed Image')
    plt.imshow(reconstructed[i].reshape(128, 128), cmap='gray')

    plt.show()

Practical Example: Flood Detection in Urban Areas - not so much practical to me but a good study from book below:

Fano, G., & Blinder, S. M. (2020). Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers: Mathematical Methods and Conceptual Foundations. Springer International Publishing. https://doi.org/10.1007/978-3-030-34783-2

Objective: Detect anomalies in spatial data representing potential flooding areas in an urban environment.

Steps: 1. Data Collection: - Collect satellite imagery and elevation data of the urban area. - Gather historical flood data and weather patterns.

  1. Data Preprocessing:
    • Normalize the satellite images.
    • Transform elevation data to a consistent grid format.
  2. Autoencoder Training:
    • Train the autoencoder using normal (non-flooded) satellite images.
  3. Anomaly Detection:
    • Use the autoencoder to reconstruct new satellite images during a rainy season.
    • Calculate reconstruction errors and flag significant deviations.
  4. Analysis:
    • Map the detected anomalies to identify potential flooding areas.

    • Cross-reference with weather data to validate findings.

Hadamard Matrix in Quantum Computing with Shor’s Algorithm

Shor’s algorithm is renowned for its ability to factor large integers exponentially faster than the best-known classical algorithms. One of the key components in constructing the quantum circuits for Shor’s algorithm is the Hadamard gate (or matrix), which is crucial for creating initial superposition of states and is also used in the QFT to extract periodicity information. Bell states are essential in quantum information theory and have applications in quantum teleportation, superdense coding and quantum key distribution. By levervging quantum superposition and entanglement, quantum algs achieve significant speedups & enable secure communication protocols.

Step-by-Step Explanation of the Hadamard Matrix in Shor’s Algorithm

  1. Initialization and Superposition:

    • Hadamard Gate (H): The Hadamard gate is used to transform the initial state \(|0\rangle^{\otimes n}\) into an equal superposition of all possible \(2^n\) basis states.

      \[ H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \]

      For an \(n\)-qubit system, the Hadamard operation \(H^{\otimes n}\) creates the superposition:

      \[ |0\rangle^{\otimes n} \xrightarrow{H^{\otimes n}} \frac{1}{\sqrt{2^n}} \sum_{x=0}^{2^n-1} |x\rangle \]

  2. Modular Exponentiation:
    • In Shor’s algorithm, a quantum circuit is used to perform modular exponentiation, which is essential for creating the periodic function required for period finding. This step doesn’t directly involve the Hadamard gate but prepares the state for the Quantum Fourier Transform (QFT).
  3. Quantum Fourier Transform (QFT):

    • After modular exponentiation, the quantum state encodes information about the period of the function \(f(x) = a^x \mod N\). The QFT is applied to extract this period information. The QFT uses Hadamard gates along with controlled phase shifts.

      \[ QFT|x\rangle = \frac{1}{\sqrt{2^n}} \sum_{k=0}^{2^n-1} e^{2\pi i x k / 2^n} |k\rangle \]

  4. Measurement and Classical Post-Processing:
    • After applying the QFT, the qubits are measured. The measurement results are processed classically to determine the period \(r\). If \(r\) is even, classical algorithms are used to find the factors of \(N\).

Example: Simple 2-Qubit System

  1. Initialization:
    • Start with the state \(|00\rangle\).
  2. Apply Hadamard Gates:

    • Apply Hadamard gates to both qubits to create an equal superposition.

      \[ H \otimes H |00\rangle = \frac{1}{2} (|00\rangle + |01\rangle + |10\rangle + |11\rangle) \]

  3. Modular Exponentiation:
    • Assume a unitary operation \(U_f\) that maps \(|x\rangle |0\rangle\) to \(|x\rangle |f(x)\rangle\). For simplicity, let’s say \(f(x)\) has a periodic pattern.
  4. Apply QFT:

    • Apply the QFT to the superposed state.

      \[ QFT \left( \frac{1}{2} (|00\rangle + |01\rangle + |10\rangle + |11\rangle) \right) \]

      The QFT will transform this state into another superposition that encodes the period information.

  5. Measurement:
    • Measure the qubits to collapse the state to a basis state, giving information about the period.
  6. Classical Post-Processing:
    • Use the measured results to determine the period \(r\) and factorize \(N\).

Understanding Bell States

Bell states are specific quantum states of two qubits that represent the simplest and most powerful examples of quantum entanglement. They are fundamental in quantum information theory.

Types of Bell States:

  1. \(|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle)\)
  2. \(|\Phi^-\rangle = \frac{1}{\sqrt{2}} (|00\rangle - |11\rangle)\)
  3. \(|\Psi^+\rangle = \frac{1}{\sqrt{2}} (|01\rangle + |10\rangle)\)
  4. \(|\Psi^-\rangle = \frac{1}{\sqrt{2}} (|01\rangle - |10\rangle)\)

Creating a Bell State using Qiskit

from qiskit import QuantumCircuit, Aer, execute
from qiskit.visualization import plot_histogram

# Create a quantum circuit with 2 qubits
qc = QuantumCircuit(2)

# Apply Hadamard gate to the first qubit
qc.h(0)

# Apply CNOT gate, controlled by qubit 0 and targeting qubit 1
qc.cx(0, 1)

# Apply Z gate to qubit 0
qc.z(0)

# Apply X gate to qubit 1
qc.x(1)

# Measure all qubits
qc.measure_all()

# Draw the circuit
qc.draw('mpl')

# Execute the circuit
backend = Aer.get_backend('qasm_simulator')
result = execute(qc, backend, shots=1024).result()
counts = result.get_counts()

# Plot the results
plot_histogram(counts)

Decoherence in Quantum Systems

Decoherence is mathematically represented using density matrices and the Lindblad equation. Here’s a detailed look at the mathematical framework:

Density Matrix

In quantum mechanics, the state of a system can be described by a density matrix \(\rho\). For a pure state \(|\psi\rangle\), the density matrix is given by:

\[ \rho = |\psi\rangle \langle \psi| \]

For a mixed state, the density matrix is a statistical mixture of pure states:

\[ \rho = \sum_i p_i |\psi_i\rangle \langle \psi_i| \]

where \(p_i\) are the probabilities of the system being in the pure states \(|\psi_i\rangle\).

Decoherence and Reduced Density Matrix

When a quantum system interacts with its environment, we can describe the total system (system + environment) using a combined density matrix \(\rho_{total}\). If the system and environment are initially in a product state \(|\psi\rangle \otimes |\phi\rangle\), the density matrix for the total system is:

\[ \rho_{total} = \rho_{system} \otimes \rho_{environment} \]

After interaction, the system becomes entangled with the environment, and we obtain the reduced density matrix for the system by tracing out the environmental degrees of freedom:

\[ \rho_{system} = \text{Tr}_{environment}(\rho_{total}) \]

This partial trace operation sums over the environmental states, effectively “averaging out” the environmental degrees of freedom and leaving the reduced density matrix for the system.

Lindblad Equation

The time evolution of the density matrix, including the effects of decoherence, can be described by the Lindblad equation (or master equation). The Lindblad equation for a density matrix \(\rho\) is:

\[ \frac{d\rho}{dt} = -\frac{i}{\hbar} [H, \rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2} \{ L_k^\dagger L_k, \rho \} \right) \]

Here, - \(H\) is the Hamiltonian of the system. - \(L_k\) are the Lindblad operators representing the interaction with the environment. - \([H, \rho]\) is the commutator of \(H\) and \(\rho\). - \(\{ L_k^\dagger L_k, \rho \}\) is the anticommutator of \(L_k^\dagger L_k\) and \(\rho\).

The first term \(-\frac{i}{\hbar} [H, \rho]\) describes the unitary evolution of the system, while the second term \(\sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2} \{ L_k^\dagger L_k, \rho \} \right)\) accounts for the non-unitary evolution due to the environment, leading to decoherence.

Example: Decoherence in a Two-Level System (Qubit)

Consider a two-level system (qubit) interacting with its environment. The density matrix for a qubit can be written as:

\[ \rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \end{pmatrix} \]

Under decoherence, the off-diagonal elements (\(\rho_{01}\) and \(\rho_{10}\)) decay over time, representing the loss of coherence. This can be modeled by a Lindblad operator \(L = \sqrt{\gamma} \sigma_z\), where \(\gamma\) is the decoherence rate and \(\sigma_z\) is the Pauli z-matrix. The Lindblad equation for this system simplifies to:

[ = - [H, ] + (_z _z - ) ]

This equation describes how the qubit’s coherence (off-diagonal elements) decays over time, leading to a diagonal density matrix in the long-time limit, corresponding to a classical probabilistic mixture of states.

Point to Note

The mathematical representation of decoherence involves the use of density matrices to describe the quantum state of a system, and the Lindblad equation to model the time evolution of the density matrix under the influence of the environment. This framework captures the transition from quantum coherence to classical behavior, providing a detailed understanding of the decoherence process.

Example Calculation

Exponential Growth (E_n): \(E_n = 3E_{n-1} + 2\)

  • Base Case: \(E_0 = 1\)
  • First Iteration: \(E_1 = 3 \times 1 + 2 = 5\)
  • Second Iteration: \(E_2 = 3 \times 5 + 2 = 17\)
  • Third Iteration: \(E_3 = 3 \times 17 + 2 = 53\)

Fibonacci Sequence (F_n): \(F_n = F_{n-1} + F_{n-2}\)

  • Base Cases: \(F_0 = 0, F_1 = 1\)
  • First Iteration: \(F_2 = 1 + 0 = 1\)
  • Second Iteration: \(F_3 = 1 + 1 = 2\)
  • Third Iteration: \(F_4 = 2 + 1 = 3\)

Axiomatic Subjectivity Scale (X): \(X = \frac{Y_s}{Y_o}\)

  • Example: \(Y_s = 4, Y_o = 5\)
  • Calculation: \(X = \frac{4}{5} = 0.8\)

TimeSphere (Z): \(Z = \frac{n}{T}\)

  • Example: \(n = 5, T = 10\)
  • Calculation: \(Z = \frac{5}{10} = 0.5\)

Combined Equation: \(Intelligence_n = E_n \times (1 + F_n) \times X \times Y \times Z \times (A \times B \times C)\)

  • Example:
    • \(E_3 = 53\)
    • \(F_4 = 3\)
    • \(X = 0.8\)
    • \(Y = 0.8\)
    • \(Z = 0.5\)
    • \(A = 0.9, B = 0.85, C = 0.8\)
    • Combined: \(Intelligence_n = 53 \times (1 + 3) \times 0.8 \times 0.8 \times 0.5 \times (0.9 \times 0.85 \times 0.8)\)

This calculation shows how each component interacts dynamically, reflecting the comprehensive nature of the Universal Axiom framework.

Decoherence in Quantum Systems

Decoherence is mathematically represented using density matrices and the Lindblad equation. Here’s a detailed look at the mathematical framework:

Density Matrix

In quantum mechanics, the state of a system can be described by a density matrix \(\rho\). For a pure state \(|\psi\rangle\), the density matrix is given by:

\[ \rho = |\psi\rangle \langle \psi| \]

For a mixed state, the density matrix is a statistical mixture of pure states:

\[ \rho = \sum_i p_i |\psi_i\rangle \langle \psi_i| \]

where \(p_i\) are the probabilities of the system being in the pure states \(|\psi_i\rangle\).

Decoherence and Reduced Density Matrix: The mathematical representation of decoherence provides a detailed understanding of the transition from quantum coherence to classical behavior, essential for developing robust QGAN models.

When a quantum system interacts with its environment, we can describe the total system (system + environment) using a combined density matrix \(\rho_{total}\). If the system and environment are initially in a product state \(|\psi\rangle \otimes |\phi\rangle\), the density matrix for the total system is:

\[ \rho_{total} = \rho_{system} \otimes \rho_{environment} \]

After interaction, the system becomes entangled with the environment, and we obtain the reduced density matrix for the system by tracing out the environmental degrees of freedom:

\[ \rho_{system} = \text{Tr}_{environment}(\rho_{total}) \]

This partial trace operation sums over the environmental states, effectively “averaging out” the environmental degrees of freedom and leaving the reduced density matrix for the system.

Lindblad Equation

The time evolution of the density matrix, including the effects of decoherence, can be described by the Lindblad equation (or master equation). The Lindblad equation for a density matrix \(\rho\) is:

\[ \frac{d\rho}{dt} = -\frac{i}{\hbar} [H, \rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2} \{ L_k^\dagger L_k, \rho \} \right) \]

Here, - \(H\) is the Hamiltonian of the system. - \(L_k\) are the Lindblad operators representing the interaction with the environment. - \([H, \rho]\) is the commutator of \(H\) and \(\rho\). - \(\{ L_k^\dagger L_k, \rho \}\) is the anticommutator of \(L_k^\dagger L_k\) and \(\rho\).

The first term \(-\frac{i}{\hbar} [H, \rho]\) describes the unitary evolution of the system, while the second term \(\sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2} \{ L_k^\dagger L_k, \rho \} \right)\) accounts for the non-unitary evolution due to the environment, leading to decoherence.

Example: Decoherence in a Two-Level System (Qubit)

Consider a two-level system (qubit) interacting with its environment. The density matrix for a qubit can be written as:

\[ \rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \end{pmatrix} \]

Under decoherence, the off-diagonal elements (\(\rho_{01}\) and \(\rho_{10}\)) decay over time, representing the loss of coherence. This can be modeled by a Lindblad operator \(L = \sqrt{\gamma} \sigma_z\), where \(\gamma\) is the decoherence rate and \(\sigma_z\) is the Pauli z-matrix. The Lindblad equation for this system simplifies to:

\[ \frac{d\rho}{dt} = -\frac{i}{\hbar} [H, \rho] + \gamma (\sigma_z \rho \sigma_z - \rho) \]

This equation describes how the qubit’s coherence (

off-diagonal elements) decays over time, leading to a diagonal density matrix in the long-time limit, corresponding to a classical probabilistic mixture of states.

Mathematical Representation

Exponential Growth (E_n)

\[ E_n = 3E_{n-1} + 2 \] - Base Case: \(E_0 = 1\) - First Iteration: \(E_1 = 3 \times 1 + 2 = 5\) - Second Iteration: \(E_2 = 3 \times 5 + 2 = 17\) - Third Iteration: \(E_3 = 3 \times 17 + 2 = 53\)

Fibonacci Sequence (F_n)

\[ F_n = F_{n-1} + F_{n-2} \] - Base Cases: \(F_0 = 0, F_1 = 1\) - First Iteration: \(F_2 = 1 + 0 = 1\) - Second Iteration: \(F_3 = 1 + 1 = 2\) - Third Iteration: \(F_4 = 2 + 1 = 3\)

Axiomatic Subjectivity Scale (X)

\[ X = \frac{Y_s}{Y_o} \] - Example: \(Y_s = 4, Y_o = 5\) - Calculation: \(X = \frac{4}{5} = 0.8\)

TimeSphere (Z)

\[ Z = \frac{n}{T} \] - Example: \(n = 5, T = 10\) - Calculation: \(Z = \frac{5}{10} = 0.5\)

Combined Equation

\[ \text{Intelligence}_n = E_n \times (1 + F_n) \times X \times Y \times Z \times (A \times B \times C) \] - Example: - \(E_3 = 53\) - \(F_4 = 3\) - \(X = 0.8\) - \(Y = 0.8\) - \(Z = 0.5\) - \(A = 0.9, B = 0.85, C = 0.8\) - Combined Calculation: \[ \text{Intelligence}_n = 53 \times (1 + 3) \times 0.8 \times 0.8 \times 0.5 \times (0.9 \times 0.85 \times 0.8) \]

Decoherence in Quantum Systems

Density Matrix

In quantum mechanics, the state of a system is described by a density matrix \(\rho\). For a pure state \(|\psi\rangle\), the density matrix is: \[ \rho = |\psi\rangle \langle \psi| \]

For a mixed state, it is a statistical mixture of pure states: \[ \rho = \sum_i p_i |\psi_i\rangle \langle \psi_i| \]

Reduced Density Matrix

When a quantum system interacts with its environment, the total system (system + environment) is described by a combined density matrix \(\rho_{total}\): \[ \rho_{total} = \rho_{system} \otimes \rho_{environment} \]

The reduced density matrix for the system is obtained by tracing out the environmental degrees of freedom: \[ \rho_{system} = \text{Tr}_{environment}(\rho_{total}) \]

Lindblad Equation

The time evolution of the density matrix, including the effects of decoherence, is described by the Lindblad equation: \[ \frac{d\rho}{dt} = -\frac{i}{\hbar} [H, \rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2} \{ L_k^\dagger L_k, \rho \} \right) \]

Where: - \(H\) is the Hamiltonian of the system. - \(L_k\) are the Lindblad operators representing the interaction with the environment. - \([H, \rho]\) is the commutator of \(H\) and \(\rho\). - \(\{ L_k^\dagger L_k, \rho \}\) is the anticommutator of \(L_k^\dagger L_k\) and \(\rho\).

Example: Decoherence in a Two-Level System (Qubit)

The density matrix for a qubit can be written as: \[ \rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \end{pmatrix} \]

Under decoherence, the off-diagonal elements (\(\rho_{01}\) and \(\rho_{10}\)) decay over time, representing the loss of coherence. This can be modeled by a Lindblad operator \(L = \sqrt{\gamma} \sigma_z\), where \(\gamma\) is the decoherence rate and \(\sigma_z\) is the Pauli z-matrix. The Lindblad equation for this system simplifies to: \[ \frac{d\rho}{dt} = -\frac{i}{\hbar} [H, \rho] + \gamma (\sigma_z \rho \sigma_z - \rho) \]

This equation describes how the qubit’s coherence decays over time, leading to a diagonal density matrix in the long-time limit, corresponding to a classical probabilistic mixture of states.

NOTE TO SELF - Make sure to include Asymptotic Analysis

ARTICLES REGARDING: QGAN (Quantum Generative Adversarial Networks) and GaN (Gallium Nitride) materials in biodetection and biosensing:

  1. Dielectrically-Modulated GANFET Biosensor for Label-Free Detection of DNA and Avian Influenza Virus: Proposal and Modeling
    • Authors: S. Yadav, A. Das, S. Rewari
    • Publication: ECS Journal of Solid State Science, 2024
    • Link to Article
  2. High sensitivity label-free detection of HER2 using an Al–GaN/GaN high electron mobility transistor-based biosensor
    • Authors: S. Mishra, P. Kachhawa, A. K. Jain, R. R. Thakur
    • Publication: Lab on a Chip, 2022
    • Link to Article
  3. Rapid detection of biomolecules in a dielectric modulated GaN MOSHEMT
    • Authors: Shaveta, H. M. M. Ahmed, R. Chaujar
    • Publication: Journal of Materials Science: Materials, 2020
    • Link to Article
    • PDF Version
  4. A highly sensitive Nano Gap Embedded AlGaN/GaN HEMT sensor for Anti-IRIS antibody detection
    • Authors: R. Poonia, A. M. Bhat, C. Periasamy, C. Sahu
    • Publication: Micro and Nanostructures, 2022
    • Link to Article
  5. Plasmon-Coupled GaN Microcavity for WGM Lasing and Label-Free SERS Sensing of Biofluids
    • Authors: J. Sun, W. Mao, C. Xia, W. Wang, Q. Cui
    • Publication: Advanced Optical Materials, 2024
    • Link to Article
  6. Construction of AlGaN/GaN high-electron-mobility transistor-based biosensor for ultrasensitive detection of SARS-CoV-2 spike proteins and virions
    • Authors: C. Yang, J. Sun, Y. Zhang, J. Tang, Z. Liu, T. Zhan
    • Publication: Biosensors and Bioelectronics, 2024
    • Link to Article
  7. Dual gate AlGaN/GaN MOS-HEMT biosensor for electrical detection of biomolecules-analytical model
    • Authors: R. Mann, S. Rewari, S. Sharma
    • Publication: Semiconductor Science and Technology, 2023
    • Link to Article
  8. Modeling and simulation of AlGaN/GaN MOS-HEMT for biosensor applications
  9. Open gate AlGaN/GaN HEMT biosensor: Sensitivity analysis and optimization
    • Authors: P. Pal, Y. Pratap, M. Gupta, S. Kabra
    • Publication: Superlattices and Microstructures, 2021
    • Link to Article
  10. A Salivary Urea Sensor Based on Microsieve Disposable Gate AlGaN/GaN High Electron Mobility Transistor
    • Authors: G. Yang, B. Xu, H. Chang, Z. Gu, J. Li
    • Publication: Analytical Methods, 2024
    • Link to Article
  11. Detection of biological reactions by AlGaN/GaN biosensor
    • Authors: A. Podolska, R.M. Seeber, U.K. Mishra, K. Pfleger
    • Conference: Optoelectronic and Microelectronic Materials & Devices (COMMAD), 2012
    • Link to Article
  12. Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers: Mathenmatical Methods and Conceptual Foundations
    • Authors: Fano, G., & Blinder, S. M. (Eds.). (2001).
    • Book: Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers: Mathematical Methods and Conceptual Foundations.
    • Publisher: Springer International Publishing. https://doi.org/10.1007/978-3-030-34783-2**

Starting Code: python w qiskit:

import qiskit
import torch
import torch.nn as nn
import torch.optim as optim
from qiskit import Aer, transpile, assemble
from qiskit.circuit.library import TwoLocal

# Quantum generator
def quantum_generator(params, shots=1024):
    qc = qiskit.QuantumCircuit(1)
    qc.ry(params[0], 0)
    qc.measure_all()
    backend = Aer.get_backend('qasm_simulator')
    t_qc = transpile(qc, backend)
    qobj = assemble(t_qc, shots=shots)
    result = backend.run(qobj).result()
    counts = result.get_counts(qc)
    return counts

# Classical discriminator
class Discriminator(nn.Module):
    def __init__(self):
        super(Discriminator, self).__init__()
        self.fc = nn.Sequential(
            nn.Linear(1, 10),
            nn.ReLU(),
            nn.Linear(10, 1),
            nn.Sigmoid()
        )
    
    def forward(self, x):
        return self.fc(x)

# Training
discriminator = Discriminator()
optimizer = optim.Adam(discriminator.parameters(), lr=0.001)
criterion = nn.BCELoss()

for epoch in range(1000):
    real_data = torch.tensor([[1.0], [0.0]])
    fake_data = torch.tensor([[0.5], [0.5]])  # Simplified example

    real_labels = torch.ones(2, 1)
    fake_labels = torch.zeros(2, 1)
    
    optimizer.zero_grad()
    
    real_output = discriminator(real_data)
    real_loss = criterion(real_output, real_labels)
    
    fake_output = discriminator(fake_data)
    fake_loss = criterion(fake_output, fake_labels)
    
    loss = real_loss + fake_loss
    loss.backward()
    optimizer.step()
    
    if epoch % 100 == 0:
        print(f'Epoch {epoch}, Loss: {loss.item()}')

print("Training completed.")

Part 2 and 3 - code workings - in progress:

# PART 2


import numpy as np
from tensorflow.keras.models import Sequential, Model
from tensorflow.keras.layers import Dense, LeakyReLU, BatchNormalization, Reshape, Flatten, Dropout, Conv2D, Conv2DTranspose
from tensorflow.keras.optimizers import Adam
import matplotlib.pyplot as plt

def preprocess_image(image_path):
    # Load and preprocess the image
    image = Image.open(image_path).convert('L')
    image = image.resize((64, 64))  # Resize to smaller dimensions for easier processing
    image_array = np.array(image)
    image_array = (image_array - 127.5) / 127.5  # Normalize to range [-1, 1]
    return image_array

# Preprocess immages
image_array = preprocess_image(image_file_path)
image_array = np.expand_dims(image_array, axis=-1)  # Add a channel dimension

# Define generator model
def build_generator():
    model = Sequential()
    model.add(Dense(256*8*8, activation="relu", input_dim=100))
    model.add(Reshape((8, 8, 256)))
    model.add(Conv2DTranspose(128, kernel_size=4, strides=2, padding="same"))
    model.add(LeakyReLU(alpha=0.01))
    model.add(BatchNormalization(momentum=0.8))
    model.add(Conv2DTranspose(64, kernel_size=4, strides=2, padding="same"))
    model.add(LeakyReLU(alpha=0.01))
    model.add(BatchNormalization(momentum=0.8))
    model.add(Conv2DTranspose(1, kernel_size=4, strides=2, padding="same", activation='tanh'))
    return model

# Define discriminator model
def build_discriminator():
    model = Sequential()
    model.add(Conv2D(64, kernel_size=4, strides=2, input_shape=(64, 64, 1), padding="same"))
    model.add(LeakyReLU(alpha=0.01))
    model.add(Dropout(0.3))
    model.add(Conv2D(128, kernel_size=4, strides=2, padding="same"))
    model.add(LeakyReLU(alpha=0.01))
    model.add(Dropout(0.3))
    model.add(Flatten())
    model.add(Dense(1, activation='sigmoid'))
    return model

# Construct & compile  GAN
discriminator = build_discriminator()
discriminator.compile(loss='binary_crossentropy', optimizer=Adam(0.0002, 0.5), metrics=['accuracy'])

generator = build_generator()

z = tf.keras.Input(shape=(100,))
img = generator(z)

discriminator.trainable = False
valid = discriminator(img)

combined = Model(z, valid)
combined.compile(loss='binary_crossentropy', optimizer=Adam(0.0002, 0.5))

# maybe training loop
def train(epochs, batch_size=128):
    X_train = np.array([image_array])  # Training on the single provided image
    valid = np.ones((batch_size, 1))
    fake = np.zeros((batch_size, 1))

    for epoch in range(epochs):
        idx = np.random.randint(0, X_train.shape[0], batch_size)
        imgs = X_train[idx]

        noise = np.random.normal(0, 1, (batch_size, 100))
        gen_imgs = generator.predict(noise)

        d_loss_real = discriminator.train_on_batch(imgs, valid)
        d_loss_fake = discriminator.train_on_batch(gen_imgs, fake)
        d_loss = 0.5 * np.add(d_loss_real, d_loss_fake)

        g_loss = combined.train_on_batch(noise, valid)

        print(f"{epoch} [D loss: {d_loss[0]:.4f}, acc.: {100*d_loss[1]:.2f}%] [G loss: {g_loss:.4f}]")

# Train GAN
train(epochs=10000, batch_size=32)



# PART 3
import numpy as np
from tensorflow.keras.models import Sequential, Model
from tensorflow.keras.layers import Dense, LeakyReLU, BatchNormalization, Reshape, Flatten, Dropout, Conv2D, Conv2DTranspose
from tensorflow.keras.optimizers import Adam
import matplotlib.pyplot as plt

def preprocess_image(image_path):
    # Load and preprocess the image
    image = Image.open(image_path).convert('L')
    image = image.resize((64, 64))  # Resize to smaller dimensions for easier processing
    image_array = np.array(image)
    image_array = (image_array - 127.5) / 127.5  # Normalize to range [-1, 1]
    return image_array

# Preprocess images
image_array = preprocess_image(new_image_file_path)
image_array = np.expand_dims(image_array, axis=-1)  # Add a channel dimension

# Define generator model
def build_generator():
    model = Sequential()
    model.add(Dense(256*8*8, activation="relu", input_dim=100))
    model.add(Reshape((8, 8, 256)))
    model.add(Conv2DTranspose(128, kernel_size=4, strides=2, padding="same"))
    model.add(LeakyReLU(alpha=0.01))
    model.add(BatchNormalization(momentum=0.8))
    model.add(Conv2DTranspose(64, kernel_size=4, strides=2, padding="same"))
    model.add(LeakyReLU(alpha=0.01))
    model.add(BatchNormalization(momentum=0.8))
    model.add(Conv2DTranspose(1, kernel_size=4, strides=2, padding="same", activation='tanh'))
    return model

# Define discriminator model
def build_discriminator():
    model = Sequential()
    model.add(Conv2D(64, kernel_size=4, strides=2, input_shape=(64, 64, 1), padding="same"))
    model.add(LeakyReLU(alpha=0.01))
    model.add(Dropout(0.3))
    model.add(Conv2D(128, kernel_size=4, strides=2, padding="same"))
    model.add(LeakyReLU(alpha=0.01))
    model.add(Dropout(0.3))
    model.add(Flatten())
    model.add(Dense(1, activation='sigmoid'))
    return model

# Construct & compile the GAN
discriminator = build_discriminator()
discriminator.compile(loss='binary_crossentropy', optimizer=Adam(0.0002, 0.5), metrics=['accuracy'])

generator = build_generator()

z = tf.keras.Input(shape=(100,))
img = generator(z)

discriminator.trainable = False
valid = discriminator(img)

combined = Model(z, valid)
combined.compile(loss='binary_crossentropy', optimizer=Adam(0.0002, 0.5))

# maybe  training loop
def train(epochs, batch_size=128):
    X_train = np.array([image_array])  # Training on the single provided image
    valid = np.ones((batch_size, 1))
    fake = np.zeros((batch_size, 1))

    for epoch in range(epochs):
        idx = np.random.randint(0, X_train.shape[0], batch_size)
        imgs = X_train[idx]

        noise = np.random.normal(0, 1, (batch_size, 100))
        gen_imgs = generator.predict(noise)

        d_loss_real = discriminator.train_on_batch(imgs, valid)
        d_loss_fake = discriminator.train_on_batch(gen_imgs, fake)
        d_loss = 0.5 * np.add(d_loss_real, d_loss_fake)

        g_loss = combined.train_on_batch(noise, valid)

        print(f"{epoch} [D loss: {d_loss[0]:.4f}, acc.: {100*d_loss[1]:.2f}%] [G loss: {g_loss:.4f}]")

# Train  GAN
train(epochs=10000, batch_size=32)

From Larson’s whiteboard QGAN - basic portrayal of elements and structure shown - translated (err sort of) into a beginnings of qgan implementation

Interpretation of the Whiteboard Diagram

  1. Components Overview:
    • D: Discriminator network.
    • G: Generator network (potentially an encoder-decoder setup).
    • E: Encoder.
    • A1, A2: Possibly indicating layers or specific parts of the network.
    • Input and Output Data: The flow of data through the system.
    • Latent Space/Z: Representation of latent space for the generator.
  2. Flow and Interaction:
    • Data flows from the input, through the encoder, and into the discriminator.
    • There is interaction between different parts of the network, suggesting a feedback loop or adversarial training.

Implementation Steps for QGAN

Step 1: Define the Network Architecture

  1. Generator (G):
    • Encoder to compress input data into a latent representation \(Z\).
    • Decoder to generate synthetic data from \(Z\).
  2. Discriminator (D):
    • Classifies data as real or synthetic.
    • Works adversarially with the generator to improve both networks.
  3. Encoder (E):
    • Part of the generator that compresses data into latent space.
    • Represents features of the input data.
  4. Latent Space (Z):
    • Represents the distribution from which the generator draws samples to create synthetic data.

Step 2: Data Collection and Preprocessing

  • Collect geological and atmospheric data.
  • Preprocess data for input into the quantum circuits.
  • Normalize and encode data for consistency.

Step 3: Quantum Circuit Design for Generator

  • Use quantum gates and circuits to design the generator.
  • Implement the quantum encoder and decoder.

Step 4: Classical-Quantum Hybrid Discriminator

  • Use classical neural networks for the discriminator.
  • Integrate quantum circuits to enhance feature detection.

Step 5: Training Process

  1. Adversarial Training:
    • Train the generator to produce realistic synthetic data.
    • Train the discriminator to distinguish between real and synthetic data.
  2. Loss Functions:
    • Use loss functions suitable for GANs, such as binary cross-entropy.
  3. Optimization:
    • Use gradient descent and quantum gradient descent methods.

Code outline version 4 using Python with TensorFlow and Qiskit:

import tensorflow as tf
from qiskit import QuantumCircuit, Aer, transpile
from qiskit.providers.aer import AerSimulator
from qiskit.utils import QuantumInstance
from qiskit_machine_learning.algorithms import VQC
from qiskit_machine_learning.neural_networks import CircuitQNN
from qiskit.circuit.library import TwoLayerQNN

# Define quantum generator
def create_generator():
    # Quantum circuit for the generator
    generator_circuit = QuantumCircuit(2)
    generator_circuit.h([0, 1])
    generator_circuit.cx(0, 1)
    return generator_circuit

# Define classical discriminator
def create_discriminator(input_shape):
    model = tf.keras.Sequential([
        tf.keras.layers.Dense(128, activation='relu', input_shape=input_shape),
        tf.keras.layers.Dense(64, activation='relu'),
        tf.keras.layers.Dense(1, activation='sigmoid')
    ])
    return model

# Instantiate  generator
generator_circuit = create_generator()
simulator = AerSimulator()
quantum_instance = QuantumInstance(simulator)
qnn = CircuitQNN(generator_circuit, quantum_instance=quantum_instance)
generator = VQC(qnn, optimizer=tf.keras.optimizers.Adam(learning_rate=0.01))

# Define input shape
input_shape = (128,)

# Instantiate discriminator
discriminator = create_discriminator(input_shape)

# Compile discriminator
discriminator.compile(loss='binary_crossentropy', optimizer='adam', metrics=['accuracy'])

# Adversarial model combining generator & discriminator
def combined_model(generator, discriminator):
    discriminator.trainable = False
    model = tf.keras.Sequential([generator, discriminator])
    return model

# Compile combined model
combined = combined_model(generator, discriminator)
combined.compile(loss='binary_crossentropy', optimizer='adam')

# Training loop (simplified - dare I say too simplified? :( )
epochs = 1000
batch_size = 32
for epoch in range(epochs):
    # Generate synthetic data
    noise = tf.random.normal([batch_size, 128])
    generated_data = generator.predict(noise)

    # Get real data and combine with generated data
    real_data = get_real_data(batch_size)  # Function to fetch real data
    combined_data = tf.concat([real_data, generated_data], axis=0)

    # Labels for real & synthetic data
    labels = tf.concat([tf.ones((batch_size, 1)), tf.zeros((batch_size, 1))], axis=0)

    # Train discriminator
    discriminator.train_on_batch(combined_data, labels)

    # Train generator via combined model
    noise = tf.random.normal([batch_size, 128])
    misleading_labels = tf.ones((batch_size, 1))
    combined.train_on_batch(noise, misleading_labels)

    if epoch % 100 == 0:
        print(f'Epoch: {epoch}, Discriminator Loss: {discriminator.loss}, Generator Loss: {combined.loss}')

Moving along - additional data preprocessing, hyperparameter tuning, and integration with GIS tools till needed

Mathematical Representation

  1. Data Input (D)
    • Includes geological & atmospheric data (fossil particle data, temperature, pressure, historical oil deposits).
  2. Classical Preprocessing (P)
    • Cleaning, normalizing, and feature extraction: \(P(X)\)
  3. Quantum Data Encoding (E)
    • Encoding classical data into quantum states: \(\ket{\psi} = E(P(X))\)
  4. Quantum Generator (G)
    • Generates synthetic data: \(G(\theta_G)\)
    • \(\theta_G\) represents the parameters of the generator
  5. Quantum Discriminator (D)
    • Evaluates authenticity: \(D(\theta_D)\)
    • \(\theta_D\) represents the parameters of the discriminator
  6. Feedback Loop
    • Discriminator provides feedback to the generator to improve the generation of synthetic data.
  7. Classical Autoencoder (A)
    • Used for anomaly detection: \(A(X)\)
    • Compares input data with reconstructed data to detect anomalies.
  8. Environmental Simulation (S)
    • Simulates environmental conditions and their impact on particle generation and movement.
  9. GIS Integration (I)
    • Visualizes QGAN results: anomalies and simulation outputs.

Python Code

import torch
import torch.nn as nn
import torch.optim as optim
from qiskit import QuantumCircuit, Aer, transpile
from qiskit.utils import QuantumInstance
from qiskit.circuit.library import TwoLocal
from qiskit_machine_learning.algorithms import QGAN

# Define the generator model
class Generator(nn.Module):
    def __init__(self, input_size, hidden_size, output_size):
        super(Generator, self).__init__()
        self.main = nn.Sequential(
            nn.Linear(input_size, hidden_size),
            nn.ReLU(),
            nn.Linear(hidden_size, output_size),
            nn.Tanh()
        )

    def forward(self, input):
        return self.main(input)

# Define the discriminator model
class Discriminator(nn.Module):
    def __init__(self, input_size, hidden_size, output_size):
        super(Discriminator, self).__init__()
        self.main = nn.Sequential(
            nn.Linear(input_size, hidden_size),
            nn.ReLU(),
            nn.Linear(hidden_size, output_size),
            nn.Sigmoid()
        )

    def forward(self, input):
        return self.main(input)

# Parameters
input_size = 100
hidden_size = 256
output_size = 1

# Instantiate the models
generator = Generator(input_size, hidden_size, output_size)
discriminator = Discriminator(output_size, hidden_size, 1)

# Optimizers
g_optimizer = optim.Adam(generator.parameters(), lr=0.0002)
d_optimizer = optim.Adam(discriminator.parameters(), lr=0.0002)

# Loss function
criterion = nn.BCELoss()

# Quantum Instance
quantum_instance = QuantumInstance(backend=Aer.get_backend('qasm_simulator'), shots=1024)

# QGAN
qgan = QGAN(generator=generator, discriminator=discriminator, quantum_instance=quantum_instance)

# Training loop
num_epochs = 10000
for epoch in range(num_epochs):
    # Train Discriminator
    real_data = torch.randn(batch_size, output_size)
    fake_data = generator(torch.randn(batch_size, input_size)).detach()
    real_labels = torch.ones(batch_size, 1)
    fake_labels = torch.zeros(batch_size, 1)

    d_optimizer.zero_grad()
    outputs = discriminator(real_data)
    d_loss_real = criterion(outputs, real_labels)
    d_loss_real.backward()

    outputs = discriminator(fake_data)
    d_loss_fake = criterion(outputs, fake_labels)
    d_loss_fake.backward()

    d_optimizer.step()

    # Train Generator
    noise = torch.randn(batch_size, input_size)
    g_optimizer.zero_grad()
    fake_data = generator(noise)
    outputs = discriminator(fake_data)
    g_loss = criterion(outputs, real_labels)
    g_loss.backward()
    g_optimizer.step()

    if (epoch+1) % 100 == 0:
        print(f'Epoch [{epoch+1}/{num_epochs}], d_loss: {d_loss_real.item()+d_loss_fake.item():.4f}, g_loss: {g_loss.item():.4f}')

print("Training completed")

Anomaly Detection with Classical Autoencoder

class Autoencoder(nn.Module):
    def __init__(self, input_size, hidden_size):
        super(Autoencoder, self).__init__()
        self.encoder = nn.Sequential(
            nn.Linear(input_size, hidden_size),
            nn.ReLU()
        )
        self.decoder = nn.Sequential(
            nn.Linear(hidden_size, input_size),
            nn.Sigmoid()
        )

    def forward(self, x):
        encoded = self.encoder(x)
        decoded = self.decoder(encoded)
        return decoded

# Instantiate the autoencoder
autoencoder = Autoencoder(input_size, hidden_size)
ae_optimizer = optim.Adam(autoencoder.parameters(), lr=0.001)
ae_criterion = nn.MSELoss()

# Training loop for autoencoder
num_epochs = 1000
for epoch in range(num_epochs):
    for data in dataloader:
        inputs = data
        ae_optimizer.zero_grad()
        outputs = autoencoder(inputs)
        loss = ae_criterion(outputs, inputs)
        loss.backward()
        ae_optimizer.step()
    
    if (epoch+1) % 100 == 0:
        print(f'Epoch [{epoch+1}/{num_epochs}], Loss: {loss.item():.4f}')

print("Autoencoder training completed")

# Anomaly detection
with torch.no_grad():
    for data in dataloader:
        outputs = autoencoder(data)
        loss = ae_criterion(outputs, data)
        if loss.item() > threshold:
            print("Anomaly detected")

Using Keras (Python)

Quantum GAN with Keras

import numpy as np
from keras.models import Sequential
from keras.layers import Dense, LeakyReLU, BatchNormalization, Reshape, Flatten
from keras.optimizers import Adam
from keras.datasets import mnist

# Load and preprocess the data
(X_train, _), (_, _) = mnist.load_data()
X_train = (X_train.astype(np.float32) - 127.5) / 127.5
X_train = X_train.reshape(X_train.shape[0], 784)

# Define the generator
def build_generator():
    model = Sequential()
    model.add(Dense(256, input_dim=100))
    model.add(LeakyReLU(alpha=0.2))
    model.add(BatchNormalization(momentum=0.8))
    model.add(Dense(512))
    model.add(LeakyReLU(alpha=0.2))
    model.add(BatchNormalization(momentum=0.8))
    model.add(Dense(1024))
    model.add(LeakyReLU(alpha=0.2))
    model.add(BatchNormalization(momentum=0.8))
    model.add(Dense(784, activation='tanh'))
    model.add(Reshape((28, 28, 1)))
    return model

# Define the discriminator
def build_discriminator():
    model = Sequential()
    model.add(Flatten(input_shape=(28, 28, 1)))
    model.add(Dense(512))
    model.add(LeakyReLU(alpha=0.2))
    model.add(Dense(256))
    model.add(LeakyReLU(alpha=0.2))
    model.add(Dense(1, activation='sigmoid'))
    return model

# Build and compile the models
optimizer = Adam(0.0002, 0.5)
generator = build_generator()
discriminator = build_discriminator()
discriminator.compile(loss='binary_crossentropy', optimizer=optimizer, metrics=['accuracy'])

# Combined model
discriminator.trainable = False
gan = Sequential([generator, discriminator])
gan.compile(loss='binary_crossentropy', optimizer=optimizer)

# Training the GAN
def train(epochs, batch_size=128, save_interval=50):
    X_train = (np.random.rand(60000, 784) - 0.5) * 2

    half_batch = int(batch_size / 2)

    for epoch in range(epochs):
        # Train discriminator
        idx = np.random.randint(0, X_train.shape[0], half_batch)
        real_imgs = X_train[idx]
        noise = np.random.normal(0, 1, (half_batch, 100))
        fake_imgs = generator.predict(noise)

        d_loss_real = discriminator.train_on_batch(real_imgs, np.ones((half_batch, 1)))
        d_loss_fake = discriminator.train_on_batch(fake_imgs, np.zeros((half_batch, 1)))
        d_loss = 0.5 * np.add(d_loss_real, d_loss_fake)

        # Train generator
        noise = np.random.normal(0, 1, (batch_size, 100))
        g_loss = gan.train_on_batch(noise, np.ones((batch_size, 1)))

        # Print progress
        if epoch % save_interval == 0:
            print(f"{epoch} [D loss: {d_loss[0]}, acc.: {100*d_loss[1]}%] [G loss: {g_loss}]")

# Run the training
train(epochs=10000, batch_size=64, save_interval=200)

Autoencoder Keras

from keras.models import Model
from keras.layers import Input, Dense

# Define the autoencoder model
input_dim = X_train.shape[1]
encoding_dim = 32

input_layer = Input(shape=(input_dim,))
encoder = Dense(encoding_dim, activation="relu")(input_layer)
decoder = Dense(input_dim, activation="sigmoid")(encoder)

autoencoder = Model(inputs=input_layer, outputs=decoder)
autoencoder.compile(optimizer='adam', loss='binary_crossentropy')

# Train the autoencoder
autoencoder.fit(X_train, X_train, epochs=50, batch_size=256, shuffle=True)

# Anomaly detection
reconstructed = autoencoder.predict(X_train)
losses = np.mean(np.power(X_train - reconstructed, 2), axis=1)
threshold = np.percentile(losses, 95)
anomalies = losses > threshold

print(f"Detected {np.sum(anomalies)} anomalies")

Using MATLAB

GAN in MATLAB

% Define the generator network
layersGenerator = [
    imageInputLayer([1 1 100],'Normalization','none')
    fullyConnectedLayer(7*7*128)
    reluLayer
    transposedConv2dLayer(7,128,'Cropping','same')
    batchNormalizationLayer
    reluLayer
    transposedConv2dLayer(4,64,'Cropping','same','Stride',2)
    batchNormalizationLayer
    reluLayer
    transposedConv2dLayer(4,1,'Cropping','same','Stride',2)
    tanhLayer];

lgraphGenerator = layerGraph(layersGenerator);

% Define the discriminator network
layersDiscriminator = [
    imageInputLayer([28 28 1])
    convolution2dLayer(4,64,'Stride',2,'Padding',1)
    leakyReluLayer(0.2)
    convolution2dLayer(4,128,'Stride',2,'Padding',1)
    batchNormalizationLayer
    leakyReluLayer(0.2)
    fullyConnectedLayer(1)
    sigmoidLayer];

lgraphDiscriminator = layerGraph(layersDiscriminator);

% Training options
options = trainingOptions('adam', ...
    'InitialLearnRate',0.0002, ...
    'MaxEpochs',100, ...
    'MiniBatchSize',128);

% Train the GAN
[netG,netD] = trainGAN(lgraphGenerator, lgraphDiscriminator, options);

Autoencoder for Anomaly Detection in MATLAB

% Load the data
[XTrain, ~] = digitTrain4DArrayData;

% Define the autoencoder
inputSize = [28 28 1];
encodingDimension = 64;

layers = [
    imageInputLayer(inputSize)
    fullyConnectedLayer(encodingDimension)
    reluLayer
    fullyConnectedLayer(prod(inputSize))
    reshapeLayer(inputSize)];

autoenc = trainAutoencoder(layers, XTrain, ...
    'MaxEpochs', 100, ...
    'L2WeightRegularization', 0.004, ...
    'SparsityRegularization', 4, ...
    'SparsityProportion', 0.15);

% Anomaly detection
XReconstructed = predict(autoenc, XTrain);
loss = mean((XTrain(:) - XReconstructed(:)).^2);
threshold = prctile(loss, 95);
anomalies = loss > threshold;

disp(['Detected ', num2str(sum(anomalies)), ' anomalies']);

TensorFlow Quantum (TFQ)

TensorFlow Quantum allows me iintegrating quantum computing layers w classical machine learning models in TensorFlow. Attempting ti implement a simple Quantum GAN using tfQ

QGAN TFQ

import tensorflow as tf
import tensorflow_quantum as tfq
import cirq
import sympy
import numpy as np

# Create a simple quantum generator
def create_quantum_generator(qubits):
    circuit = cirq.Circuit()
    for qubit in qubits:
        circuit.append(cirq.rx(sympy.Symbol(f"theta_{qubit}")).on(qubit))
    return circuit

# Create a simple quantum discriminator
def create_quantum_discriminator(qubits):
    circuit = cirq.Circuit()
    for qubit in qubits:
        circuit.append(cirq.ry(sympy.Symbol(f"phi_{qubit}")).on(qubit))
    return circuit

# Create qubits
qubits = [cirq.GridQubit(0, i) for i in range(4)]

# Create the generator and discriminator circuits
generator_circuit = create_quantum_generator(qubits)
discriminator_circuit = create_quantum_discriminator(qubits)

# Define the quantum data encoding
def quantum_data_encoding(data):
    circuit = cirq.Circuit()
    for i, value in enumerate(data):
        circuit.append(cirq.rx(value).on(qubits[i]))
    return circuit

# Define the generator model
generator = tf.keras.Sequential([
    tf.keras.layers.Input(shape=(4,)),
    tfq.layers.PQC(generator_circuit, sympy.symbols([f"theta_{qubit}" for qubit in qubits]))
])

# Define the discriminator model
discriminator = tf.keras.Sequential([
    tf.keras.layers.Input(shape=(), dtype=tf.dtypes.string),
    tfq.layers.PQC(discriminator_circuit, sympy.symbols([f"phi_{qubit}" for qubit in qubits]))
])

# Define the loss and optimizer
bce = tf.keras.losses.BinaryCrossentropy(from_logits=True)
generator_optimizer = tf.keras.optimizers.Adam(1e-4)
discriminator_optimizer = tf.keras.optimizers.Adam(1e-4)

# Training step
@tf.function
def train_step(real_data):
    noise = tf.random.normal([batch_size, 4])

    with tf.GradientTape() as gen_tape, tf.GradientTape() as disc_tape:
        generated_data = generator(noise, training=True)
        real_output = discriminator(tfq.convert_to_tensor([quantum_data_encoding(x) for x in real_data]), training=True)
        fake_output = discriminator(tfq.convert_to_tensor([quantum_data_encoding(x) for x in generated_data]), training=True)

        gen_loss = bce(tf.ones_like(fake_output), fake_output)
        disc_loss = bce(tf.ones_like(real_output), real_output) + bce(tf.zeros_like(fake_output), fake_output)

    gradients_of_generator = gen_tape.gradient(gen_loss, generator.trainable_variables)
    gradients_of_discriminator = disc_tape.gradient(disc_loss, discriminator.trainable_variables)

    generator_optimizer.apply_gradients(zip(gradients_of_generator, generator.trainable_variables))
    discriminator_optimizer.apply_gradients(zip(gradients_of_discriminator, discriminator.trainable_variables))

    return gen_loss, disc_loss

# Training loop
epochs = 1000
batch_size = 32

for epoch in range(epochs):
    real_data = np.random.uniform(-1, 1, size=(batch_size, 4))
    gen_loss, disc_loss = train_step(real_data)
    if epoch % 100 == 0:
        print(f"Epoch {epoch}: Generator Loss: {gen_loss.numpy()}, Discriminator Loss: {disc_loss.numpy()}")

MATLAB Deep Learning Toolbox - giving it a go.

GAN in MATLAB with DLT

NOTE this is a basic implementation of a QgAN using TensorFlow Quantum & a classical GAN autoencoder using MATLAb’s Deep Learning Toolbox. The TFQ example integrates quantum computing layer. MATLAB examole uses classical deep learning layers 2 test using similar goals

% Define the generator network
layersGenerator = [
    imageInputLayer([1 1 100],'Normalization','none')
    fullyConnectedLayer(7*7*128)
    reluLayer
    transposedConv2dLayer(7,128,'Cropping','same')
    batchNormalizationLayer
    reluLayer
    transposedConv2dLayer(4,64,'Cropping','same','Stride',2)
    batchNormalizationLayer
    reluLayer
    transposedConv2dLayer(4,1,'Cropping','same','Stride',2)
    tanhLayer];

lgraphGenerator = layerGraph(layersGenerator);

% Define the discriminator network
layersDiscriminator = [
    imageInputLayer([28 28 1])
    convolution2dLayer(4,64,'Stride',2,'Padding',1)
    leakyReluLayer(0.2)
    convolution2dLayer(4,128,'Stride',2,'Padding',1)
    batchNormalizationLayer
    leakyReluLayer(0.2)
    fullyConnectedLayer(1)
    sigmoidLayer];

lgraphDiscriminator = layerGraph(layersDiscriminator);

% Training options
options = trainingOptions('adam', ...
    'InitialLearnRate',0.0002, ...
    'MaxEpochs',100, ...
    'MiniBatchSize',128);

% Train the GAN
[netG,netD] = trainGAN(lgraphGenerator, lgraphDiscriminator, options);

AAD in MATLAB (autencode anomaly detect)

% Load the data
[XTrain, ~] = digitTrain4DArrayData;

% Define the autoencoder
inputSize = [28 28 1];
encodingDimension = 64;

layers = [
    imageInputLayer(inputSize)
    fullyConnectedLayer(encodingDimension)
    reluLayer
    fullyConnectedLayer(prod(inputSize))
    reshapeLayer(inputSize)];

autoenc = trainAutoencoder(layers, XTrain, ...
    'MaxEpochs', 100, ...
    'L2WeightRegularization', 0.004, ...
    'SparsityRegularization', 4, ...
    'SparsityProportion', 0.15);

% Anomaly detection
XReconstructed = predict(autoenc, XTrain);
loss = mean((XTrain(:) - XReconstructed(:)).^2);
threshold = prctile(loss, 95);
anomalies = loss > threshold;

disp(['Detected ', num2str(sum(anomalies)), ' anomalies']);