title: “mqpa” author: “Jessica McPhaul” date: “2024-07-19” output: html_document
MQPA: A (Potentially) New Paradigm in Quantum Computing
In practical terms, particle dynamics exhibit free movement, akin to the natural flow of water and wind. Extreme weather events like tsunamis, tornadoes, and hurricanes showcase diverse and unpredictable movement patterns, significantly impacting society. Similar to the butterfly effect, wherein one physical force influences another, particles intrinsically react in a non-linear fashion. It’s reasonable to hypothesize that particles can move both forwards and backwards, with arbitrary gates marking Eigenvalues at each “step” or movement, despite the lack of formal mathematical proof. Current theories and algorithms do not fully elucidate the juxtaposition of particle effects while maintaining entanglement and continuous superposition. In scenarios involving quantum tunneling, particles may traverse different benchmarks simultaneously from distinct Eigensmarks, with one particle moving backwards and another moving laterally. Specifically, the resulting interactions are quantified by the particle’s spin, examining how the spin influences the entanglement and what alterations occur with each movement—whether additional particles attach or the overall state evolves from the initial Eigenstate to the present configuration.
In the real world, particles move around freely, much like water flowing back and forth or the wind blowing. Extreme weather events such as tsunamis, tornadoes, and hurricanes move unpredictably and cause significant disruptions. Just like in the butterfly effect, where a small change can influence larger events, particles interact in a complex way, not just in straight lines. I can assume that particles can move in different directions, and the checkpoints I use to measure their movement are just markers for their energy levels, even though this isn’t fully proven by math yet.
I haven’t found any theories or algorithms that explain how particles stay connected and influence each other while being in a state of constant uncertainty. However, even with quantum tunneling, particles can cross different checkpoints at the same time from different starting points—one might move backwards while another moves sideways. The effects of this movement are measured by the particle’s spin, showing how the connection between particles changes with each movement, whether other particles join in, and how the situation changes from the start to the current state. This scientific supposition is the basis for building my theory.
Particles move in complex, non-linear ways. Water ebbs and flows, winds swirl, and storms surge. These dynamic movements, often unpredictable and interconnected, are reminiscent of the behavior of quantum particles. Current quantum algorithms, however, tend to be linear and don’t fully capture this inherent dynamism.
MQPA (McPhaul Quantum Pathway Algorithm) proposes a radical shift in quantum computing. It embraces the non-linearity and interconnectedness of quantum particles, allowing quantum gates (operations that change the state of qubits, the quantum equivalent of bits) to be applied dynamically and adaptively.
Instead of following a predetermined sequence, MQPA determines which gate to apply based on the qubit’s current state and movement. This means qubits can interact with gates multiple times, move backward, or even change direction, mirroring the fluidity of natural phenomena.
Initialization: Qubits are prepared in their initial states, and quantum gates are placed at arbitrary points.
Dynamic Gate Application: At each step, the algorithm analyzes the qubit’s state and movement to determine which gate to apply next. This allows for a highly flexible and adaptive computational process.
State Update and Measurement: The qubit’s state is updated after each gate application, and measurements are taken to track its evolution.
Distance and Relationship Analysis: MQPA calculates the distances and relationships between qubit states after each set of gate operations. This reveals patterns, correlations, and anomalies that can provide insights into the underlying quantum system.
Multi-Qubit Interactions: The algorithm can be extended to handle interactions between multiple qubits, further enriching the analysis.
Probabilistic and Exponential Insights: MQPA leverages statistical methods to calculate probabilities and explore exponential growth or decay in qubit states and interactions.
MQPA offers several potential advantages over traditional quantum algorithms:
Flexibility: Its dynamic gate application allows for more adaptability and versatility in solving complex problems.
Efficiency: By optimizing gate sequences based on the qubit’s behavior, MQPA could potentially reduce computational overhead.
Deeper Insights: MQPA’s focus on measuring distances and relationships between qubit states could uncover new patterns and behaviors in quantum systems.
Applications: MQPA could have applications in fields such as quantum chemistry, materials science, optimization, and machine learning.
MQPA is still a theoretical proposal, but it represents a promising new direction in quantum computing. Further research and development will be needed to validate its potential and explore its full range of applications.
Implementing an MQPA-inspired approach on the QM9 dataset, followed by training an autoencoder for anomaly detection. This will be done using Python, TensorFlow, and other relevant libraries. The following steps will take me through the process:
First, let’s set up the environment and import the necessary libraries.
# Install the necessary libraries
!pip install tensorflow pandas scikit-learn matplotlib
# Import libraries
import tensorflow as tf
from tensorflow.keras.layers import Input, Dense, Lambda
from tensorflow.keras.models import Model
from tensorflow.keras import backend as K
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_errorI will load the QM9 dataset and preprocess it. The QM9 dataset is a collection of molecular structures and their properties.
# Load the QM9 dataset
# Note: Replace the path with the actual path to my QM9 dataset file
data = pd.read_csv('qm9.csv')
# Drop non-numeric columns if any
data = data.select_dtypes(include=[np.number])
# Standardize the data
scaler = StandardScaler()
data_scaled = scaler.fit_transform(data)
# Split the data into training and testing sets
X_train, X_test = train_test_split(data_scaled, test_size=0.2, random_state=42)I will define an autoencoder model for anomaly detection.
# Define the autoencoder model
input_dim = X_train.shape[1]
encoding_dim = 32 # This can be adjusted
input_layer = Input(shape=(input_dim,))
encoded = Dense(encoding_dim, activation='relu')(input_layer)
decoded = Dense(input_dim, activation='sigmoid')(encoded)
autoencoder = Model(input_layer, decoded)
autoencoder.compile(optimizer='adam', loss='mse')I will train the autoencoder on the training data.
# Train the autoencoder
history = autoencoder.fit(X_train, X_train,
epochs=50,
batch_size=256,
shuffle=True,
validation_data=(X_test, X_test),
verbose=1)I will evaluate the autoencoder and detect anomalies in the test set.
# Get the reconstruction loss
X_train_pred = autoencoder.predict(X_train)
train_loss = np.mean(np.square(X_train - X_train_pred), axis=1)
X_test_pred = autoencoder.predict(X_test)
test_loss = np.mean(np.square(X_test - X_test_pred), axis=1)
# Set the threshold for anomaly detection
threshold = np.percentile(train_loss, 95) # 95th percentile
# Identify anomalies
anomalies = test_loss > threshold
# Print the results
print(f'Number of anomalies detected: {np.sum(anomalies)}')
# Plot the reconstruction loss
plt.figure(figsize=(10, 6))
plt.hist(test_loss, bins=50)
plt.axvline(threshold, color='r', linestyle='dashed', linewidth=2)
plt.xlabel('Reconstruction loss')
plt.ylabel('Number of samples')
plt.title('Reconstruction Loss for Test Data')
plt.show()Here is a simplified example that simulates MQPA-inspired dynamic gate application within the autoencoder framework. This example focuses on dynamically adjusting the encoding dimension based on the data.
# Define a function to dynamically adjust the encoding dimension
def dynamic_encoding_dim(data_point):
# Example: Adjust the encoding dimension based on the mean of the data point
mean_val = np.mean(data_point)
if mean_val < -1:
return 16
elif mean_val < 0:
return 32
else:
return 64
# Define the autoencoder model with dynamic encoding dimension
def create_autoencoder(input_dim, encoding_dim):
input_layer = Input(shape=(input_dim,))
encoded = Dense(encoding_dim, activation='relu')(input_layer)
decoded = Dense(input_dim, activation='sigmoid')(encoded)
autoencoder = Model(input_layer, decoded)
autoencoder.compile(optimizer='adam', loss='mse')
return autoencoder
# Train the autoencoder with dynamic encoding dimension
for epoch in range(50): # Number of epochs
for batch_start in range(0, X_train.shape[0], 256): # Batch size
batch_end = min
(batch_start + 256, X_train.shape[0])
X_batch = X_train[batch_start:batch_end]
encoding_dim = dynamic_encoding_dim(np.mean(X_batch, axis=0))
autoencoder = create_autoencoder(input_dim, encoding_dim)
autoencoder.fit(X_batch, X_batch, epochs=1, verbose=0)This example provides a high-level overview of how I might approach implementing MQPA and autoencoder-based anomaly detection. I can further refine and expand this approach to suit my specific needs and data characteristics.
If the current environment does not support TensorFlow, let’s look at and examine the expected output of each step in the code. I can run the code on my local machine to observe the actual results.
Training Output: During training, I should see output similar to this for each epoch, showing the loss and validation loss:
Epoch 1/10
4/4 [==============================] - 0s 30ms/step - loss: 0.2954 - val_loss: 0.2518
Epoch 2/10
4/4 [==============================] - 0s 6ms/step - loss: 0.2408 - val_loss: 0.2047
...Number of Anomalies Detected: The output should show the number of anomalies detected in the test data:
Number of anomalies detected: XWhere X is the number of anomalies found based on the
reconstruction loss threshold.
Reconstruction Loss Histogram: The plot will display the histogram of the reconstruction loss for the test data, with a vertical line indicating the threshold for anomaly detection. The anomalies are the samples with reconstruction loss above this threshold.
Here is the complete code again to run on my local machine:
import tensorflow as tf
from tensorflow.keras.layers import Input, Dense
from tensorflow.keras.models import Model
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import train_test_split
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
# Generate synthetic data similar to QM9 dataset for demonstration purposes
np.random.seed(42)
data = np.random.rand(1000, 100) # 1000 samples, 100 features
# Standardize the data
scaler = StandardScaler()
data_scaled = scaler.fit_transform(data)
# Split the data into training and testing sets
X_train, X_test = train_test_split(data_scaled, test_size=0.2, random_state=42)
# Define the autoencoder model
input_dim = X_train.shape[1]
encoding_dim = 32
input_layer = Input(shape=(input_dim,))
encoded = Dense(encoding_dim, activation='relu')(input_layer)
decoded = Dense(input_dim, activation='sigmoid')(encoded)
autoencoder = Model(input_layer, decoded)
autoencoder.compile(optimizer='adam', loss='mse')
# Train the autoencoder
history = autoencoder.fit(X_train, X_train,
epochs=10, # Reduced epochs for demo
batch_size=256,
shuffle=True,
validation_data=(X_test, X_test),
verbose=1)
# Get the reconstruction loss
X_train_pred = autoencoder.predict(X_train)
train_loss = np.mean(np.square(X_train - X_train_pred), axis=1)
X_test_pred = autoencoder.predict(X_test)
test_loss = np.mean(np.square(X_test - X_test_pred), axis=1)
# Set the threshold for anomaly detection
threshold = np.percentile(train_loss, 95)
# Identify anomalies
anomalies = test_loss > threshold
# Print the number of anomalies detected
num_anomalies = np.sum(anomalies)
print(f'Number of anomalies detected: {num_anomalies}')
# Plot the reconstruction loss
plt.figure(figsize=(10, 6))
plt.hist(test_loss, bins=50)
plt.axvline(threshold, color='r', linestyle='dashed', linewidth=2)
plt.xlabel('Reconstruction loss')
plt.ylabel('Number of samples')
plt.title('Reconstruction Loss for Test Data')
plt.show()This code provides the training process, number of detected anomalies, and the histogram plot for reconstruction loss. Let me know if I need further assistance!
If the current environment does not support TensorFlow, I can run the code on my local machine to observe the actual results.
Training Output: During training, I should see output similar to this for each epoch, showing the loss and validation loss:
Epoch 1/10
4/4 [==============================] - 0s 30ms/step - loss: 0.2954 - val_loss: 0.2518
Epoch 2/10
4/4 [==============================] - 0s 6ms/step - loss: 0.2408 - val_loss: 0.2047
...Number of Anomalies Detected: The output should show the number of anomalies detected in the test data:
Number of anomalies detected: XWhere X is the number of anomalies found based on the
reconstruction loss threshold.
Reconstruction Loss Histogram: The plot will display the histogram of the reconstruction loss for the test data, with a vertical line indicating the threshold for anomaly detection. The anomalies are the samples with reconstruction loss above this threshold.
code for local machgine
import tensorflow as tf
from tensorflow.keras.layers import Input, Dense
from tensorflow.keras.models import Model
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import train_test_split
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
# Generate synthetic data similar to QM9 dataset for demonstration purposes
np.random.seed(42)
data = np.random.rand(1000, 100) # 1000 samples, 100 features
# Standardize the data
scaler = StandardScaler()
data_scaled = scaler.fit_transform(data)
# Split the data into training and testing sets
X_train, X_test = train_test_split(data_scaled, test_size=0.2, random_state=42)
# Define the autoencoder model
input_dim = X_train.shape[1]
encoding_dim = 32
input_layer = Input(shape=(input_dim,))
encoded = Dense(encoding_dim, activation='relu')(input_layer)
decoded = Dense(input_dim, activation='sigmoid')(encoded)
autoencoder = Model(input_layer, decoded)
autoencoder.compile(optimizer='adam', loss='mse')
# Train the autoencoder
history = autoencoder.fit(X_train, X_train,
epochs=10, # Reduced epochs for demo
batch_size=256,
shuffle=True,
validation_data=(X_test, X_test),
verbose=1)
# Get the reconstruction loss
X_train_pred = autoencoder.predict(X_train)
train_loss = np.mean(np.square(X_train - X_train_pred), axis=1)
X_test_pred = autoencoder.predict(X_test)
test_loss = np.mean(np.square(X_test - X_test_pred), axis=1)
# Set the threshold for anomaly detection
threshold = np.percentile(train_loss, 95)
# Identify anomalies
anomalies = test_loss > threshold
# Print the number of anomalies detected
num_anomalies = np.sum(anomalies)
print(f'Number of anomalies detected: {num_anomalies}')
# Plot the reconstruction loss
plt.figure(figsize=(10, 6))
plt.hist(test_loss, bins=50)
plt.axvline(threshold, color='r', linestyle='dashed', linewidth=2)
plt.xlabel('Reconstruction loss')
plt.ylabel('Number of samples')
plt.title('Reconstruction Loss for Test Data')
plt.show()This code provides the training process, number of detected anomalies, and the histogram plot for reconstruction loss.
Running an autoencoder for anomaly detection on a quantum computer involves different steps compared to classical computing.
Here’s a simplified example using the qiskit library to
demonstrate a basic quantum autoencoder. Please note that current
quantum hardware is not yet capable of handling large-scale machine
learning tasks, so this example is for educational purposes and may not
directly correlate with the performance of classical models.
set up the environment and import the necessary libraries.
# Install the necessary libraries
!pip install qiskit
# Import libraries
from qiskit import QuantumCircuit, transpile, Aer, execute
from qiskit.visualization import plot_histogram
import numpy as np
import matplotlib.pyplot as plt
# Set up a quantum simulator
backend = Aer.get_backend('qasm_simulator')we will define a simple quantum autoencoder circuit. For simplicity, we’ll use a small number of qubits.
# Define a simple quantum autoencoder circuit
def quantum_autoencoder():
qc = QuantumCircuit(3, 3)
# Encoder
qc.h(0)
qc.cx(0, 1)
qc.cx(1, 2)
# Decoder
qc.cx(1, 2)
qc.cx(0, 1)
qc.h(0)
# Measurement
qc.measure([0, 1, 2], [0, 1, 2])
return qc
# Create the circuit
qc = quantum_autoencoder()
# Transpile the circuit for the simulator
qc = transpile(q
c, backend)
# Execute the circuit
job = execute(qc, backend, shots=1024)
result = job.result()
# Get the counts
counts = result.get_counts(qc)
print(counts)
# Plot the results
plot_histogram(counts)
plt.show()For the purpose of anomaly detection, we will simulate how the quantum autoencoder behaves with normal and anomalous data.
# Simulate normal data
normal_counts = {'000': 512, '111': 512}
# Simulate anomalous data
anomalous_counts = {'001': 512, '110': 512}
# Define a threshold for anomaly detection
threshold = 100
# Detect anomalies based on the counts
def detect_anomaly(counts, threshold):
anomaly_score = sum(counts.get(key, 0) for key in ['001', '010', '011', '100', '101', '110'])
return anomaly_score > threshold
# Check normal data
is_anomaly = detect_anomaly(normal_counts, threshold)
print(f'Normal data anomaly detected: {is_anomaly}')
# Check anomalous data
is_anomaly = detect_anomaly(anomalous_counts, threshold)
print(f'Anomalous data anomaly detected: {is_anomaly}')To run this code on real quantum hardware, I will need access to IBM Quantum Experience and replace the simulator with a real quantum device. Here are the steps:
Set up IBM Quantum Experience Account:
from qiskit import IBMQ
IBMQ.save_account('my_IBM_QUANTUM_API_TOKEN')
IBMQ.load_account()
provider = IBMQ.get_provider('ibm-q')
backend = provider.get_backend('ibmq_quito') # Replace with my preferred backendExecute on Real Quantum Device:
qc = transpile(qc, backend)
job = execute(qc, backend, shots=1024)
job_monitor(job)
result = job.result()
counts = result.get_counts(qc)
print(counts)
plot_histogram(counts)
plt.show()Replace 'my_IBM_QUANTUM_API_TOKEN' with my actual IBM
Quantum Experience API token.
This example provides a basic introduction to running a quantum autoencoder for anomaly detection. The current quantum hardware limitations mean that more complex and practical quantum machine learning models are still in the research phase.
Applying the McPhaul Quantum Pathway Algorithm (MQPA) concept to the quantum autoencoder involves dynamically adjusting the gates based on the state of the qubits. This example will illustrate a simple form of MQPA by modifying the quantum autoencoder circuit to include conditional operations.
Ensure I have the necessary libraries installed and imported.
!pip install qiskit
from qiskit import QuantumCircuit, transpile, Aer, execute
from qiskit.visualization import plot_histogram
import numpy as np
import matplotlib.pyplot as plt
from qiskit.providers.aer import AerSimulatorwe define a quantum circuit that uses dynamic gate applications based on the current state of the qubits.
# Define the MQPA-inspired quantum autoencoder circuit
def mqpa_quantum_autoencoder():
qc = QuantumCircuit(3, 3)
# Encoder: Apply Hadamard and CNOT gates
qc.h(0)
qc.cx(0, 1)
qc.cx(1, 2)
# Dynamic gate application based on qubit state (simplified example)
qc.h(2) # Example dynamic operation
# Decoder: Reverse the encoding process
qc.cx(1, 2)
qc.cx(0, 1)
qc.h(0)
# Measurement
qc.measure([0, 1, 2], [0, 1, 2])
return qc
# Create the circuit
qc = mqpa_quantum_autoencoder()
# Transpile the circuit for the simulator
backend = AerSimulator()
qc = transpile(qc, backend)
# Execute the circuit
job = execute(qc, backend, shots=1024)
result = job.result()
# Get the counts
counts = result.get_counts(qc)
print(counts)
# Plot the results
plot_histogram(counts)
plt.show()For the purpose of anomaly detection, we’ll simulate how the MQPA-inspired quantum autoencoder behaves with normal and anomalous data.
```python # Simulate normal data normal_counts = {‘000’: 512, ‘111’: 512}
anomalous_counts = {‘001’: 512, ‘110’: 512}
threshold = 100
def detect_anomaly(counts, threshold): anomaly_score = sum(counts.get key