Quantum Computing Explained

Quantum Computing

• Classical computers use bits representing 0 or 1
• Quantum computers use qubits
• Qubits use superposition, representing 0, 1, or proportions of both
• Entanglement links qubits to correlate their states

Applications

• Drug discovery and materials science benefit from simulating molecules accurately
• Optimization problems such as supply chain and financial modeling can be solved
• Cryptography is being researched for quantum-resistant methods due to potential breaches

Challenges

• Qubit stability is a concern due to sensitivity to environment, causing decoherence
• Error correction is a significant challenge due to error prone quantum computations
• Scalability is an engineering challenge to build large-scale quantum computers

Quantum Hardware

• Superconducting qubits use superconducting electronic circuits, pursued by Google and IBM
• Trapped ions use individual ions controlled with electromagnetic fields, used by IonQ
• Photonic qubits use photons (light particles) to encode quantum information

Programming Quantum Computers

• Quantum circuits represent quantum programs as sequences of quantum gates on qubits
• Quantum algorithms are designed for quantum computers, such as Shor's and Grover's
• Quantum Software Development Kits (SDKs) write quantum programs, like Qiskit (IBM), Cirq (Google), and PennyLane (Xanadu)

The Future

• Quantum computing is still in its early stages
• It has the potential to revolutionize many fields

Static Typing

• Type checking occurs at compile time
• Variable types are declared
• It is stricter and catches errors earlier
• Examples include C++, Java, Fortran, Pascal

Dynamic Typing

• Type checking occurs at run time
• Variable types are not declared
• It is more flexible, but errors are caught later
• Examples include Python, Lisp, Ruby, JavaScript

Hybrid Typing

• Some languages allow both static and dynamic typing
• Examples include C#, TypeScript, Go, and Swift

Introduction to Non-Linear Classifiers

• This includes Decision Trees, Random Forests, Boosting, Support Vector Machines (SVM), and k-Nearest Neighbors (k-NN)

Decision Trees

• Decision trees classify data by recursively partitioning the feature space
• Start at the root node, choose a feature to split the data, and create branches based on feature values
• Repeat the process for each branch until a stopping criterion is met
• Assign a class label to each leaf node

Decision Tree for Continuous Attributes

• Splits have the form $x_i > t$, where $x_i$ is a continuous attribute and t is a threshold

Stopping Criteria

• All data in the node have the same class label
• The node contains only a small number of data points
• The tree reaches a maximum depth

Prediction

• To predict the class label for a new data point, traverse the tree from the root node to a leaf node
• Follow the branches that correspond to the data point's feature values

Advantage

• Easy to interpret and visualize
• Can handle both numerical and categorical data
• Non-parametric

Disadvantage

• Can easily overfit the data

How to prevent overfitting in Decision Trees?

• Pre-pruning: Stop growing the tree early
• Post-pruning: Grow a full tree, then prune it back

Random Forests

• Random forests are an ensemble learning method that combines multiple decision trees to improve accuracy and reduce overfitting
• Create multiple decision trees using a random subset of the data and a random subset of the features
• For each tree, grow it fully without pruning
• To predict the class label for a new data point, feed it to each tree
• Take the majority vote of the trees

Advantages

• More accurate than a single decision tree
• Can handle high-dimensional data
• Less prone to overfitting

Disadvantages

• More difficult to interpret than a single decision tree
• More computationally expensive

Boosting

• Boosting is an ensemble learning method that combines multiple weak learners to create a strong learner
• Train a weak learner on the original data and assign weights to the data points, giving higher weights to the misclassified points
• Train a second weak learner on the weighted data
• Repeat the process until a stopping criterion is met
• Combine the weak learners into a strong learner by weighting their predictions
• Example: AdaBoost

Advantages

• More accurate than a single weak learner
• Can handle both numerical and categorical data

Disadvantages

• Sensitive to noisy data and outliers
• More computationally expensive

Support Vector Machines

• SVMs classify data by finding the optimal hyperplane that separates the data points into different classes
• Map the data points into a high-dimensional feature space using a kernel function
• Find the optimal hyperplane that separates the data points into different classes
• The hyperplane is chosen to maximize the margin, which is the distance between the hyperplane and the nearest data points
• The data points that are closest to the hyperplane are called support vectors

Kernel Functions

• Linear
• Polynomial
• Radial Basis Function (RBF)

Advantages

• Effective in high-dimensional spaces
• Relatively memory efficient

Disadvantages

• Prone to overfitting
• Difficult to choose the optimal kernel function and parameters
• Not easily interpretable

k-Nearest Neighbors

• k-NN classifies data by finding the k nearest neighbors to a data point and assigning the data point to the class that is most common among its neighbors
• Choose a value for k
• For each data point, find the k nearest neighbors
• Assign the data point to the class that is most common among its neighbors

Distance Metrics

• Euclidean distance
• Manhattan distance
• Minkowski distance

Advantages

• Simple to implement and understand
• Non-parametric

Disadvantages

• Computationally expensive
• Sensitive to the choice of k and the distance metric
• Prone to overfitting