Quantum Computing with Qiskit

Below are the modules,

1. Fundamentals of Quantum Computing with Qiskit
2. Superdense Coding
3. Teleportation
4. Bernstein Vazirani Algorithm: Replacement for classical for-loop algorithm.
5. Deutsch Algorithm: Replacement to determine if function is constant or not.
6. Grover Algorithm: Replacement for classical search algorithm
7. Shor Algorithm: Breaking RSA encryption.
8. Variational Quantum Classifier
9. Quantum Kernel-based Classifier
10. Hybrid Neural Network (Quantum + Classical)
11. Solving Chemistry Problem (through Simulation)
12. Portfolio Optimization using Quantum Annealing

Summary of Basics

• Superposition: A qubit can exist in multiple states (0 and 1) at the same time until it is measured.

  • For example, if a qubit is in a superposition of |0› and |1›, it holds a probability distribution of both states

• Entanglement: When two or more qubits are entangled, changing the state of one instantly affects the state of the other, no matter how far apart they are

• Application: Used in quantum cryptography and secure communications

• Quantum Gates: Manipulate qubits, just as classical logic gates (AND, OR, NOT) manipulate bits

  • Z gate (π phase flip): Adds a 180° phase to |1⟩, leaving probabilities unchanged unless interference occurs later.
  • S gate (π/2 phase shift): Adds a 90° phase to |1⟩, not affecting immediate probabilities but altering interference in subsequent gates.
  • T gate (π/4 phase shift): Adds a 45° phase to |1⟩, preserving measurement probabilities alone while subtly influencing later interference outcomes.

• Hadamard (H): Creates superposition (0 → 0 & 1 at the same time)

• Pauli-X (X): Flips the qubit (0 → 1, 1 → 0)

• CNOT (Controlled-NOT): Creates entanglement between qubits

• Phase Shift (S, T, Z): Rotates the phase of the qubit state

• Toffoli (CCNOT): Quantum version of classical AND gate

Quantum Computing Mathematics

A Qubit state is reprsented as, where $α$ and $β$ are complex numbers representing probabilities - where $|0⟩$ (north-pole) and $|1⟩$ (south-pole).

$|ψ⟩ = α|0⟩ + β|1⟩$

The probabilities must sum to $1$:

$|α|² + |β|² = 1$

For example, a Hadamard gate applied to $|0⟩$ creates an equal superposition state,

$|ψ⟩ = 1/√2 (|0⟩ + |1⟩)$

The Wave function is represented as $ψ$, which describes the Quantum system state,

$P(x) = |\psi(x)|^2$, $\int |\psi(x)|^2 , dx = 1$

A general qubit state is

$|ψ⟩ = cos(θ/2) |0⟩ + e^{iφ} sin(θ/2) |1⟩$

• $θ$ (theta) determines the probability of measuring $0$ or $1$
• $φ$ (phi) determines the phase of the state

Differences from Quantum to Classical Computing

Feature	Classical Computing	Quantum Computing
Unit of Information	Bit (0 or 1)	Qubit (0, 1, or both simultaneously)
Processing Speed	Sequential processing	Parallel processing using superposition
Scalability	Limited for large problems	Exponentially faster for complex computations
Key Principle	Boolean Logic	Quantum Mechanics