1. Mathematical Foundations (Non-negotiable)
Linear Algebra
- Vector spaces (real vs complex)
- Basis, dimension
- Inner product
- Norms
- Orthogonality
- Eigenvalues & eigenvectors
- Spectral decomposition
- Unitary matrices
- Hermitian operators
- Tensor products (VERY IMPORTANT)
Probability Theory
- Random variables
- Conditional probability
- Bayes theorem
- Expectation & variance
Complex Numbers
- Euler’s formula
- Polar form
- Complex conjugates
2. Quantum Mechanics Foundations
Postulates of Quantum Mechanics
- State representation (wavefunction/state vector)
- Measurement postulate
- Evolution (Schrödinger equation intuition)
- Observables as operators
Dirac Notation
- Bra ⟨ψ| and Ket |ψ⟩
- Inner product ⟨ψ|φ⟩
- Outer product |ψ⟩⟨φ|
Quantum States
- Pure states
- Mixed states
- Density matrices
3. Qubits and Multi-Qubit Systems
Single Qubit
- Representation: α|0⟩ + β|1⟩
- Normalisation condition
- Measurement probabilities
Bloch Sphere
- State as a point on a sphere
- Rotations = quantum gates
Multi-Qubit Systems
- Tensor product states
- Basis states (|00⟩, |01⟩, etc.)
- State explosion (2ⁿ-dimensional space)
Entanglement
- Bell states
- Non-separability
- Difference from classical correlation
4. Quantum Gates
Single-Qubit Gates
- Pauli Gates (X, Y, Z)
- Hadamard (H)
- Phase gate (S, T)
Multi-Qubit Gates
- CNOT (controlled NOT)
- Toffoli gate
- Controlled-U gates
Properties
- Unitary transformations
- Reversibility
5. Quantum Circuits
Circuit Model
- Wires = qubits
- Gates = operations
Concepts
- Circuit depth
- Parallelism
- Measurement at end
Universal Gate Sets
- Gate completeness
6. Quantum Algorithms
Deutsch-Jozsa Algorithm
- First demonstration of quantum advantage
Grover’s Algorithm
- Unstructured search
- Quadratic speedup
Shor’s Algorithm
- Integer factorization
- Cryptography implications
Quantum Fourier Transform (QFT)
- Core subroutine for many algorithms
Variational Quantum Algorithms
- VQE (Variational Quantum Eigensolver)
- QAOA (Quantum Approximate Optimisation Algorithm)
7. Quantum Machine Learning (QML)
Data Encoding
- Basis encoding
- Amplitude encoding
- Angle encoding
Quantum Models
- Quantum Neural Networks (QNN)
- Variational circuits
- Quantum kernels
Hybrid Models
- Classical + Quantum pipelines
Open Problems
- Data loading bottleneck
- Noise sensitivity
- Lack of clear advantage
8. Quantum Hardware
Physical Implementations
- Superconducting qubits
- Trapped ions
- Photonic systems
Challenges
- Decoherence
- Noise
- Error rates
9. Quantum Error Correction
- Bit flip error
- Phase flip error
- Shor code
- Surface codes
10. Industry Landscape
- IBM (Qiskit)
- Google (Quantum AI, Sycamore)
- Microsoft (Azure Quantum)
- Rigetti Computing
- IonQ
11. Limitations & Misconceptions
- Not all problems get speedup
- Quantum ≠ universally faster
- Hardware still immature
12. Future Directions
- Fault-tolerant quantum computing
- Quantum internet
- Quantum advantage in ML
13. Tools & Frameworks
- Qiskit
- Cirq
- PennyLane
14. Research-Level Topics
- Quantum complexity classes (BQP, NP)
- Quantum supremacy vs advantage
- Hamiltonian simulation
- Adiabatic quantum computing
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