Quantum Computing

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 Basi