Introduction to Quantum Computing

From one qubit to Shor, Grover and error correction.

About this course

A fourteen-lecture introduction to quantum computation. It begins with the qubit and the gates that act on it, builds through entanglement, the circuit model, measurement and decoherence, then turns to the protocols that have no classical analogue — no-cloning, superdense coding and teleportation. The algorithmic core covers quantum parallelism, the quantum Fourier transform, phase estimation, and from those foundations Shor's factoring and Grover's search. It ends on what it takes to build a real machine: error correction, hardware, and the current frontier.

What you will be able to do

  • Represent qubit states and apply single- and multi-qubit gates in the circuit model
  • Explain entanglement and use it in superdense coding and teleportation
  • Describe measurement, mixed states and decoherence, and why they limit real devices
  • Trace how the quantum Fourier transform and phase estimation underpin Shor's algorithm
  • Explain Grover's quadratic speedup and the principles of quantum error correction

What this course covers

  • Course spine and eighth clause introduction
  • Classical discrete Fourier transform and FFT cost
  • Quantum Fourier transform: definition and sameness with DFT
  • Exponential gate advantage of the QFT over the FFT
  • The readout caveat: QFT is not a faster FFT for classical data
  • QFT as a period detector
  • QFT circuit: product form, Hadamards, and controlled rotations
  • Worked two-qubit QFT example
  • QFT as engine of phase estimation and Shor's algorithm
  • The No-Cloning Theorem
  • Quantum Cryptography and BB84
  • Superdense Coding
  • Quantum Teleportation Protocol
  • Teleportation Does Not Violate No-Cloning
  • No Faster-Than-Light Signalling via Teleportation
  • Worked Calculation: Z Gate on a Bell Pair
  • Entanglement as a Tradeable Resource: Unifying the Three Results
  • Three obstacles to quantum error correction
  • Three-qubit bit-flip code
  • Syndrome measurement: error without data
  • Phase-flip code and Hadamard-basis duality
  • Shor's nine-qubit code and error digitisation
  • Stabilizer formalism
  • Threshold theorem and fault-tolerant scaling
  • Worked example: bit-flip on qubit two
  • Qubit overhead and the timeline to fault tolerance
  • Information is physical: course spine
  • The qubit and superposition
  • Interference: amplitudes vs. probabilities
  • The Bloch sphere
  • Measurement and the Born rule
  • Storage vs. readout limit (Holevo's theorem)
  • Physical realizations of a qubit
  • The Quantum Oracle
  • Quantum Parallelism and Its Catch
  • Interference as the Real Resource
  • The Phase Kick Trick
  • The Query Model for Measuring Speedup
  • Deutsch's Algorithm
  • Deutsch-Jozsa Algorithm
  • The Four-Step Template and Road to Future Algorithms
  • The Global-vs-Relative Phase Paradox
  • The Eigenphase Estimation Problem
  • Phase Kickback: Making the Invisible Relative
  • The Phase Estimation Circuit
  • Inverse QFT Readout and Precision Scaling
  • Phase Estimation as the Master Subroutine
  • Worked Example: Three-Qubit Phase Estimation
  • The Quantum Ruler Picture
  • Unitary evolution and reversibility of quantum gates
  • Gates versus measurement: two distinct operations
  • Pauli gates X, Y, and Z as half-turn rotations
  • The Bloch sphere globe analogy for single-qubit gates
  • The Hadamard gate and quantum interference
  • Phase gates S and T, and global versus relative phase
  • Every single-qubit gate as a rotation; universality of H and T
  • HZH = X: basis conjugation as a design technique
  • The DiVincenzo Criteria and Isolation–Control Tension
  • Superconducting Qubits and the Cold Puzzle
  • Trapped Ions, Photonics, and Platform Trade-offs
  • Metrics That Actually Decide Quantum Capability
  • The NISQ Era: Noisy Intermediate-Scale Quantum
  • Physical vs. Logical Qubits: The RSA Gap
  • Worked Calculation: Operations Before Decoherence
  • The Five-Step Engineering Arc of Any Quantum Computer
  • The Thirteenth Clause: Information Can Be Built
  • Quantum advantage vs. useful quantum advantage
  • The NISQ-era quantum computing ecosystem
  • Variational quantum algorithms (VQE and QAOA)
  • Feynman's quantum simulation idea
  • Honest ledger: what quantum computers are and aren't good at
  • The road to fault tolerance and the hype verdict
  • The 2^N exponential memory argument for quantum simulation
  • Quantum computing as interference choreography — the fourteenth clause
  • Two-qubit state space and the tensor product
  • Product states versus entangled states
  • The Bell states and their construction
  • EPR paradox and spooky action at a distance
  • The no-signaling principle
  • Proving |Φ+⟩ is entangled by failed factorization
  • Quantum teleportation preview
  • RSA encryption and the hardness of factoring
  • Reduction of factoring to period finding
  • Period finding as quantum phase estimation
  • The Shor circuit: modular exponentiation and inverse QFT
  • Continued fractions recovery of the period
  • Shor's algorithm as a classical-quantum hybrid loop
  • Polynomial vs sub-exponential speed-up and real-world threat assessment
  • The tenth clause: periodicity becomes a phase, becomes a broken lock
  • Reversibility paradox: classical vs. quantum gates
  • The CNOT gate
  • Controlled-U gates and the Toffoli gate
  • The quantum circuit model
  • Universal gate sets and the density argument
  • Worked circuit execution: building a Bell state
  • Resolution of the reversibility paradox
  • Unstructured search and the classical cost
  • The quantum oracle: phase flip and its limits
  • The Grover iteration: oracle plus diffusion
  • Geometric picture: rotation in the good-bad plane
  • Iteration count and the overshoot trap
  • Optimality: root N is the proven quantum floor
  • Cryptographic impact: quadratic vs. exponential speed-up
  • Worked example: N = 4, one marked item
  • Measurement formalized: basis, Born rule, collapse
  • The measurement problem and interpretations
  • The density matrix and quantum state purity
  • Schrödinger's cat and macroscopic superpositions
  • Partial trace and the reduced density matrix
  • Decoherence: coherence loss through environmental entanglement
  • Worked calculation: superposition vs. classical mixture
  • Near-absolute-zero hardware as decoherence engineering
  • Module one capstone: the five-clause spine of quantum information

Syllabus

  1. Foundations of Quantum Information

    1. The Qubit
    2. Single-Qubit Gates
    3. Two Qubits & Entanglement
    4. Multi-Qubit Gates & the Circuit Model
    5. Measurement, Mixed States & Decoherence

  2. Quantum Protocols & Primitives

    1. No-Cloning, Superdense Coding & Teleportation
    2. Quantum Parallelism & the First Speedup

  3. The Great Algorithms

    1. The Quantum Fourier Transform
    2. Quantum Phase Estimation
    3. Shor's Algorithm
    4. Grover's Search

  4. Making It Real

    1. Quantum Error Correction
    2. Building a Quantum Computer
    3. The Frontier