Classical vs Quantum Computing
Classical computers manipulate bits — discrete 0s and 1s — using deterministic logic. Quantum computers exploit quantum mechanics to process information in fundamentally new ways, enabling computational power that scales exponentially.
| Aspect | Classical Computer | Quantum Computer |
|---|---|---|
| Information Unit | Bit — 0 or 1 (definite) | Qubit — 0 and 1 (superposition) |
| Processing | Sequential, one state at a time | Parallel — 2ᴺ states simultaneously |
| Operations | Deterministic logic gates (AND, OR, NOT) | Probabilistic unitary quantum gates |
| State Space | N bits = N values at a time | N qubits = 2ᴺ amplitudes at once |
| Error Tolerance | Highly stable, digital noise immunity | Fragile — decoherence is a major challenge |
| Current Scale | Billions of transistors on a chip | 100s–1000s of qubits (NISQ era) |
| Best For | General-purpose everyday computation | Optimization, simulation, cryptography |
The Qubit
A qubit (quantum bit) is the fundamental unit of quantum information. Unlike a classical bit that is strictly 0 or 1, a qubit can exist in a continuous superposition of both states simultaneously — until it is measured.
What makes a qubit special?
A classical bit is always definitively 0 or 1 — like a light switch. A qubit is in a weighted combination of both until measured — like a coin spinning in the air. The amplitudes α and β encode this combination, with their squares giving measurement probabilities.
⚡ Superposition
A qubit simultaneously exists as |0⟩ and |1⟩. This isn't the qubit "being random" — it's a genuine physical quantum state. Applying a Hadamard gate puts |0⟩ into equal superposition: |+⟩ = (|0⟩ + |1⟩)/√2, with 50% probability of each outcome.
🔗 Entanglement
Two qubits can be entangled so their states are correlated. Measuring one instantly determines the other's outcome, regardless of distance. This is not faster-than-light communication but a deep correlation with no classical analogue.
〰 Interference
Quantum amplitudes behave like waves — they can add constructively (amplifying correct answers) or destructively (cancelling wrong answers). Quantum algorithms are carefully designed to exploit this so that the right answer has the highest measurement probability.
The Bloch Sphere
The Bloch sphere is the geometric representation of all possible single-qubit states. Every point on its surface corresponds to a valid qubit state. Classical bits occupy only the two poles; qubits can be anywhere on the sphere.
Classical bit 0 — the "spin-up" state. Pure |0⟩ with 100% probability.
Classical bit 1 — the "spin-down" state. Pure |1⟩ with 100% probability.
The arrow from center to sphere surface represents the current qubit state. Defined by θ (polar) and φ (azimuthal) angles.
Equal superposition states — 50/50 chance of measuring 0 or 1. The Hadamard gate maps |0⟩ to |+⟩.
|ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩
0 ≤ θ ≤ π, 0 ≤ φ < 2π
Quantum Measurement
Measurement is the process of extracting classical information from a qubit. It is irreversible and probabilistic — it collapses the quantum superposition into one definite classical outcome, with probabilities determined by the amplitudes.
📐 How It Works
Given |ψ⟩ = α|0⟩ + β|1⟩:
• Probability of measuring 0 = |α|²
• Probability of measuring 1 = |β|²
• After measurement, the state collapses — the qubit is now fully 0 or 1.
• Repeating the measurement will always give the same result.
⚠️ Collapse is Irreversible
Once a qubit is measured, its superposition is permanently destroyed. This is why quantum algorithms must complete all quantum operations before measuring. Premature measurement destroys the quantum advantage — you'd just get a random classical answer.
Irreversible
Wavefunction collapse cannot be undone. Superposition is permanently destroyed upon observation.
Probabilistic
You can only predict outcome probabilities, not exact outcomes, until measurement occurs.
Basis-Dependent
Results depend on the measurement basis chosen. Different bases reveal different aspects of the state.
Post-Measurement
After measurement, the qubit is in a definite classical state. Subsequent measurements give the same result.
Dirac (Bra-Ket) Notation
Invented by physicist Paul Dirac, bra-ket notation is the standard mathematical language of quantum mechanics. It provides a compact, elegant way to write quantum states and operations using linear algebra.
| Notation | Name | Type | Meaning |
|---|---|---|---|
| |ψ⟩ | Ket | Column vector | Represents a quantum state. E.g., |0⟩ = [1, 0]ᵀ, |1⟩ = [0, 1]ᵀ |
| ⟨ψ| | Bra | Row vector | The conjugate transpose of the ket. ⟨0| = [1, 0], ⟨1| = [0, 1] |
| ⟨φ|ψ⟩ | Bracket / Inner Product | Complex scalar | Probability amplitude for measuring state |φ⟩ given state |ψ⟩. |⟨φ|ψ⟩|² is the probability. |
| |0⟩, |1⟩ | Computational Basis | Orthonormal basis | Standard basis vectors of the qubit Hilbert space (ℂ²). Analogous to unit vectors î and ĵ. |
| |+⟩, |−⟩ | Hadamard Basis | Superposition states | |+⟩ = (|0⟩ + |1⟩)/√2 |−⟩ = (|0⟩ − |1⟩)/√2. Equal superposition with different phases. |
| Â|ψ⟩ | Operator Application | State transformation | Quantum gate  applied to state |ψ⟩. Quantum gates are unitary matrices: † = I. |
Quantum Entanglement
Entanglement is a quantum correlation between two or more qubits that has no classical equivalent. Einstein called it "spooky action at a distance." The 2022 Nobel Prize in Physics was awarded for experimentally proving it.
🌍 Non-Local Correlation
Entangled qubits exhibit perfect correlations regardless of the physical distance between them. Measure A on Earth; B on Mars collapses instantaneously. Distance is irrelevant.
🚫 No FTL Signaling
Despite instantaneous correlations, entanglement cannot transmit information faster than light. The measurement outcomes are random — you can't control which value A gets, so you can't encode a message.
💍 Monogamy
A qubit cannot be maximally entangled with two qubits simultaneously. Entanglement is a shared, exclusive resource — it cannot be "copied" or "cloned" (No-Cloning Theorem).
🔑 Quantum Resource
Entanglement powers: quantum teleportation (state transfer), QKD (quantum key distribution for secure communication), superdense coding (2 classical bits via 1 qubit), and quantum error correction.
Quantum Hardware Overview
Physical qubits can be implemented in several ways. Each approach has distinct trade-offs in coherence time, gate fidelity, scalability, and operating temperature. No single technology has yet "won" — all are actively researched.
⚡ Superconducting Qubits
~15 mKJosephson junctions cooled to ~15 millikelvin (colder than deep space). The most mature and widely deployed platform. IBM's Condor processor has 1,121 qubits. Fast gate times (~50 ns) but limited coherence (~100 μs).
🔵 Trapped Ion Qubits
Room TempIndividual ions (like ytterbium or barium) held in electromagnetic traps and manipulated with laser pulses. Highest gate fidelity of any platform (>99.9% two-qubit gate). Slower gates (~1 ms) but long coherence times.
💡 Photonic Qubits
Room TempPhotons (light particles) encode quantum information. Naturally suited for quantum communication and networking. Main challenge: photon-photon interactions are weak, making two-qubit gates difficult without special hardware.
🔬 Topological Qubits
~mK rangeUses exotic quasiparticles called Majorana fermions. The topological encoding provides intrinsic error protection — errors must affect both ends of a quasiparticle simultaneously, which is physically unlikely. Still experimental.
💎 Silicon Spin Qubits
~mK rangeElectron spins in silicon quantum dots — leveraging decades of CMOS fabrication expertise. Potentially the most scalable approach due to compatibility with existing chip manufacturing processes.
📊 NISQ Era
NowNoisy Intermediate-Scale Quantum — the current era where devices have 50–1000 qubits but without full error correction. IBM's roadmap targets fault-tolerant quantum computing beyond 2030, with intermediate milestones each year.
Quantum Programming Platforms
Quantum programming frameworks let you design circuits, simulate them classically, and run on real quantum hardware — all from Python. Qiskit is the primary platform for this course.
The most widely-used quantum SDK. Full-stack: design circuits, simulate, transpile, and run on IBM quantum hardware. Excellent documentation, tutorials, and an active global community.
Designed for near-term NISQ devices with fine-grained control over gate placement and timing. Primary tool for Google's Sycamore experiments. Great for quantum error research.
Specializes in quantum machine learning. Integrates seamlessly with PyTorch, TensorFlow, and JAX — enabling automatic differentiation of quantum circuits (quantum gradients).
A dedicated quantum programming language (not just a library). Deeply integrated with Azure Quantum. Targets both simulated and topological hardware. Good for algorithmic research.
Cloud platform giving a single unified API to access hardware from multiple vendors: IonQ, Rigetti, OQC, and simulators — all on AWS infrastructure. Good for vendor comparison.
# Install: pip install qiskit qiskit-aer
from qiskit import QuantumCircuit
# Create a 2-qubit circuit
qc = QuantumCircuit(2, 2)
# Hadamard on qubit 0 → puts it in superposition |+⟩
qc.h(0)
# CNOT (controlled-X) on qubit 1, controlled by qubit 0
# Creates a Bell state: (|00⟩ + |11⟩) / √2
qc.cx(0, 1)
# Measure both qubits
qc.measure([0, 1], [0, 1])
# Draw the circuit
print(qc.draw())
# Simulate with Aer
from qiskit_aer import AerSimulator
sim = AerSimulator()
job = sim.run(qc, shots=1024)
counts = job.result().get_counts()
print(counts) # {'00': ~512, '11': ~512} — Bell state confirmed!
Learning Outcomes
After completing Unit I, you should be able to achieve the following outcomes from the course syllabus (CO1).
-
1
Explain differences between classical and quantum computing
Articulate the shift from deterministic bit-based computation to probabilistic qubit-based computation, covering state space, operations, error models, and suitable problem domains.
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2
Interpret the mathematical representation of qubits and quantum states
Use Dirac notation correctly, write qubit states as |ψ⟩ = α|0⟩ + β|1⟩, apply the normalization constraint, and visualize states on the Bloch sphere using θ and φ.
-
3
Understand the three core quantum phenomena
Describe superposition (concurrent existence of states), entanglement (non-local quantum correlation), and interference (amplitude cancellation/amplification) and their role in quantum algorithms.
-
4
Describe quantum measurement and its consequences
Explain wavefunction collapse, calculate measurement probabilities from amplitudes, and understand why measurement is irreversible and why it destroys superposition.
-
5
Identify quantum hardware types and programming platforms
Compare superconducting, trapped ion, photonic, and topological qubit technologies. Set up Qiskit and write basic quantum circuits that demonstrate superposition and entanglement.
🧠 Quick Knowledge Check
Test your understanding of Unit I concepts. Click an option to see if you're right.
Textbooks & Resources
📘 Nielsen & Chuang
Quantum Computation and Quantum Information — The definitive reference textbook. Chapters 1–2 cover everything in Unit I. Available in most university libraries.
📗 Jack Hidary
Quantum Computing: An Applied Approach — Practical, code-first introduction. All examples use Qiskit and Cirq. Great complement to Nielsen & Chuang.
📙 Eleanor Rieffel & Wolfgang Polak
Quantum Computing: A Gentle Introduction — More accessible than Nielsen & Chuang. Recommended for students new to quantum mechanics.
🌐 Ronald de Wolf Lecture Notes
Free, comprehensive lecture notes covering quantum algorithms, complexity, and theory. Available at CWI Amsterdam. Excellent theoretical depth.
⚙️ IBM Quantum Learning
Free interactive courses, textbook, and labs at learning.quantum.ibm.com. Runs Qiskit directly in the browser — no local installation needed.
🔬 Scott Aaronson
Quantum Computing Since Democritus — Accessible, witty, and deep. Connects quantum computing to philosophy, complexity theory, and physics. Great reading for intellectually curious students.