Foundations

Introduction to
Quantum Computing

From classical bits to quantum superposition — exploring the foundational concepts of quantum computation, including qubits, quantum gates, and circuit design, and why quantum computers represent a fundamentally different paradigm.

Qubits Superposition Quantum Gates Bloch Sphere Quantum Circuits Entanglement (intro)

The Qubit

From Bits to Qubits

A classical computer stores information as bits — each one strictly 0 or 1. A quantum computer uses qubits (quantum bits), which can exist in any combination of 0 and 1 simultaneously. This is called superposition.

Mathematically, a qubit's state is written as |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers called amplitudes. The probabilities of measuring 0 or 1 are |α|² and |β|² respectively, and they must sum to 1.

Physical implementations of qubits include: superconducting circuits (Google, IBM), trapped ions (IonQ, Quantinuum), photonic systems, and silicon spin qubits — each with different coherence times and error rates.

Visualising a Qubit

The Bloch Sphere: every point on the surface represents a valid pure qubit state. The north pole is |0⟩, the south pole is |1⟩, and all equatorial points are equal superpositions with different phases. Quantum gates correspond to rotations about the sphere's axes. Source

Core Principles

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Superposition

A qubit in superposition encodes both 0 and 1 simultaneously with different amplitudes. The Hadamard gate creates equal superposition: H|0⟩ = (|0⟩ + |1⟩)/√2. This enables a quantum computer to explore many solutions in parallel.

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Entanglement

Two or more qubits can be correlated in ways that have no classical analogue. Measuring one instantly determines the state of the other, regardless of distance. Entanglement is a key resource for quantum algorithms and quantum communication.

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Interference

Like waves, quantum amplitudes can interfere constructively (amplifying correct answers) or destructively (cancelling wrong ones). Quantum algorithms are designed to harness interference — not raw speed — as the source of computational advantage.

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Measurement

Measuring a qubit collapses its superposition into a definite 0 or 1, with probabilities given by its amplitudes. This irreversible process is the final step of any quantum computation — the quantum state must be cleverly arranged so the right answer is the most probable outcome.

Quantum Gates

Operations on Qubits

Quantum gates are the quantum analogue of logic gates. Unlike classical gates, they must be reversible (unitary matrices), meaning no information is ever destroyed during computation. This is a fundamental requirement of quantum mechanics.

Single-qubit gates: The Pauli gates X (bit flip — like a quantum NOT), Y, and Z (phase flip) rotate the Bloch vector 180° around the respective axes. The Hadamard gate H creates superposition. The S and T gates apply phase rotations.

Two-qubit gates: The CNOT (Controlled-NOT) flips the target qubit only if the control qubit is |1⟩. Together with single-qubit gates, CNOT forms a universal gate set — any quantum computation can be decomposed into these operations.

Quantum Circuit Diagrams

A quantum circuit diagram. Each horizontal line represents one qubit, initialised to |0⟩ on the left. Boxes are gate operations applied in left-to-right order. The meter symbol at the end represents measurement. Time flows left to right. Source

Classical vs Quantum Computing

Property Classical Quantum
Basic unit Bit (0 or 1) Qubit (superposition of 0 and 1)
State space (n units) 1 of 2ⁿ states at a time Superposition across all 2ⁿ states
Operations Boolean logic (AND, OR, NOT) Unitary transformations (reversible)
Error source Thermal noise, hardware faults Decoherence, gate errors, measurement collapse
Speedup mechanism N/A (parallelism via multiple cores) Superposition + interference + entanglement
Best problem types General-purpose computation Factoring, simulation, optimisation, search
The key insight: Quantum speedup does not come from doing things faster — it comes from quantum interference, which amplifies the probability of correct answers and suppresses incorrect ones before measurement.