Chapter 09
Quantum information and computing
Quantum mechanics as a resource: entanglement, Bell inequalities, algorithms, error correction.
- Sections
- 10
- Finalised
- 10/10
- Simulations
- 0
- Estimated time
- 4 hours
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- Chapter 03Chapter 05
- 9.1Qubits and the Bloch sphereSpin 1/2 gets a new name: the qubit. The pure states of a two-level system exactly tile a sphere, and every gate, entangled state and algorithm in this chapter lives on that map.
- 9.2Single-qubit and two-qubit gatesQuantum operations = rotations of the Bloch sphere: Pauli, H, S and T each own one rotation, CNOT wires two qubits together, and a small handful of gates assembles every unitary there is.
- 9.3Quantifying entanglementFitting entanglement with a measuring stick: the more mixed the reduced density matrix, the deeper the entanglement — von Neumann entropy and concurrence upgrade "is it entangled?" to "how entangled is it?".
- 9.4The Bell inequality and CHSHWrite the worldview "the outcomes were fixed all along" as an inequality S ≤ 2; quantum mechanics computes 2√2, loophole-free experiments confirm it — and local realism is out.
- 9.5Quantum teleportationOne Bell pair plus one two-bit phone call delivers an unknown quantum state to the far side intact: four measurement outcomes, four correction gates, all worked by hand — with the no-cloning theorem proved and the not-faster-than-light point cleared up along the way.
- 9.6Superdense codingShare a Bell pair in advance and mailing one qubit carries two classical bits: the four Pauli gates tune one entangled pair through the four orthogonal Bell states — teleportation's mirror image.
- 9.7The Deutsch-Jozsa algorithmThe first clean example of quantum speedup: phase kickback writes the function values into phases, and interference reads out "constant or balanced" in one go — classically 2ⁿ⁻¹ + 1 queries in the worst case, quantumly just 1.
- 9.8Grover's search algorithmTwo mirrors squeeze out a rotation: an oracle reflection plus an inversion-about-the-mean turns the state 2θ toward the target each round, hitting it in about (π/4)√N rounds — a quadratic speedup, proven to be the limit.
- 9.9The idea behind Shor’s algorithmFactoring → period finding → reading the period with the QFT: every leg of the triple jump hand-computed on N = 15; the gap between polynomial and sub-exponential, hanging over RSA.
- 9.10An introduction to quantum error correctionNo cloning, no peeking — and yet errors can be corrected: the three-qubit code catches them with choral measurements that ask only about parity, never content; the stabiliser formalism turns the trick into a system, and the surface code and threshold theorem carry it into engineering.