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Suggested sequence

From a loop to a quantum computer.

A guided route through the mathematical prerequisites, experimentally observable consequences, and engineering limitations.

Stage 01

Start with the topology of spaces

Learn continuity, homotopy, winding, manifolds and bundles.

Topology versus geometry Homotopy and winding Manifolds, spheres and tori Fiber bundles and quantum eigenstates

Stage 02

Understand quantum-state geometry

Derive Berry phases, curvature and holonomy.

Projective Hilbert space Berry phase and adiabatic evolution Berry curvature as effective magnetic field Holonomy and Wilson loops

Stage 03

Connect invariants to transport

Follow Chern number to Hall conductivity and edge modes.

Chern number from Berry curvature TKNN quantization and Hall conductance Landau levels and integer quantum Hall effect Bulk–boundary correspondence

Stage 04

Explore symmetry and interactions

Compare Z2 insulators, fractional Hall states, and anyons.

Time-reversal symmetry and Z2 topology Fractional quantum Hall effect Anyons and braid statistics Topological order and long-range entanglement

Stage 05

Build toward quantum technology

Investigate Majoranas, codes, sensing, and experiments.

Majorana zero modes Kitaev chain and topological superconductivity Topological quantum sensing Cold atoms and synthetic dimensions Quantum computing with topological codes

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