Unitary Dynamics and Quantum Circuits

Stub. The state-evolution axiom: closed quantum systems evolve by unitary operators; SU(2) and its concrete action as rotations of the Bloch sphere, giving every single-qubit gate a clean geometric picture; and multi-qubit gates and quantum circuits, the standard way to describe a sequence of unitary operations (and, later, measurements) acting on several qubits.

10/6/2026 · 1 min read · 151 words

Corresponds to Lectures 6–8. The second dynamical axiom: a closed quantum system’s state evolves under a unitary operator $U$ ( $\rho \mapsto U\rho U^\dagger$ ), which is reversible and norm-preserving — unitarity is the quantum analogue of a classical deterministic, invertible evolution. For a single qubit, every unitary (up to a global phase) corresponds to an element of $SU(2)$ , and $SU(2)$ acts on the Bloch sphere exactly as the rotation group $SO(3)$ acts on an ordinary sphere — so every single-qubit gate is literally a rotation of the Bloch ball. Multi-qubit gates (like CNOT) extend this to several qubits at once, and quantum circuits are the standard diagrammatic language for composing gates (and later, measurements) into an algorithm.

Outline (TODO — flesh out each)

The state evolution axiom — unitary dynamics
$SU(2)$ and Bloch sphere rotations
Multi-qubit gates
Quantum circuits

Outline (TODO — flesh out each)

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