Non-Abelian Berry Phase

A degenerate quantum system transported adiabatically around loops in parameter space acquires a non-commuting (matrix) holonomy. Two loops in different orders give different final states — the heart of topological quantum gates.

Loop Parameters

Loop 1 radius: 0.50 Loop 2 radius: 0.50 Loop tilt θ₁: 0° Loop tilt θ₂: 90° Speed: 1.5

Holonomy Matrices

Computing...

Non-commutativity

|U₁U₂ - U₂U₁|: -

Order: U₁ then U₂

Phase: Loop 1