Institute for Quantum Information (IQI) Weekly Seminar

Quantum computation has taken much from the scientific fields it sprouted from. Occasionally, it has also given back. I will discuss two recent results, both of which employ basic methods and ideas from quantum computation to prove a new theorem about low-dimensional topology. In the first result, we show the existence of 3-manifold diagrams which cannot be made ``very thin'' via local transformations. The key to the proof is establishing the #P-hardness of certain 3-manifold invariants, which we achieve via an application of the Solovay-Kitaev universality theorem with exponential precision. In the second result, we prove a relationship between the distinguishing power of a link invariant, and the entangling power of the linear operator that describes braiding. More precisely, we show that link invariants derived from non-entangling solutions to the Yang-Baxter equation are trivial.

The former is joint work with Catharine Lo (Caltech), and the latter is joint work with Stephen Jordan and Michael Jarett (UMD).