Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)

Quantum Algorithms from Topological Quantum Field Theories Alagic, Gorjan


Topological Quantum Field Theories (or TQFTs) are abstract constructions from category theory and mathematical physics. Their conception was originally motivated by the search for a physical theory that unifies general relativity and quantum mechanics. At its core, a TQFT is a map from manifolds (e.g., spacetimes) to linear maps (e.g., quantum operations) that satisfies some physically sensible properties. For instance, the disjoint union of two manifolds must be mapped to the tensor product of the two corresponding linear maps. To manifolds without boundary, a TQFT assigns a topologically invariant number called a quantum invariant. This discovery added a beautiful new direction in the study of manifold invariants in pure mathematics. For this reason and many others, this area has seen a tremendous amount of work in the past two decades, from physicists and mathematicians alike. In this talk, we will discuss how this theory can be applied to design quantum algorithms for approximating certain quantum invariants. The aim of the talk is to give an accessible introduction to some of the ideas in this area, and to motivate quantum computation enthusiasts to study it further. We will begin with the simplest two-dimensional state-sum models. These examples are quite attractive, since they can be described in a combinatorial manner by means of triangulations. We will then define a three-dimensional state-sum TQFT, called the Turaev-Viro theory. Finally, we will discuss a recent result (joint with Stephen Jordan, Robert Koenig, and Ben Reichardt) showing that approximating the Turaev-Viro quantum invariant is a universal problem for quantum computation.

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