Quantum Algorithms for Invariants of Triangulated Manifolds

TL;DR

This paper explores quantum algorithms for approximating topological invariants of triangulated manifolds, leveraging tensor network contractions and specific models like finite groups and Fibonacci anyons.

Contribution

It introduces quantum algorithms for invariant approximation of triangulated manifolds using tensor networks, with explicit cases for surfaces and 3-manifolds.

Findings

Quantum algorithms efficiently approximate manifold invariants.

Invariant formulas involve group representations and Fibonacci anyon models.

Results include closed-form expressions for surface invariants and Turaev-Viro invariants.

Abstract

One of the apparent advantages of quantum computers over their classical counterparts is their ability to efficiently contract tensor networks. In this article, we study some implications of this fact in the case of topological tensor networks. The graph underlying these networks is given by the triangulation of a manifold, and the structure of the tensors ensures that the overall tensor is independent of the choice of internal triangulation. This leads to quantum algorithms for additively approximating certain invariants of triangulated manifolds. We discuss the details of this construction in two specific cases. In the first case, we consider triangulated surfaces, where the triangle tensor is defined by the multiplication operator of a finite group; the resulting invariant has a simple closed-form expression involving the dimensions of the irreducible representations of the group and the Euler characteristic of the surface. In the second case, we consider triangulated 3-manifolds, where the tetrahedral tensor is defined by the so-called Fibonacci anyon model; the resulting invariant is the well-known Turaev-Viro invariant of 3-manifolds.