P.L. homeomorphic manifolds are equivalent by elementary shellings
European Journal of Combinatorics
Diagonal flips of triangulations on closed surfaces preserving specified properties
Journal of Combinatorial Theory Series B
A polynomial quantum algorithm for approximating the Jones polynomial
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Algorithms for quantum computation: discrete logarithms and factoring
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Quantum Computation and the Evaluation of Tensor Networks
SIAM Journal on Computing
The Jones polynomial: quantum algorithms and applications in quantum complexity theory
Quantum Information & Computation
Estimating Jones polynomials is a complete problem for one clean qubit
Quantum Information & Computation
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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.