Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation

TL;DR

This paper presents a quantum algorithm that approximates Turaev-Viro invariants of 3-manifolds and demonstrates that this task is universal for quantum computation, linking topology and quantum computational power.

Contribution

It introduces a quantum algorithm for approximating Turaev-Viro invariants and proves this problem is universal for quantum computation, establishing a new connection between topology and quantum complexity.

Findings

The algorithm approximates Turaev-Viro invariants with nontrivial accuracy.

Approximating these invariants is as powerful as any quantum computation.

This links 3-manifold topology problems to quantum computational universality.

Abstract

The Turaev-Viro invariants are scalar topological invariants of compact, orientable 3-manifolds. We give a quantum algorithm for additively approximating Turaev-Viro invariants of a manifold presented by a Heegaard splitting. The algorithm is motivated by the relationship between topological quantum computers and (2+1)-D topological quantum field theories. Its accuracy is shown to be nontrivial, as the same algorithm, after efficient classical preprocessing, can solve any problem efficiently decidable by a quantum computer. Thus approximating certain Turaev-Viro invariants of manifolds presented by Heegaard splittings is a universal problem for quantum computation. This establishes a novel relation between the task of distinguishing non-homeomorphic 3-manifolds and the power of a general quantum computer.