Approximating the Turaev-Viro Invariant of Mapping Tori is Complete for One Clean Qubit

TL;DR

This paper demonstrates that approximating the Fibonacci Turaev-Viro invariant of mapping tori is a complete problem for the DQC1 complexity class, linking topological invariants with quantum computational complexity.

Contribution

It establishes the DQC1-completeness of estimating the Fibonacci Turaev-Viro invariant for mapping tori, extending the understanding of quantum complexity in topological invariants.

Findings

Estimating the Fibonacci Turaev-Viro invariant of mapping tori is DQC1-complete.

Estimating the Turaev-Viro invariant for arbitrary manifolds is BQP-complete.

Draws an analogy with the computational complexity of the Jones polynomial.

Abstract

The Turaev-Viro invariants are scalar topological invariants of three-dimensional manifolds. Here we show that the problem of estimating the Fibonacci version of the Turaev-Viro invariant of a mapping torus is a complete problem for the one clean qubit complexity class (DQC1). This complements a previous result showing that estimating the Turaev-Viro invariant for arbitrary manifolds presented as Heegaard splittings is a complete problem for the standard quantum computation model (BQP). We also discuss a beautiful analogy between these results and previously known results on the computational complexity of approximating the Jones polynomial.