Quantum Invariants of 3-manifolds and NP vs #P

TL;DR

This paper proves that computing the SO(3) Witten-Reshetikhin-Turaev invariant of 3-manifolds is #P-hard using quantum computation tools, and explores implications for Heegaard splittings and related invariants.

Contribution

It establishes the #P-hardness of WRT invariant computation and links this complexity to properties of Heegaard splittings and other topological invariants.

Findings

WRT invariant of 3-manifolds is #P-hard to compute.

Existence of infinitely many Heegaard splittings resistant to simplification under WRT-preserving moves, assuming complexity class separations.

Approximation of WRT invariants remains #P-hard, extending the hardness results.

Abstract

The computational complexity class #P captures the difficulty of counting the satisfying assignments to a boolean formula. In this work, we use basic tools from quantum computation to give a proof that the SO(3) Witten-Reshetikhin-Turaev (WRT) invariant of 3-manifolds is #P-hard to calculate. We then apply this result to a question about the combinatorics of Heegaard splittings, motivated by analogous work on link diagrams by M. Freedman. We show that, if $\#\text{P}\neq\text{FP}^\text{NP}$, then there exist infinitely many Heegaard splittings which cannot be made logarithmically thin by local WRT-preserving moves, except perhaps via a superpolynomial number of steps. We also outline two extensions of the above results. First, adapting a result of Kuperberg, we show that any presentation-independent approximation of WRT is also #P-hard. Second, we sketch out how all of our results can be translated to the setting of triangulations and Turaev-Viro invariants.