



@article {2619,
	title = {3-manifold diagrams and NP vs P},
	journal = {Quantum Information \& Computation },
	volume = {17},
	year = {2017},
	pages = {125-141 },
	abstract = {<p>The computational complexity class $\#$P captures the di_culty 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 $\#$P \&sube; FPNP, 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.</p>},
	url = {https://dl.acm.org/doi/abs/10.5555/3179483.3179491},
	author = {Gorjan Alagic and C. Lo}
}
