A Complexity Dichotomy for Quantum Invariants of 3-Manifolds

TL;DR

This paper establishes precise complexity classifications for computing two quantum invariants of 3-manifolds, showing when these computations are feasible or computationally hard based on categorical properties.

Contribution

It provides exact dichotomies for the computational complexity of Reshetikhin--Turaev and Turaev--Viro invariants, linking complexity to categorical data and introducing a genus-one graph family for reductions.

Findings

Computing Reshetikhin--Turaev invariant is in FP iff the category is pointed.

Computing Turaev--Viro invariant is in FP iff the Drinfeld center is pointed.

Otherwise, both invariants are -hard.

Abstract

We prove exact complexity dichotomies for two quantum invariants of closed oriented three-manifolds, with the categorical data fixed. For a modular category $\mathcal{C}$, computing the Reshetikhin--Turaev invariant $Z_{\mathcal{C}}(M)$ from a framed-link surgery presentation is in $\mathrm{FP}$ exactly when $\mathcal{C}$ is pointed, that is, when all simple objects are invertible under tensor product; otherwise it is $\#\mathrm{P}$-hard. For a spherical fusion category $\mathcal{A}$, computing the Turaev--Viro invariant $|M|_{\mathcal{A}}$ from a triangulation, equivalently from a skeleton, is in $\mathrm{FP}$ exactly when its Drinfeld center $\mathcal{Z}(\mathcal{A})$ is pointed, equivalently when $\mathcal{A}$ is trivializable pointed; otherwise it is $\#\mathrm{P}$-hard. The polynomial-time cases reduce to finite abelian linear algebra and Gauss sums. The reductions are based on a genus-one graph-manifold family $M_G$, indexed by graphs $G$. In the anomaly-free case, the core calculation expresses $Z_{\mathcal{C}}(M_G)$ as the weighted graph homomorphism partition function with matrix $A_{\mathcal{C}}(i,j)=S_{i,j^*}/(d_i d_j)$, where $S$ is the modular matrix, $d_i$ is the quantum dimension of $i$, and $j^*$ is the dual label. Combining this formula with the Cai--Govorov dichotomy gives the hard side; the remaining Reshetikhin--Turaev and Turaev--Viro cases then follow by passing to Drinfeld centers.