Quantum topology provides various frameworks for defining and computing invariants of manifolds inspired by quantum theory. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work we analyze the computational complexity of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. While these categories are the simplest source of state sum invariants beyond finite abelian groups (whose invariants can be computed in polynomial time) their computational complexities are yet to be fully understood. We first establish that the invariants arising from even the smallest Tambara-Yamagami categories are #P-hard to compute, so that one expects the same to be true of the whole family. Our main result is then the existence of a fixed parameter tractable algorithm to compute these 3-manifold invariants, where the parameter is the first Betti number of the 3-manifold with Z/2Z coefficients. Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is computable in polynomial time. Thus while one generally expects quantum invariants to be hard to compute classically, our results suggest that the hardness of computing state sum invariants from Tambara-Yamagami categories arises from classical 3-manifold topology rather than the quantum nature of the algebraic input.

翻译：量子拓扑学提供了多种受量子理论启发的框架来定义和计算流形不变量。在数学和物理学领域均受到广泛关注的一个此类框架是Turaev-Viro-Barrett-Westbury状态和构造，该构造利用球面融合范畴的数据，通过张量网络缩并来定义三角剖分3-流形的拓扑不变量。本文中，我们分析了源自Tambara-Yamagami范畴的3-流形状态和不变量的计算复杂性。虽然这些范畴是超越有限阿贝尔群（其不变量可在多项式时间内计算）的最简单状态和不变量来源，但其计算复杂性尚未被完全理解。我们首先证明，即使是最小的Tambara-Yamagami范畴产生的不变量也是#P难计算的，因此可以预期整个族系具有相同的复杂性。我们的主要结果是：存在一种固定参数可处理算法来计算这些3-流形不变量，其中参数是3-流形在Z/2Z系数下的一阶贝蒂数。与计算拓扑学的其他领域（如曲面上的图论问题）不同，在3-流形拓扑学中，已知仅有极少数困难问题允许具有拓扑参数的FPT算法。然而，此类算法具有特殊意义，因为其复杂性仅依赖于输入组合表示的多项式函数，而与规模或组合宽度无关。此外，对于贝蒂数参数，其本身可在多项式时间内计算。因此，虽然通常预期量子不变量在经典计算中难以处理，但我们的结果表明：计算Tambara-Yamagami范畴状态和不变量的困难性源于经典3-流形拓扑学，而非代数输入的量子特性。