We show that a topological quantum computer based on the evaluation of a Witten-Reshetikhin-Turaev TQFT invariant of knots can always be arranged so that the knot diagrams one compiles are hyperbolic. Furthermore, the diagrams can be arranged to have additional nice properties, such as being alternating with minimal crossing number. Various complexity-theoretic hardness results regarding the calculation of quantum invariants of knots follow as corollaries. In particular, we argue that the hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.

翻译：我们显示,基于对Witten-Reshetikhin-Turaev TQFT的结节变数评估的地形量计计算机总是可以安排的,这样结图就是一个双曲的编译。此外,图表也可以安排具有其他的好属性,例如与最小的过量数交替。计算结结数变数的各种复杂理论和硬性结果作为卷轴。特别是,我们认为结数的双曲几何不可能用于地形量量计算。