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 with which one computes are diagrams of hyperbolic knots. The diagrams can even be arranged to have additional nice properties, such as being alternating with minimal crossing number. Moreover, the reduction is polynomially uniform in the self-braiding exponent of the coloring object. 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：我们证明，基于Witten-Reshetikhin-Turaev TQFT扭结不变量评估的拓扑量子计算机始终可以安排为，用于计算的扭结图均为双曲扭结的图。这些图甚至可以进一步具备额外优良性质，例如具有最小交叉数的交错性。此外，该归约在着色对象的自编织指数上是多项式均匀的。作为推论，可得出关于扭结量子不变量计算的若干复杂度理论下的困难结果。特别地，我们论证扭结的双曲几何不太可能对拓扑量子计算有所助益。