Braiding defects in topological stabiliser codes has been widely studied as a promising approach to fault-tolerant quantum computing. Here, we explore the potential and limitations of such schemes in codes of all spatial dimensions. We prove that a universal gate set for quantum computing cannot be realised by supplementing locality-preserving logical operators with defect braiding, even in more than two dimensions. However, notwithstanding this no-go theorem, we demonstrate that higher dimensional defect-braiding schemes have the potential to play an important role in realising fault-tolerant quantum computing. Specifically, we present an approach to implement the full Clifford group via braiding in any code possessing twist defects on which a fermion can condense. We explore three such examples in higher dimensional codes, specifically: in self-dual surface codes; the three dimensional Levin-Wen fermion mode; and the checkerboard model. Finally, we show how our no-go theorems can be circumvented to provide a universal scheme in three-dimensional surface codes without magic state distillation. Specifically, our scheme employs adaptive implementation of logical operators conditional on logical measurement outcomes to lift a combination of locality-preserving and braiding logical operators to universality.

中文翻译：

---

具有拓扑稳定器代码缺陷的容错量子门

拓扑稳定器代码中的编织缺陷已被广泛研究为一种有前途的容错量子计算方法。在这里，我们探索所有空间维度的代码中这种方案的潜力和局限性。我们证明，即使在二维以上的情况下，通过使用缺陷编织对保留位置的逻辑运算符进行补充也无法实现用于量子计算的通用门集。然而，尽管有这个不做定理，但我们证明了高维缺陷编织方案在实现容错量子计算中具有潜在的重要作用。具体来说，我们提出了一种通过编织任何具有扭曲缺陷的代码来实现完整的Clifford组的方法，费米子可以在扭曲缺陷上进行冷凝。我们将在高维代码中探索三个这样的示例，特别是：以自对偶的表面代码；三维莱文温费米子模式；和棋盘模型。最后，我们展示了如何绕开我们的不动定理，以在没有魔术状态蒸馏的情况下提供三维表面代码的通用方案。具体而言，我们的方案采用了以逻辑测量结果为条件的逻辑运算符的自适应实现，以将保留位置和编织逻辑运算符的组合提升为通用性。