In a model of fault-tolerant quantum computation with quick and noiseless polyloglog-time auxiliary classical computation, we construct a fault tolerance protocol with constant-space and $\widetilde{O}(\log N)$-time overhead, where $\widetilde{O}(\cdot)$ hides sub-polylog factors. Our construction utilizes constant-rate quantum locally testable codes (qLTC), new fault-tolerant gadgets on qLTCs and qLDPC codes, and a new analysis framework. In particular, 1) we develop a new simple and self-contained construction of magic state distillation for qubits using qudit quantum Reed-Solomon codes with $(\log \frac{1}{\varepsilon})^{\gamma}$ spacetime overhead, where $\gamma \rightarrow 0$. 2) We prove that the recent family of almost-good qLTCs of Dinur-Lin-Vidick admit parallel single-shot decoders against adversarial errors of weight scaling with the code distance. 3) We develop logical state preparation and logical gate procedures with $\widetilde{O}(1)$-spacetime overhead on qLTCs. 4) To combine these ingredients, we introduce a new framework of fault tolerance analysis called the weight enumerator formalism. The framework permits easy formal composition of fault-tolerant gadgets, so we expect it to be of independent interest. Our work gives the lowest spacetime overhead to date, which, for the first time, matches that of classical fault tolerance up to sub-polylog factors. We conjecture this is optimal up to sub-polylog factors.

翻译：在一种具备快速且无噪声的多重对数时间辅助经典计算的容错量子计算模型中，我们构建了一种具有常数空间与$\widetilde{O}(\log N)$时间开销的容错协议，其中$\widetilde{O}(\cdot)$隐藏了亚多重对数因子。我们的构造利用了常数速率量子局部可测试码（qLTC）、基于qLTC与qLDPC码的新型容错组件，以及一套新的分析框架。具体而言：1）我们开发了一种新颖、简洁且自包含的量子比特魔术态蒸馏方案，该方案采用量子dit的量子Reed-Solomon码，其时空开销为$(\log \frac{1}{\varepsilon})^{\gamma}$，其中$\gamma \rightarrow 0$；2）我们证明了Dinur-Lin-Vidick近期提出的近乎最优qLTC族，在面对权重随码距线性增长的对抗性错误时，允许并行单轮解码器的存在；3）我们在qLTC上开发了具有$\widetilde{O}(1)$时空开销的逻辑态制备与逻辑门操作流程；4）为整合这些要素，我们引入了一种名为权重枚举器形式体系的新型容错分析框架。该框架允许容错组件的简便形式化组合，因此我们预期其具有独立的研究价值。本工作实现了迄今最低的时空开销，首次在亚多重对数因子范围内达到了与经典容错相当的水平。我们推测该结果在亚多重对数因子范围内是最优的。