We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive generic formula for a transversal $T$ gate of color codes on general 3-manifolds, which acts as collective non-Clifford logical CCZ gates on any triplet of logical qubits with their logical-$X$ membranes having a $\mathbb{Z}_2$ triple intersection at a single point. The triple intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory: the $\mathbb{Z}_2^3$ gauge theory. Moreover, the transversal $S$ gate of the color code corresponds to a higher-form symmetry supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical CZ gates. We have developed a generic formalism to compute the triple intersection invariants for 3-manifolds and also study the scaling of the Betti number and systoles with volume for various 3-manifolds, which translates to the encoding rate and distance. We further develop three types of LDPC codes supporting such logical gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface and a circle, with almost-constant rate $k/n=O(1/\log(n))$ and $O(\log(n))$ distance; (2) A homological fibre bundle code with $O(1/\log^{\frac{1}{2}}(n))$ rate and $O(\log^{\frac{1}{2}}(n))$ distance; (3) A specific family of 3D hyperbolic codes: the Torelli mapping torus code, constructed from mapping tori of a pseudo-Anosov element in the Torelli subgroup, which has constant rate while the distance scaling is currently unknown. We then show a generic constant-overhead scheme for applying a parallelizable universal gate set with the aid of logical-$X$ measurements.

翻译：我们研究了定义在三维流形上的同调量子低密度奇偶校验（LDPC）码族的并行容错量子计算架构，这些码具有常速率或近常速率编码特性。针对一般三维流形上的颜色码，我们推导了横向$T$门的通用公式，该门作用于任意逻辑量子比特三元组，表现为集体非克利福德逻辑CCZ门，其逻辑-$X$膜在单点处具有$\mathbb{Z}_2$三重相交。该三重相交数是一个拓扑不变量，在拓扑量子场论（即$\mathbb{Z}_2^3$规范理论）中，也会作为涌现高阶对称性算符路径积分的一部分出现。此外，颜色码的横向$S$门对应于余维数为1的子流形上的高阶形式对称性，可产生指数级可寻址且可并行化的逻辑CZ门。我们建立了计算三维流形三重相交不变量的通用形式体系，并研究了不同三维流形中贝蒂数和极小周长随体积的标度关系——这些量直接对应编码速率与距离。进一步地，我们构造了支持此类逻辑门的三类LDPC码：（1）由二维双曲曲面与圆环乘积构成的准双曲码，具有近常速率$k/n=O(1/\log(n))$和$O(\log(n))$距离；（2）同调纤维丛码，具有$O(1/\log^{\frac{1}{2}}(n))$速率和$O(\log^{\frac{1}{2}}(n))$距离；（3）特定家族的三维双曲码：托雷利映射环面码，通过托雷利子群中伪阿诺索夫元的映射环面构造，具有常速率但距离标度关系尚不明确。最终，我们展示了一种利用逻辑-$X$测量的通用恒定开销方案，可实现可并行化的通用量子门集合。