高维流形上的量子码
Quantum Codes from High-Dimensional Manifolds
摘要：
我们在各种高维流形上构造环面码。假设几何中的一个猜想，我们在$N$量子位上找到了量子CSS稳定器代码家族，对于任何$ε>0$，该代码具有对数权重稳定器和距离$N^{1-ε}$。猜想存在一个常数$C>0$，使得对于任何$n$维环面${\mathbb T}^n={\mathbb R}^n/Λ$，其中$Λ$是一个格，最小体积无取向$n/2$维曲面（使用欧几里德度量）表示非平凡同源性的超平面的体积至少为$C^n$乘以表示非平凡同源性的最小体积$n/2$维超平面的体积;事实上，对于$Λ$，有这个结果就足够了，$Λ$是一个积分晶格，其表面被单位超立方体限制在立方体的表面上。主要的技术结果是对某些随机格的Rankin不变量\cite{rankin}的估计，表明在某种意义上它们是最优的。此外，我们构造代码的平方根距离，对数重量稳定剂，和逆多对数健全的因素（被认为是量子局部可测试的代码\cite{qltc}）。我们还提供了一个简短的替代证明，证明格的外幂中的最短向量可能是非分裂的\cite{coulangeon}。
英文摘要：
We construct toric codes on various high-dimensional manifolds. Assuming a conjecture in geometry we find families of quantum CSS stabilizer codes on $N$ qubits with logarithmic weight stabilizers and distance $N^{1-ε}$ for any $ε>0$. The conjecture is that there is a constant $C>0$ such that for any $n$-dimensional torus ${\mathbb T}^n={\mathbb R}^n/Λ$, where $Λ$ is a lattice, the least volume unoriented $n/2$-dimensional surface (using the Euclidean metric) representing nontrivial homology has volume at least $C^n$ times the volume of the least volume $n/2$-dimensional hyperplane representing nontrivial homology; in fact, it would suffice to have this result for $Λ$ an integral lattice with the surface restricted to faces of a cubulation by unit hypercubes. The main technical result is an estimate of Rankin invariants\cite{rankin} for certain random lattices, showing that in a certain sense they are optimal. Additionally, we construct codes with square-root distance, logarithmic weight stabilizers, and inverse polylogarithmic soundness factor (considered as quantum locally testable codes\cite{qltc}). We also provide an short, alternative proof that the shortest vector in the exterior power of a lattice may be non-split\cite{coulangeon}.
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