奇异时空R4的量子计算和测量
Quantum computation and measurements from an exotic space-time R4
摘要：
作者利用具有关系的自由群$G$的子群陪集结构，建立了一个泛量子计算模型。在$G$中索引$d$的有效子群$H$导致在$d$维希尔伯特空间中的“魔术”状态$\left|ψ\right\rangle$，其编码最小信息上完全的量子测量（或MIC），可能携带有限的“上下文”几何。在目前的工作中，我们选择$G$作为一个奇异的$4$流形$V$的基本群$π_1(V)$，更准确地说，是一个“小奇异”（时空）$R^4$（即同胚和等距，但不同胚的欧几里德$\mathbb{R}^4$）。我们选择的例子，由于S。Akbulut和R.~ E. Gompf有两个显著的特性：（i）它显示了标准上下文几何的出现，例如Fano平面（在索引$7$），Mermin的五角星（在索引$10$处），双量子比特换向图$GQ(2,2)$（在索引$15$处）以及组合格拉斯曼Gr $(2,8)$（在索引$28$处），（ii）它允许将MIC测量解释为由这种外来（时空）$R^4$引起。我们将拓扑量子计算和奇异时空联系起来的新图景也旨在成为“量子引力”的一种方法。
英文摘要：
The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group $G$ with relations. A valid subgroup $H$ of index $d$ in $G$ leads to a 'magic' state $\left|ψ\right\rangle$ in $d$-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite 'contextual' geometry. In the present work, we choose $G$ as the fundamental group $π_1(V)$ of an exotic $4$-manifold $V$, more precisely a 'small exotic' (space-time) $R^4$ (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean $\mathbb{R}^4$). Our selected example, due to to S. Akbulut and R.~E. Gompf, has two remarkable properties: (i) it shows the occurence of standard contextual geometries such as the Fano plane (at index $7$), Mermin's pentagram (at index $10$), the two-qubit commutation picture $GQ(2,2)$ (at index $15$) as well as the combinatorial Grassmannian Gr$(2,8)$ (at index $28$) , (ii) it allows the interpretation of MICs measurements as arising from such exotic (space-time) $R^4$'s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of 'quantum gravity'.
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