Instantaneous braids and Dehn twists in topologically ordered states
Instantaneous braids and Dehn twists in topologically ordered states
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DOI：
10.1103/physrevb.102.075105
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发表时间：
2018-06
影响因子：
3.7
通讯作者：
Guanyu Zhu;A. Lavasani;M. Barkeshli
中科院分区：
文献类型：
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作者：
Guanyu Zhu;A. Lavasani;M. Barkeshli
拓扑有序态的一个定义特征是在具有非平凡拓扑的空间上存在局部不可区分的态。这些简并态形成了空间的映射类群（MCG）的表示，它是由缺陷或任意子的辫子，以及沿沿着非收缩圈的Dehn扭曲产生的。这些操作可以被看作拓扑量子纠错码和拓扑量子计算中的容错逻辑门。在这里，我们表明，辫子和德恩扭曲一般可以实现一个恒定的深度量子电路，具有独立的代码距离$d$和系统大小的深度。该电路由一个恒定深度的局部量子电路（LQC），实现量子态的局部几何变形，然后对量子位进行置换（重新标记）。置换需要置换量子比特，这些量子比特之间的距离为$d$阶;它可以通过移动的量子比特的集体经典运动来实现，也可以通过在im移动的量子比特上使用长距离SWAP操作（范围设置为$d$）来实现。将这些结果应用于某些非阿贝尔量子纠错码，演示了如何仅使用恒定深度酉电路在编码量子比特上实现通用逻辑门集。
A defining feature of topologically ordered states of matter is the existence of locally indistinguishable states on spaces with non-trivial topology. These degenerate states form a representation of the mapping class group (MCG) of the space, which is generated by braids of defects or anyons, and by Dehn twists along non-contractible cycles. These operations can be viewed as fault-tolerant logical gates in the context of topological quantum error correcting codes and topological quantum computation. Here we show that braids and Dehn twists can in general be implemented by a constant depth quantum circuit, with a depth that is independent of code distance $d$ and system size. The circuit consists of a constant depth local quantum circuit (LQC) implementing a local geometry deformation of the quantum state, followed by a permutation on (relabelling of) the qubits. The permutation requires permuting qubits that are separated by a distance of order $d$; it can be implemented by collective classical motion of mobile qubits or as a constant depth circuit using long-range SWAP operations (with a range set by $d$) on immobile qubits. Applying these results to certain non-Abelian quantum error correcting codes demonstrates how universal logical gate sets can be implemented on encoded qubits using only constant depth unitary circuits.