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Instantaneous braids and Dehn twists in topologically ordered states

机译：在拓扑上有序状态下瞬间辫子和dehn扭曲

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A defining feature of topologically ordered states of matter is the existence of locally indistinguishable states on spaces with nontrivial 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 noncontractible 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 (relabeling 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. We further show that (i) applying a given braid or Dehn twist k times can be achieved with O(log k) time overhead, independent of code distance and system size, which implies an exponential speedup for certain logical gate sequences by trading space for time, and (ⅱ) an arbitrary element of the MCG can be implemented by a constant depth (independent of d) LQC followed by a permutation, where in this case the range of interactions of the LQC grows with the number of generators in the presentation of the group element. 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.
机译：拓扑排序状态的定义特征是存在与非活动拓扑的空间上的局部难以区分状态。这些退化状态形成了空间的映射类组（MCG）的表示，其由缺陷或任何缺陷或任何缺陷的辫子生成，并且沿着不可转换的循环由Dehn曲折。在拓扑量子误差校正码和拓扑量子计算的上下文中，可以将这些操作视为容错逻辑栅极。这里，我们示出了辫子和Dehn曲线通常可以由恒定深度量子电路实现，其深度与代码距离D和系统尺寸无关。该电路由实现量子状态的局部几何变形的恒定深度局部量子电路（LQC）组成，然后是QUBITS的置换（重新标记）。置换需要置换由订单D距离分开的Qubits;它可以通过移动QUBITS的集体经典运动来实现，或者使用远程交换操作（在Immobile Qubits上的长距离交换操作（具有d）设置的恒定深度电路来实现。我们进一步示出了应用给定编织的（I）可以使用O（log k）时间开销，与代码距离和系统大小无关，这意味着通过交易空间来实现某些逻辑门序列的指数加速度来实现o（log k）时间开销。时间，和（Ⅱ）MCG的任意元素可以通过恒定的深度（独立于D）LQC来实现，然后在这种情况下，LQC的相互作用的范围随着演示中的发电机的数量而增长群体元素。将这些结果应用于某些非阿比越量误差校正码，演示了如何在使用恒定深度酉电路的编码Qubits上实现通用逻辑门机的方式。

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  • 来源
    《Physical review》 |2020年第7期|075105.1-075105.44|共44页
  • 作者

    Guanyu Zhu; Ali Lavasani; Maissam Barkeshli;

  • 作者单位

    Department of Physics. Condensed Matter Theory Center University of Maryland College Park Maryland 20742 USA and Joint Quantum Institute University of Maryland College Park Maryland 20742 USA;

    Department of Physics. Condensed Matter Theory Center University of Maryland College Park Maryland 20742 USA and Joint Quantum Institute University of Maryland College Park Maryland 20742 USA;

    Department of Physics. Condensed Matter Theory Center University of Maryland College Park Maryland 20742 USA and Joint Quantum Institute University of Maryland College Park Maryland 20742 USA;

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