论文标题
投影定理用于离散时间量子步行
Projection Theorem for Discrete-Time Quantum Walks
论文作者
论文摘要
我们对观察结果进行了概括，即在步行图上划分的离散时间量子步行的概率幅度与步进操作员一致的距离概率振幅会导致在还原图上的单一进化，这也是量子行走的。由于预测步行的有效步行图不一定比原始的简单，因此使用经过详细研究的euclidean lattices等经过详尽研究的案例中的已知结果，这可能会给某些量子步行的动态带来新的见解。我们使用对步行空间和步行者位移的抽象处理，目的是提出陈述的一般性。使用这种方法，我们还确定了一些病理案例，其中投影映射分解。对于晶格上的步行，该操作通常会导致具有超维硬币空间的量子步行。相反，这样的步行可以看作是在无法访问，更大的空间上的步行预测，并且可以从父母的步行中推断出它们的特性。我们表明，懒惰的量子步行，漫步的漫步是很大的连贯跳跃，并在圆形的边界条件上散步。我们还讨论了该理论与量子步行的光学光学实现的关系。此外，在某些情况下，这种明显的不可逆操作可以撤消，并且可以从其一组投影中重建量子行走。
We make and generalize the observation that summing of probability amplitudes of a discrete-time quantum walk over partitions of the walking graph consistent with the step operator results in a unitary evolution on the reduced graph which is also a quantum walk. Since the effective walking graph of the projected walk is not necessarily simpler than the original, this may bring new insights into the dynamics of some kinds of quantum walks using known results from thoroughly studied cases like Euclidean lattices. We use abstract treatment of the walking space and walker displacements in aim for a generality of the presented statements. Using this approach we also identify some pathological cases in which the projection mapping breaks down. For walks on lattices, the operation typically results in quantum walks with hyper-dimensional coin spaces. Such walks can, conversely, be viewed as projections of walks on inaccessible, larger spaces, and their properties can be inferred from the parental walk. We show that this is is the case for a lazy quantum walk, a walk with large coherent jumps and a walk on a circle with a twisted boundary condition. We also discuss the relation of this theory to the time-multiplexing optical implementations of quantum walks. Moreover, this manifestly irreversible operation can, in some cases and with a minor adjustment, be undone, and a quantum walk can be reconstructed from a set of its projections.
