For quantum search via the continuous-time quantum walk, the evolution of the whole system is usually limited in a small subspace. In this paper, we discuss how the symmetries of the graphs are related to the existence of such an invariant subspace, which also suggests a dimensionality reduction method based on group representation theory. We observe that in the one-dimensional subspace spanned by each desired basis state which assembles the identically evolving original basis states, we always get a trivial representation of the symmetry group. So, we could find the desired basis by exploiting the projection operator of the trivial representation. Besides being technical guidance in this type of problem, this discussion also suggests that all the symmetries are used up in the invariant subspace and the asymmetric part of the Hamiltonian is very important for the purpose of quantum search.

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对称性在通过连续时间量子游走的量子搜索中的作用

对于通过连续时间量子游走的量子搜索，整个系统的演化通常被限制在一个小的子空间中。在本文中，我们讨论了图的对称性如何与这种不变子空间的存在相关联，这也提出了一种基于群表示理论的降维方法。我们观察到，在由每个期望的基态组成的一维子空间中，这些基态组合了相同演化的原始基态，我们总是得到对称群的平凡表示。因此，我们可以通过利用平凡表示的投影算子找到所需的基础。除了作为此类问题的技术指导外，