论文标题
基于Bernoulli功能和完美状态转移的连续时间量子步行的摘要模型
Abstract Model of Continuous-Time Quantum Walk Based on Bernoulli Functionals and Perfect State Transfer
论文作者
论文摘要
在本文中，我们提出了基于Bernoulli功能的连续时间量子步行（CTQW）的抽象模型，并表明该模型具有完美的状态转移（PST）等。令$ \ mathfrak {h} $为Square Antekable Complect-valued Bernoulli功能的空间，该功能是无限维度的。首先，我们在给定的子空间$ \ mathfrak {h} _l \ subset \ mathfrak {h} $通过$ \ mathfrak {h} $上的规范统一互动，以及通过分析其频谱结构的所有eigenvalues。然后，我们以$ \ mathfrak {h} _l $作为其状态空间介绍了CTQW的抽象模型，该模型由以$Δ_L$作为Hamiltonian的schrödinger方程来管理。我们定义模型的时间平均概率分布，获得分布的明确表达，尤其是我们发现分布允许对称属性。我们还通过向操作员$Δ_L$以及对模型本身提供图理论解释来证明模型是合理的。最后，我们证明该模型在时间时具有PST $ t = \fracπ{2} $。该模型也证明了其他一些有趣的结果。
In this paper, we present an abstract model of continuous-time quantum walk (CTQW) based on Bernoulli functionals and show that the model has perfect state transfer (PST), among others. Let $\mathfrak{h}$ be the space of square integrable complex-valued Bernoulli functionals, which is infinitely dimensional. First, we construct on a given subspace $\mathfrak{h}_L \subset \mathfrak{h}$ a self-adjoint operator $Δ_L$ via the canonical unitary involutions on $\mathfrak{h}$, and by analyzing its spectral structure we find out all its eigenvalues. Then, we introduce an abstract model of CTQW with $\mathfrak{h}_L$ as its state space, which is governed by the Schrödinger equation with $Δ_L$ as the Hamiltonian. We define the time-average probability distribution of the model, obtain an explicit expression of the distribution, and, especially, we find the distribution admits a symmetry property. We also justify the model by offering a graph-theoretic interpretation to the operator $Δ_L$ as well as to the model itself. Finally, we prove that the model has PST at time $t=\fracπ{2}$. Some other interesting results are also proven of the model.
