Entanglement in coined quantum walks on regular graphs
Entanglement in coined quantum walks on regular graphs
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DOI：
10.1088/1367-2630/7/1/156
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发表时间：
2005-07-12
影响因子：
3.3
通讯作者：
Knight, PL
中科院分区：
文献类型：
--
作者：
Carneiro, I;Loo, M;Knight, PL
量子行走，无论是离散（创造的）时间还是连续时间，构成了最近几种量子算法的基础。在这里，我们使用数值模拟来研究离散的、创造的量子行走的属性。我们通过计算硬币约简密度矩阵的熵来研究硬币与粒子位置之间纠缠的变化。我们考虑正则图上 2 到 8 维硬币的动态演化和渐近极限。对于低硬币尺寸，传播得更快的量子游走（通过其分布与均匀分布的均方偏差来测量）也表现出更快地收敛到硬币和粒子位置之间纠缠的渐近值。对于高维币，DFT 币算子的传播效率比 Grover 币更高。我们研究了硬币在规则有限图（例如周期）上的纠缠，并且还表明，在完全二部图上，格罗弗硬币的量子游走始终具有周期为四的周期性。我们将 Childs 等人（2003 Proc. STOC，第 59-68 页）使用的“粘合树”图推广到更高的分支率（扇出），并验证分支率和树深度的缩放比例是多项式。
Quantum walks, both discrete (coined) and continuous time, form the basis of several recent quantum algorithms. Here we use numerical simulations to study the properties of discrete, coined quantum walks. We investigate the variation in the entanglement between the coin and the position of the particle by calculating the entropy of the reduced density matrix of the coin. We consider both dynamical evolution and asymptotic limits for coins of dimensions from two to eight on regular graphs. For low coin dimensions, quantum walks which spread faster (as measured by the mean square deviation of their distribution from uniform) also exhibit faster convergence towards the asymptotic value of the entanglement between the coin and particle's position. For high-dimensional coins, the DFT coin operator is more efficient at spreading than the Grover coin. We study the entanglement of the coin on regular finite graphs such as cycles, and also show that on complete bipartite graphs, a quantum walk with a Grover coin is always periodic with period four. We generalize the 'glued trees' graph used by Childs et al (2003 Proc. STOC, pp 59-68) to higher branching rate (fan out) and verify that the scaling with branching rate and with tree depth is polynomial.