Average position in quantum walks with an arbitrary initial state

Li Min, Cheng ZaiJun, Wang LingJie, Huang HaiBo

Published 2016-03-20Version 1

We investigated discrete-time quantum walks with an arbitrary initial state \mid\Psi_{0}(\theta,\phi,\varphi)\rangle=\cos\theta ^{i\phi}\mid0L\rangle+\sin\theta e^{i\varphi}\mid0R\rangle with a U(2) coin U(\alpha,\beta,\gamma) . We discover that the average position \overline{x}=\max(\overline{x})\cos(\alpha+\gamma+\phi-\varphi), with coin operator U_{C}(\alpha,\pi/4,\gamma) and initial state \mid\Phi_{0}(\pi/4,\phi,\varphi)\rangle=(e^{i\phi}\mid0L\rangle+e^{i\varphi}\mid0R\rangle)\sqrt{2}/2 .If we set initial state and coin operator to \mid\Phi_{0}\rangle(\theta,\pi/2,0)=i\cos\theta\mid0L\rangle+\sin\theta\mid0R\rangle) and coin operators U_{C}(0,\pi/4,0), for \alpha+\gamma+\phi-\varphi=\pi/2, we discover that \overline{x}=-\max(\overline{x})\cos(2\theta). Last we verify the result above, and obtain the summarize properties of quantum walks with an arbitrary state. We get that \overline{x}\theta,\phi,\varphi,\alpha,\beta,\gamma,t)=\cos2\theta*\overline{x}_{|0L\rangle}(\beta,t)+\sin2\theta*\cos(\alpha+\gamma+\phi-\varphi)*\overline{x}_{(\mid0L\rangle+\mid0R\rangle)\sqrt{2}/2}(\alpha=\gamma=0,\beta,t) . If the average positions \overline{x} with initial state |0L\rangle and \mid\Psi_{0}\rangle=(\mid0L\rangle+\mid0R\rangle)\sqrt{2}/2 are known, we can get the average position result for an arbitrary initial state.