Average position of quantum walks with an arbitrary initial state

Abstract

We investigated discrete time quantum walks with an arbitrary initial state $\midΨ_{0}(θ,ϕ,φ)\rangle=\cosθe^{iϕ}\mid0L\rangle+\sinθe^{iφ}\mid0R\rangle$ with a U(2) coin $U(α,β,γ)$. We discover that the average position $\bar{x}=\max(\bar{x})\cos(α+γ+ϕ-φ)$, with coin operator $U(α,π/4,γ)$ and initial state $\midΦ_{0}(π/4,ϕ,φ)\rangle=(e^{iϕ}\mid0L\rangle+e^{iφ}\mid0R\rangle)\sqrt{2}/2$. If we set initial state and coin operator to $\midΦ_{0}\rangle(θ,π/2,0)=i\cosθ\mid0L\rangle+\sinθ\mid0R\rangle)$ and coin operator $U(0,π/4,0)$, for $α+γ+ϕ-φ=π/2$, we discover that $\bar{x}=-\max(\bar{x})\cos(2θ).$ Last we verify the result above, and obtain the summarize properties of quantum walks with an arbitrary state. We get that $\bar{x}(θ,ϕ,φ,α,β,γ,t)=\cos2θ*\bar{x}_{|0L\rangle}(β,t)+\sin2θ*\cos(α+γ+ϕ-φ)*\bar{x}_{(\mid0L\rangle+\mid0R\rangle)\sqrt{2}/2}(α=γ=0,β,t)$. If the average positions $\bar{x}$ with initial state $|0L\rangle$ and $\midΨ_{0}\rangle=(\mid0L\rangle+\mid0R\rangle)\sqrt{2}/2$ and coin operator $U(0,β,0)$ are known, we can get the average position result of quantum walks with an arbitrary initial state and a U(2) coin operator.