Asymptotic velocity of a position-dependent quantum walk

Asymptotic velocity of a position-dependent quantum walk We consider a position-dependent coined quantum walk on $$\mathbb {Z}$$ Z and assume that the coin operator C(x) satisfies $$\begin{aligned} \Vert C(x) - C_0 \Vert \le c_1|x|^{-1-\epsilon }, \quad x \in \mathbb {Z}\setminus \{0\} \end{aligned}$$ ‖ C ( x ) - C 0 ‖ ≤ c 1 | x | - 1 - ϵ , x ∈ Z \ { 0 } with positive $$c_1$$ c 1 and $$\epsilon $$ ϵ and $$C_0 \in U(2)$$ C 0 ∈ U ( 2 ) . We show that the Heisenberg operator $$\hat{x}(t)$$ x ^ ( t ) of the position operator converges to the asymptotic velocity operator $$\hat{v}_+$$ v ^ + so that $$\begin{aligned} \text{ s- }\lim _{t \rightarrow \infty } \mathrm{exp}\left( i \xi \frac{\hat{x}(t)}{t} \right) = \Pi _\mathrm{p}(U) + \mathrm{exp}(i \xi \hat{v}_+) \Pi _\mathrm{ac}(U) \end{aligned}$$ s- lim t → ∞ exp i ξ x ^ ( t ) t = Π p ( U ) + exp ( i ξ v ^ + ) Π ac ( U ) provided that U has no singular continuous spectrum. Here $$\Pi _\mathrm{p}(U)$$ Π p ( U ) (resp., $$\Pi _\mathrm{ac}(U)$$ Π ac ( U ) ) is the orthogonal projection onto the direct sum of all eigenspaces (resp., the subspace of absolute continuity) of U. We also prove that for the random variable $$X_t$$ X t denoting the position of a quantum walker at time $$t \in \mathbb {N}$$ t ∈ N , $$X_t/t$$ X t / t converges in law to a random variable V with the probability distribution $$\begin{aligned} \mu _V = \Vert \Pi _\mathrm{p}(U)\Psi _0\Vert ^2\delta _0 + \Vert E_{\hat{v}_+}(\cdot ) \Pi _\mathrm{ac}(U)\Psi _0\Vert ^2, \end{aligned}$$ μ V = ‖ Π p ( U ) Ψ 0 ‖ 2 δ 0 + ‖ E v ^ + ( · ) Π ac ( U ) Ψ 0 ‖ 2 , where $$\Psi _0$$ Ψ 0 is the initial state, $$\delta _0$$ δ 0 the Dirac measure at zero, and $$E_{\hat{v}_+}$$ E v ^ + the spectral measure of $$\hat{v}_+$$ v ^ + . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

Asymptotic velocity of a position-dependent quantum walk

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Publisher
Springer US
Copyright
Copyright © 2015 by Springer Science+Business Media New York
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-1183-x
Publisher site
See Article on Publisher Site

Abstract

We consider a position-dependent coined quantum walk on $$\mathbb {Z}$$ Z and assume that the coin operator C(x) satisfies $$\begin{aligned} \Vert C(x) - C_0 \Vert \le c_1|x|^{-1-\epsilon }, \quad x \in \mathbb {Z}\setminus \{0\} \end{aligned}$$ ‖ C ( x ) - C 0 ‖ ≤ c 1 | x | - 1 - ϵ , x ∈ Z \ { 0 } with positive $$c_1$$ c 1 and $$\epsilon $$ ϵ and $$C_0 \in U(2)$$ C 0 ∈ U ( 2 ) . We show that the Heisenberg operator $$\hat{x}(t)$$ x ^ ( t ) of the position operator converges to the asymptotic velocity operator $$\hat{v}_+$$ v ^ + so that $$\begin{aligned} \text{ s- }\lim _{t \rightarrow \infty } \mathrm{exp}\left( i \xi \frac{\hat{x}(t)}{t} \right) = \Pi _\mathrm{p}(U) + \mathrm{exp}(i \xi \hat{v}_+) \Pi _\mathrm{ac}(U) \end{aligned}$$ s- lim t → ∞ exp i ξ x ^ ( t ) t = Π p ( U ) + exp ( i ξ v ^ + ) Π ac ( U ) provided that U has no singular continuous spectrum. Here $$\Pi _\mathrm{p}(U)$$ Π p ( U ) (resp., $$\Pi _\mathrm{ac}(U)$$ Π ac ( U ) ) is the orthogonal projection onto the direct sum of all eigenspaces (resp., the subspace of absolute continuity) of U. We also prove that for the random variable $$X_t$$ X t denoting the position of a quantum walker at time $$t \in \mathbb {N}$$ t ∈ N , $$X_t/t$$ X t / t converges in law to a random variable V with the probability distribution $$\begin{aligned} \mu _V = \Vert \Pi _\mathrm{p}(U)\Psi _0\Vert ^2\delta _0 + \Vert E_{\hat{v}_+}(\cdot ) \Pi _\mathrm{ac}(U)\Psi _0\Vert ^2, \end{aligned}$$ μ V = ‖ Π p ( U ) Ψ 0 ‖ 2 δ 0 + ‖ E v ^ + ( · ) Π ac ( U ) Ψ 0 ‖ 2 , where $$\Psi _0$$ Ψ 0 is the initial state, $$\delta _0$$ δ 0 the Dirac measure at zero, and $$E_{\hat{v}_+}$$ E v ^ + the spectral measure of $$\hat{v}_+$$ v ^ + .

Journal

Quantum Information ProcessingSpringer Journals

Published: Nov 13, 2015

References

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