We define coined Quantum Walks on the infinite rooted binary tree given by unitary operators $U(C)$ on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix $C\in U(3)$ , and study their spectral properties. For circulant unitary coin matrices $C$ , we derive an equation for the Carathéodory function associated to the spectral measure of a cyclic vector for $U(C)$ . This allows us to show that for all circulant unitary coin matrices, the spectrum of the Quantum Walk has no singular continuous component. Furthermore, for coin matrices $C$ which are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum of $U(C)$ is pure point.We define coined Quantum Walks on the infinite rooted binary tree given by unitary operators $U(C)$ on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix $C\in U(3)$, and study their spectral properties. For circulant unitary coin matrices $C$, we derive an equation for the Carathéodory function associated to the spectral measure of a cyclic vector for $U(C)$. This allows us to show that for all circulant unitary coin matrices, the spectrum of the Quantum Walk has no singular continuous component. Furthermore, for coin matrices $C$ which are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum of $U(C)$ is pure point.

Jan 1, 20146 citations

Eigenvalues of Quantum Walks of Grover and Fourier Types

Takashi Komatsu, Tatsuya Tate

arXiv:1704.05236Journal of Fourier Analysis and ApplicationsJun 27, 20188 citations

Extraordinary behaviors in a two-dimensional decoherent alternative quantum walk

Tian Chen, Xiangdong Zhang

We reveal the quantum and classical behaviors of the two-dimensional (2D) alternative quantum walk (AQW) in the presence of decoherence. For different kinds of decoherence, the analytic expressions for the moments of position distribution of the AQW are obtained. Taking the broken line noise and coin decoherence as examples of decoherence, we find that when decoherence emerges in only one direction, the anisotropic position distribution pattern appears, and not all the motions of the walker exhibit the transition from quantum to classical behaviors. Considering the effect of decoherence, we reveal the anisotropic correlations between the $x\phantom{\rule{0.28em}{0ex}}(y)$ position of the 2D walker and the state of the coin in 2D AQWs.We reveal the quantum and classical behaviors of the two-dimensional (2D) alternative quantum walk (AQW) in the presence of decoherence. For different kinds of decoherence, the analytic expressions for the moments of position distribution of the AQW are obtained. Taking the broken line noise and coin decoherence as examples of decoherence, we find that when decoherence emerges in only one direction, the anisotropic position distribution pattern appears, and not all the motions of the walker exhibit the transition from quantum to classical behaviors. Considering the effect of decoherence, we reveal the anisotropic correlations between the $x\phantom{\rule{0.28em}{0ex}}(y)$ position of the 2D walker and the state of the coin in 2D AQWs.

arXiv:1601.06483Physical Review AJul 11, 201619 citations

We define coined Quantum Walks on the infinite rooted binary tree given by unitary operators

U(C)

on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix

C\in U(3)

, and study their spectral properties. For circulant unitary coin matrices

C

, we derive an equation for the Carathéodory function associated to the spectral measure of a cyclic vector for

U(C)

. This allows us to show that for all circulant unitary coin matrices, the spectrum of the Quantum Walk has no singular continuous component. Furthermore, for coin matrices

C

which are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum of

U(C)

is pure point.

x\phantom{\rule{0.28em}{0ex}}(y)

position of the 2D walker and the state of the coin in 2D AQWs.