We use simple deterministic dynamical systems as coins in studying quantum walks. These dynamical systems can be chosen to display, in the classical limit, a range of behaviors from the integrable to chaotic, or deterministically random. As an example of an integrable coin we study the Fourier walk that generalizes the Hadamard walk and show that the walker slows down with coin dimensionality, which controls the effective Planck constant. Introducing multi-Harper maps as deterministic models of random walks we study the effect of coin chaos on the quantum walk. We also demonstrate that breaking time-reversal symmetry in the coin dynamics effectively slows down the walk.We use simple deterministic dynamical systems as coins in studying quantum walks. These dynamical systems can be chosen to display, in the classical limit, a range of behaviors from the integrable to chaotic, or deterministically random. As an example of an integrable coin we study the Fourier walk that generalizes the Hadamard walk and show that the walker slows down with coin dimensionality, which controls the effective Planck constant. Introducing multi-Harper maps as deterministic models of random walks we study the effect of coin chaos on the quantum walk. We also demonstrate that breaking time-reversal symmetry in the coin dynamics effectively slows down the walk.

arXiv:quant-ph/0305026May 6, 20030 citations

Controllability of quantum walks on graphs

Francesca Albertini, Domenico D’Alessandro

Mathematics of Control Signals and SystemsMar 23, 20126 citations

Controllability of Quantum Systems

Sonia G. Schirmer, I. C. H. Pullen, A. I. Solomon

arXiv:quant-ph/0302121IFAC Proceedings VolumesApr 1, 200324 citations

Relation between random walks and quantum walks

Stefan Boettcher, Stefan Falkner, Renato Portugal

Based on studies of four specific networks, we conjecture a general relation between the walk dimensions ${d}_{w}$ of discrete-time random walks and quantum walks with the (self-inverse) Grover coin. In each case, we find that ${d}_{w}$ of the quantum walk takes on exactly half the value found for the classical random walk on the same geometry. Since walks on homogeneous lattices satisfy this relation trivially, our results for heterogeneous networks suggest that such a relation holds irrespective of whether translational invariance is maintained or not. To develop our results, we extend the renormalization-group analysis (RG) of the stochastic master equation to one with a unitary propagator. As in the classical case, the solution $\ensuremath{\rho}(x,t)$ in space and time of this quantum-walk equation exhibits a scaling collapse for a variable ${x}^{{d}_{w}}/t$ in the weak limit, which defines ${d}_{w}$ and illuminates fundamental aspects of the walk dynamics, e.g., its mean-square displacement. We confirm the collapse for $\ensuremath{\rho}(x,t)$ in each case with extensive numerical simulation. The exact values for ${d}_{w}$ themselves demonstrate that RG is a powerful complementary approach to study the asymptotics of quantum walks that weak-limit theorems have not been able to access, such as for systems lacking translational symmetries beyond simple trees.Based on studies of four specific networks, we conjecture a general relation between the walk dimensions ${d}_{w}$ of discrete-time random walks and quantum walks with the (self-inverse) Grover coin. In each case, we find that ${d}_{w}$ of the quantum walk takes on exactly half the value found for the classical random walk on the same geometry. Since walks on homogeneous lattices satisfy this relation trivially, our results for heterogeneous networks suggest that such a relation holds irrespective of whether translational invariance is maintained or not. To develop our results, we extend the renormalization-group analysis (RG) of the stochastic master equation to one with a unitary propagator. As in the classical case, the solution $\ensuremath{\rho}(x,t)$ in space and time of this quantum-walk equation exhibits a scaling collapse for a variable ${x}^{{d}_{w}}/t$ in the weak limit, which defines ${d}_{w}$ and illuminates fundamental aspects of the walk dynamics, e.g., its mean-square displacement. We confirm the collapse for $\ensuremath{\rho}(x,t)$ in each case with extensive numerical simulation. The exact values for ${d}_{w}$ themselves demonstrate that RG is a powerful complementary approach to study the asymptotics of quantum walks that weak-limit theorems have not been able to access, such as for systems lacking translational symmetries beyond simple trees.

arXiv:1410.7034Physical Review AMay 29, 201523 citations

Extreme events of quantum walks on networks

Nisarg Vyas, M. S. Santhanam

Extreme events are the ones for which the magnitude of the event is much larger than its typical values. They have been extensively investigated on classical dynamical and stochastic processes on networks. In this work, in analogy with well-studied extreme events of random walks, the extreme events of quantum walks on regular lattices, a scale-free network, and Erdős-Rényi network are studied. Due to unitary quantum evolution, the quantum walk dynamics usually differs from that of the classical random walks, and hence we expect extreme event properties to be different in classical and quantum settings. In contrast to this expectation, we report that extreme events of phase coherent quantum walks are qualitatively similar to that of classical random walks. We show that this counterintuitive behavior originates from the dominance of nodal fluctuations over internode correlations in disordered network structures. In particular, the occurrence probability for extreme events on scale-free and Erdős-Rényi networks displays a decaying power law with the degree of nodes. That is, the extreme event probability is larger for small degree nodes compared to the hubs on the network, in qualitative agreement with corresponding classical random walk results. Further, it is shown that extreme event probability scales with the threshold used to define extreme events.Extreme events are the ones for which the magnitude of the event is much larger than its typical values. They have been extensively investigated on classical dynamical and stochastic processes on networks. In this work, in analogy with well-studied extreme events of random walks, the extreme events of quantum walks on regular lattices, a scale-free network, and Erdős-Rényi network are studied. Due to unitary quantum evolution, the quantum walk dynamics usually differs from that of the classical random walks, and hence we expect extreme event properties to be different in classical and quantum settings. In contrast to this expectation, we report that extreme events of phase coherent quantum walks are qualitatively similar to that of classical random walks. We show that this counterintuitive behavior originates from the dominance of nodal fluctuations over internode correlations in disordered network structures. In particular, the occurrence probability for extreme events on scale-free and Erdős-Rényi networks displays a decaying power law with the degree of nodes. That is, the extreme event probability is larger for small degree nodes compared to the hubs on the network, in qualitative agreement with corresponding classical random walk results. Further, it is shown that extreme event probability scales with the threshold used to define extreme events.

Physical review. EMar 13, 20260 citations

Discrete-time controllability for feedback quantum dynamics

Francesca Albertini, Francesco Ticozzi

arXiv:1012.1397AutomaticaSep 7, 20118 citations

Based on studies of four specific networks, we conjecture a general relation between the walk dimensions

{d}_{w}

of discrete-time random walks and quantum walks with the (self-inverse) Grover coin. In each case, we find that

{d}_{w}

of the quantum walk takes on exactly half the value found for the classical random walk on the same geometry. Since walks on homogeneous lattices satisfy this relation trivially, our results for heterogeneous networks suggest that such a relation holds irrespective of whether translational invariance is maintained or not. To develop our results, we extend the renormalization-group analysis (RG) of the stochastic master equation to one with a unitary propagator. As in the classical case, the solution $\ensuremath{\rho}(x,t)$ in space and time of this quantum-walk equation exhibits a scaling collapse for a variable

{x}^{{d}_{w}}/t

in the weak limit, which defines

{d}_{w}

and illuminates fundamental aspects of the walk dynamics, e.g., its mean-square displacement. We confirm the collapse for $\ensuremath{\rho}(x,t)$ in each case with extensive numerical simulation. The exact values for

{d}_{w}

themselves demonstrate that RG is a powerful complementary approach to study the asymptotics of quantum walks that weak-limit theorems have not been able to access, such as for systems lacking translational symmetries beyond simple trees.