Coined Quantum Walks on Graphs

Yusuke Higuchi, Norio Konno, Iwao Sato +1 more

We clarify that coined quantum walk is determined by only the choice of local quantum coins. To do so, we characterize coined quantum walks on graph by disjoint Euler circles with respect to symmetric arcs. In this paper, we introduce a new class of coined quantum walk by a special choice of quantum coins determined by corresponding quantum graph, called quantum graph walk. We show that a stationary state of quantum graph walk describes the eigenfunction of the quantum graph.

Stéphane Attal, Francesco Petruccione, Ilya Sinayskiy

Renato Portugal

Pascal Philipp, Renato Portugal

The connection between coined and continuous-time quantum walk models has been addressed in a number of papers. In most of those studies, the continuous-time model is derived from coined quantum walks by employing dimensional reduction and taking appropriate limits. In this work, we produce the evolution of a coined quantum walk on a generic graph using a continuous-time quantum walk on a larger graph. In addition to expanding the underlying structure, we also have to switch on and off edges during the continuous-time evolution to accommodate the alternation between the shift and coin operators from the coined model. In one particular case, the connection is very natural, and the continuous-time quantum walk that simulates the coined quantum walk is driven by the graph Laplacian on the dynamically changing expanded graph.

Renato Portugal

Ivens Carneiro, Meng Loo, Xibai Xu +3 more

Quantum walks, both discrete (coined) and continuous time, form the basis of several recent quantum algorithms. Here we use numerical simulations to study the properties of discrete, coined quantum walks. We investigate the variation in the entanglement between the coin and the position of the particle by calculating the entropy of the reduced density matrix of the coin. We consider both dynamical evolution and asymptotic limits for coins of dimensions from two to eight on regular graphs. For low coin dimensions, quantum walks which spread faster (as measured by the mean square deviation of their distribution from uniform) also exhibit faster convergence towards the asymptotic value of the entanglement between the coin and particle's position. For high-dimensional coins, the DFT coin operator is more efficient at spreading than the Grover coin. We study the entanglement of the coin on regular finite graphs such as cycles, and also show that on complete bipartite graphs, a quantum walk with a Grover coin is always periodic with period four. We generalize the 'glued trees' graph used by Childs et al (2003 Proc. STOC , pp 59–68) to higher branching rate (fan out) and verify that the scaling with branching rate and with tree depth is polynomial.