Szegedy’s Quantum Walk

Raqueline A. M. Santos

Thomas G. Wong

Sergio A. Ortega, Miguel Angel Martin-Delgado

Abstract Szegedy's quantum walk is an algorithm for quantizing a general Markov chain. It has plenty of applications, such as many variants of optimizations. In order to check its properties in an error‐free environment, it is important to have a classical simulator. However, the current simulation algorithms require a great deal of memory due to the particular formulation of this quantum walk. In this paper, a memory‐saving algorithm is proposed that scales as with the size of the graph. Additional procedures are provided for simulating Szegedy's quantum walk over mixed states and also the Semiclassical Szegedy walk. With these techniques, a classical simulator in Python called SQUWALS (Szegedy QUantum WALks Simulator) has been built. It is shown that the simulator scales as in both time and memory resources. This package provides some high‐level applications for algorithms based on Szegedy's quantum walk, as for example the quantum PageRank.

Norio Konno, Kaname Matsue, Hideo Mitsuhashi +1 more

We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the way to obtain eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix.

Norie Konno

Salvador Elías Venegas-Andraca