Raqueline A. M. Santos

The staggered quantum walk model

There are at least three models of discrete-time quantum walks (QWs) on graphs currently under active development. In this work, we focus on the equivalence of two of them, known as Szegedy’s and staggered QWs. We give a formal definition of the staggered model and discuss generalized versions for searching marked vertices. Using this formal definition, we prove that any instance of Szegedy’s model is equivalent to an instance of the staggered model. On the other hand, we show that there are instances of the staggered model that cannot be cast into Szegedy’s framework. Our analysis also works when there are marked vertices. We show that Szegedy’s spatial search algorithms can be converted into search algorithms in staggered QWs. We take advantage of the similarity of those models to define the quantum hitting time in the staggered model and to describe a method to calculate the eigenvalues and eigenvectors of the evolution operator of staggered QWs.

Adjacent Vertices Can Be Hard to Find by Quantum Walks

Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element only. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs \(\varOmega (N)\) steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjacent marked vertices. The analysis is done for the two-dimensional grid and hypercube, and then is generalized for any graph.

Exceptional quantum walk search on the cycle

Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk’s hitting time. In some exceptional cases, however, the system only evolves by sign flips, staying in a uniform probability distribution for all time. We prove that the one-dimensional periodic lattice or cycle with any arrangement of marked vertices is such an exceptional configuration. Using this discovery, we construct a search problem where the quantum walk’s random sampling yields an arbitrary speedup in query complexity over the classical random walk’s hitting time. In this context, however, the mixing time to prepare the initial uniform state is a more suitable comparison than the hitting time, and then, the speedup is roughly quadratic.

Szegedy's quantum walk with queries

When searching for a marked vertex in a graph, Szegedy’s usual search operator is defined by using the transition probability matrix of the random walk with absorbing barriers at the marked vertices. Instead of using this operator, we analyze searching with Szegedy’s quantum walk by using reflections around the marked vertices, that is, the standard form of quantum query. We show we can boost the probability to 1 of finding a marked vertex in the complete graph. Numerical simulations suggest that the success probability can be improved for other graphs, like the two-dimensional grid. We also prove that, for a certain class of graphs, we can express Szegedy’s search operator, obtained from the absorbing walk, using the standard query model.

Moments of coinless quantum walks on lattices

The properties of the coinless quantum-walk model have not been as thoroughly analyzed as those of the coined model. Both evolve in discrete time steps, but the former uses a smaller Hilbert space, which is spanned merely by the site basis. Besides, the evolution operator can be obtained using a process of lattice tessellation, which is very appealing. The moments of the probability distribution play an important role in the context of quantum walks. The ballistic behavior of the mean square displacement indicates that quantum-walk-based algorithms are faster than random-walk-based ones. In this paper, we obtain analytical expressions for the moments of the coinless model on d-dimensional lattices by employing the methods of Fourier transforms and generating functions. The mean square displacement for large times is explicitly calculated for the one- and two-dimensional lattices, and using optimization methods, the parameter values that give the largest spread are calculated and compared with the equivalent ones of the coined model. Although we have employed asymptotic methods, our approximations are accurate even for small numbers of time steps.

Decoherence in quantum Markov chains

It is known that under some assumptions, the hitting time in quantum Markov chains is quadratically smaller than the hitting time in classical Markov chains. This work extends this result for decoherent quantum Markov chains. The decoherence is introduced using a percolation-like graph model, which allows us to define a decoherent quantum hitting time and to establish a decoherent-intensity range for which the decoherent quantum hitting time is quadratically smaller than the classical hitting time. The detection problem under decoherence is also solved with quadratic speedup in this range.