Quantum walks are widely and successfully used to model diverse physical processes. This leads to computation of the models, to explore their properties. Quantum walks have also been shown to be universal for quantum computing. This is a more subtle result than is often appreciated, since it applies to computations run on qubit-based quantum computers in the single walker case, and physical quantum walkers in the multi-walker case (quantum cellular automata). Nonetheless, quantum walks are powerful tools for quantum computing when correctly applied. I will explain the relationship between quantum walks as models and quantum walks as computational tools, and give some examples of their application in both contexts.[-]

Continuous quantum walks are of great interest in quantum computing and, over the last decade, my group has been studying this topic intensively. As graph theorists, one of our main goals has been to get a better understanding of the relation between the properties of a walk and the properties of the underlying graph. We have had both successes and failures. The failures lead to a number of interesting open questions, which I will present in my talk.[-]

The coined quantum walk is a discretization of the Dirac equation of relativistic quantum mechanics, and it is a useful model for developing quantum algorithms. For example, many quantum spatial search algorithms are based on coined quantum walks. In this talk, we explore a lazy version of the coined quantum walk, called a lackadaisical quantum walk, which uses a weighted self-loop at each vertex so that the walker has some amplitude of staying put. We show that lackadaisical quantum walks can solve the spatial search problem more quickly than a regular, coined quantum walk for avariety of graphs, suggesting that it is a useful tool for improving quantum algorithms.[-]

Photonic quantum systems, which comprise multiple optical modes, have become an established platform for the experimental implementation of quantum walks. However, the implementation of large systems with many modes, this means for many step operations, a high and dynamic control of many different coin operations and variable graph structures typically poses a considerable challenge.
Time-multiplexed quantum walks are a versatile tool for the implementation of a highly flexible simulation platform with dynamic control of the different graph structures and propagation properties. Our time-multiplexing techniques is based on a loop geometry ensures a extremely high homogeneity of the quantum walk system, which results in highly reliable walk statistics. By introducing optical modulators we can control the dynamics of the photonic walks as well as input and output couplings of the states at different stages during the evolution of the walk.
Here we present our recent results on our time-multiplexed quantum walk experiments.[-]