We analyze the decoherence in quantum walks in two-dimensional lattices generated by broken-link-type noise. In this type of decoherence, the links of the lattice are randomly broken with some given constant probability. We obtain the evolution equation for a quantum walker moving on two-dimensional (2D) lattices subject to this noise, and we point out how to generalize for lattices in more dimensions. In the nonsymmetric case, when the probability of breaking links in one direction is different from the probability in the perpendicular direction, we have obtained a nontrivial result. If one fixes the link-breaking probability in one direction, and gradually increases the probability in the other direction from 0 to 1, the decoherence initially increases until it reaches a maximum value, and then it decreases. This means that, in some cases, one can increase the noise level and still obtain more coherence. Physically, this can be explained as a transition from a decoherent 2D walk to a coherent 1D walk.

The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time $O\left(n\text{log}n\right)$. We show that the mean probability distribution of a discrete-time quantum walk on a hypercube mixes to a (generally nonuniform) distribution $\pi \left(x\right)$ in time $O\left(n\right)$, and the stationary distribution is determined by the initial state of the walk. An explicit expression for $\pi \left(x\right)$ is derived for the particular case of a symmetric walk. These results are consistent with those obtained previously for a continuous-time quantum walk. The effect of decoherence due to randomly breaking links between connected sites in the hypercube is also considered. We find that the probability distribution mixes to the uniform distribution as expected. However, the mixing time has a minimum at a critical decoherence rate $p\approx 0.1$. A similar effect was previously reported for a quantum walk on an $N$-cycle with decoherence from repeated measurements of position. A controlled amount of decoherence helps in obtaining—and preserving—a uniform distribution over the $2^n$ sites of the hypercube in the shortest possible time.

The spatial search problem consists of minimising the number of steps required to find a given site in a network under the restriction that only oracle queries or translations to neighbouring sites are allowed. We propose a quantum algorithm for the spatial search problem on a honeycomb lattice withNsites and torus-like boundary conditions. The search algorithm is based on a modified quantum walk on an hexagonal lattice and the general framework proposed by Ambainis, Kempe and Rivosh (Ambainiset al. 2005) is employed to show that the time complexity of this quantum search algorithm is $O(\sqrt{N \log N})$ .

The quantum walk is a powerful tool to develop quantum algorithms, which usually are based on searching for a vertex in a graph with multiple marked vertices, with Ambainis's quantum algorithm for solving the element distinctness problem being the most shining example. In this work, we address the problem of calculating analytical expressions of the time complexity of finding a marked vertex using quantum-walk-based search algorithms with multiple marked vertices on arbitrary graphs, extending previous analytical methods based on Szegedy's quantum walk, which can be applied only to bipartite graphs. Two examples based on the coined quantum walk on two-dimensional lattices and hypercubes show the details of our method.