We investigate the amount of communication that must augment classical local hidden variable models in order to simulate the behavior of entangled quantum systems. We consider the scenario where a bipartite measurement is given from a set of possibilities and the goal is to obtain exactly the same correlations that arise when the actual quantum system is measured. We show that, in the case of a single pair of qubits in a Bell state, a constant number of bits of communication is always sufficient—regardless of the number of measurements under consideration. We also show that, in the case of a system of $n$ Bell states, a constant times $2^n$ bits of communication is necessary.

We show how the quantum paradigm can be used to speed up unsupervised learning algorithms. More precisely, we explain how it is possible to accelerate learning algorithms by quantizing some of their subroutines. Quantization refers to the process that partially or totally converts a classical algorithm to its quantum counterpart in order to improve performance. In particular, we give quantized versions of clustering via minimum spanning tree, divisive clustering and k-medians that are faster than their classical analogues. We also describe a distributed version of k-medians that allows the participants to save on the global communication cost of the protocol compared to the classical version. Finally, we design quantum algorithms for the construction of a neighbourhood graph, outlier detection as well as smart initialization of the cluster centres.

Finding a phone number in a directory when one knows the name is easy; much harder is finding the name given the number. On average, a search algorithm will have to scan half the total number of entries. If the database or directory is extremely large, the search may be impossible with classical computers. New possibilities emerge, however, with quantum computing. As Brassard discusses in his Perspective, a new algorithm developed by L. K. Grover enables much faster searching of large databases if one uses a quantum computer.