I will explain how to simulate arbitrary quantum circuits on a distributed quantum computer (DQC), in which the pairs of qubits that are allowed to interact are restricted to the edges of some (connected) graph G. Even for graphs with only a modest number of long-range qubit interactions, such as the hypercube, this simulation is, in fact, efficient. Furthermore, for all graphs, the emulation scheme is very close to being optimal.
Secondly I will present an efficient quantum algorithm for parallel unrestricted memory look-up. As an application, I will show that the space-time trade off for Element Distinctness and Collision Finding can be improved.
Both results arise from applying the ideas of reversible sorting networks to quantum computing.