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We show that the depth of quantum circuits in the realistic architecturewhere a classical controller determines which local interactions to apply onthe kD grid Z^k where k >= 2 is the same (up to a constant factor) as in thestandard model where arbitrary interactions are allowed. This allowsminimum-depth circuits (up to a constant factor) for the nearest-neighborarchitecture to be obtained from minimum-depth circuits in the standardabstract model. Our work therefore justifies the standard assumption thatinteractions can be performed between arbitrary pairs of qubits. In particular,our results imply that Shor's algorithm, controlled operations and fanouts canbe implemented in constant depth, polynomial size and polynomial width in thisarchitecture.  We also present optimal non-adaptive quantum circuits for controlledoperations and fanouts on a kD grid. These circuits have depth Theta(n^(1 /k)), size Theta(n) and width Theta(n). Our lower bound also applies to a moregeneral class of operations.

Optimal Quantum Circuits for Nearest-Neighbor Architectures