Universal quantum computation via quantum controlled classical operations.

It is well known that a universal set of gates for classical computation augmented with the Hadamard gate results in universal quantum computing. While this requires the addition of a genuine quantum element to the set of passive classical gates, here we ask the following: can the same result be attained by adding a quantum control unit while keeping the circuit itself completely classical? In other words, can we get universal quantum computation by coherently controlling classical operations? In this work we provide an affirmative answer to this question, by considering a computational model that consists of $2n$ target bits together with a set of classical gates, controlled by log$(2n+1)$ ancillary qubits. We show that this model is equivalent to a quantum computer operating on $n$ qubits. Furthermore, we show that even a primitive computer that is capable of implementing only SWAP gates, can be lifted to universal quantum computing, if aided with an appropriate quantum control of logarithmic size. Our results thus exemplify the information processing power brought forth by the quantum control system.