Is it possible to build a quantum logic circuit that has feedback loop? Using classic logic gates, it is possible to make circuits with loops, i.e. with feedback. I wonder whether that is still the case using quantum gates. I do not mean sequential circuits that are synchronized using clock, but simple logic circuits with feedback loops. Thanks.
 A: Using the output of a gate as input of another, in the quantum case, amounts to applying something like a CNOT operation. Unless you want classical feedback (e.g. the decision of what ist the next gate being conditional to a previous measurement result), in which case the evolution ceases to be unitary (which doesn't mean that it cannot be done: this is common e.g. in one-way quantum computation schemes).
If instead you refer to the output being fed to the input, the question is a bit ill-posed, because when you write a quantum circuit the "wires" don't really refer to the information flowing between spatially separated gates, but rather to the information flow in time. In other words, if you were to pass your information carrier twice through the same physical device implementing some operation, you would still probably write the corresponding circuit in the normal sequential way.
If you are just asking whether it is possible to pass an information carrier through the same "quantum device" more than once, then sure this is possible. For example, if you encode a qubit in the polarisation of a photon, you can have a feedback loop with the photon passing through the same device multiple times using a fiber optic cable (and likely something to implement further operations at each loop).
