Quantum circuits cannot control unknown operations

Abstract

One of the essential building blocks of classical computer programs is the\n"if" clause, which executes a subroutine depending on the value of a control\nvariable. Similarly, several quantum algorithms rely on applying a unitary\noperation conditioned on the state of a control system. Here we show that this\ncontrol cannot be performed by a quantum circuit if the unitary is completely\nunknown. However, this no-go theorem does not prevent implementing quantum\ncontrol of unknown unitaries in practice, as any physical implementation of an\nunknown unitary provides additional information that makes the control\npossible. We then argue that one should extend the quantum circuit formalism to\ncapture this possibility in a straightforward way. This is done by allowing\nunknown unitaries to be applied to subspaces and not only to subsystems.\n