Beyond causally ordered quantum computers

We introduce a task—the classical switch of black boxes—that is easily implementable in a quantum laboratory, and we prove that it cannot be translated into a quantum circuit with fixed causal structure. The task involves assembling a circuit conditionally on the value of a classical bit. We also introduce a generalization of the same task—the quantum switch—where the control is performed by a quantum bit, that can thus become entangled with the circuit structure, and propose a scheme for its implementation.