Benchmarking indirect quantum control schemes via higher-order quantum operations

Indirect quantum control aims to manipulate a target system through interventions on an auxiliary controller with which it interacts. We formulate finite-step indirect-control protocols using the language of higher-order quantum operations, representing admissible controller manipulations as deterministic superinstruments. This representation separates the uncontrolled but fixed dynamics from the controllable operations and allows the optimal control problem to be written as a semidefinite program for a chosen figure of merit. The resulting optimum gives an operational benchmark by providing the best performance achievable by a finite sequence of control operations, including protocols with classical or quantum feed-forward. More restricted and experimentally motivated control classes, such as independent unitary controls or memoryless quantum channels, can then be compared against this benchmark. We illustrate the framework with a two-step qubit-purification task, in which an initially mixed target qubit is steered toward a pure state through a fixed interaction with a controller qubit. The example shows regimes where simple unitary strategies saturate the higher-order benchmark, as well as regimes where they are provably suboptimal. This approach provides a systematic way to study the value of control resources in indirect quantum-control schemes.