Learning When and How to Use Measurement Feedback in Dynamic Quantum Circuits

Mid-circuit measurements let a circuit steer later operations with information it has just created. Yet the feedforward rule that maps outcomes to gates is still designed by hand, one bespoke rule per protocol, while $k$ measured qubits create $2^k$ outcome contingencies. External controllers move it outside the circuit, leaving open when conditioning helps. We make the feedforward rule a trainable part of the circuit. Exact branch decomposition enumerates the measurement outcomes inside the optimization loop, so the rule angles and every circuit parameter descend one balanced branch objective together, with no sampling estimator. It also yields analytic floors that bind any parameter-matched circuit ignoring the outcome, a spectral energy gap and a chance-level classification bound, which certify before training whether the outcome can change the attainable objective. Across energy and classification tasks the learned rule crosses those floors, clears every unconditional variant, and at matched budgets and restarts ties or exceeds the best baseline on $26$ of $27$ grid cells. Under a hardware-informed deployment model its net value is task dependent, repaying the feedback latency on classification while energy recovery crosses to a static preference. A mid-circuit measurement is thus a computational resource whose value can be certified and whose use can be learned.