How Much Fault Tolerance Does a QNN Need? Rethinking Precision Through Convergence Analysis

The transition from NISQ to fault tolerance changes how a quantum neural network (QNN) is executed. Trainable rotations are no longer applied directly, but synthesized into sequences of logical gates at a chosen approximation error. Convergence behavior under such error is barely understood. As a result, the error QNN training can tolerate to converge, and the quantum resources that tolerance demands, are set by convention rather than by analysis. We derive a convergence-based theory of synthesis error that gives a principled way to set precision during fault-tolerant QNN training. From a biased-gradient convergence bound, we obtain the gradient distortion an optimizer can tolerate at a given convergence state. We translate this into closed-form tolerance laws for synthesis precision and magic-state fidelity. The laws cover both deterministic synthesis and mixed synthesis, which averages over compilations of the same rotation. We then prove a T-cost theorem for reaching a target stationarity under each strategy, with a matching lower bound for generic rotation angles. The two strategies affect the gradient differently. Deterministic synthesis introduces a first-order bias that persists during training, so the allowable error must shrink as convergence approaches. Mixed synthesis cancels this bias in expectation, leaving only a second-order residual: the error becomes gradient variance, and the same target is reached with fewer T-gates. Experiments with explicit Clifford+T synthesis and end-to-end training on three datasets confirm these predictions: mixed synthesis roughly halves the T cost per circuit relative to convergence-scheduled deterministic synthesis without loss of accuracy, and scheduling magic-state fidelity reduces distillation work by up to $28\%$. These results show that the fault-tolerant precision of QNN training can be derived from learning rather than fixed by convention.