Measurement as Training: Exploring BP-Free Learning Method of Neural Networks via Quantum Sampling

The training of neural networks, especially large language models, is dominated by iterative optimization driven by backpropagation (BP), whose compute cost grows rapidly with the model and data size, and whose training process suffers from slow convergence, loss spikes, and even sudden collapse. Quantum computing (QC) has been proposed as a promising direction for breaking this bottleneck, yet existing studies merely insert quantum components into the classical pipeline, so the BP outer loop and its cost remain untouched. In this paper, we propose measurement as training (MaT), a fundamentally different QC-based training scheme that does not require BP at all. MaT loads a coherent prior over discretized network parameters as a quantum superposition, marks the parameters whose full-batch training loss is at most $\epsilon$ through a coherent loss indicator, and applies amplitude amplification to return a certified $\epsilon$-good parameter. Theoretically, we prove that under aligned split priors the prior mass of the $\epsilon$-good set is a constant bounded away from zero, which yields $O(1)$ oracle queries up to preparation cost together with a full-batch $\epsilon$-certificate and a quadratic query advantage over classical accept/reject from the same prior. We then extend the analysis from a single linear layer to deep linear models, ReLU networks, and softmax attention, and the guarantees become progressively more conditional as the architecture grows, with negative results and a certificate gap against mini-batch SGD delimiting the scope of the framework. We hope MaT can inspire more training paradigms beyond classical BP for neural networks.