A Trainable Closed-Form Quantum Neuron: Parameter Efficiency and Trainability at Scale

Variational quantum neural networks (VQNNs) offer attractive theoretical properties: periodic response functions arising from quantum interference provide non-linear expressive power, and resource requirements can scale linearly with input dimension. However, training is costly and difficult to deploy, and most architectures lack an atomic learning primitive whose input–output map admits a closed form for rigorous analysis. We propose the Variational Quantum Neuron (VQN), a single-neuron-level closed-form solution that we prove is mathematically equivalent to its corresponding quantum circuit, and verify this equivalence independently via circuit simulation to within floating-point precision. This closed form enables training entirely without circuit simulation, at negligible computational cost. We show that a single VQN, using only $m+1$ parameters, achieves $100\%$ accuracy on the $m$-bit parity task for $m$ up to $20$, while a parameter-matched perceptron remains at chance level; matching VQN's performance with classical ReLU networks given substantially larger capacity requires over 135 times more parameters at $m=16$, with this gap widening sharply as task scale increases. Our results demonstrate that a rigorously derived, quantum-circuit-equivalent closed form can serve as a parameter-efficient, readily trainable learning primitive at practical scales.