Coherent Projector-Overlap QFNNs with Chebyshev Responses: Expressivity, Approximation, and Coherent Depth
Andrej Sum-Shik
Abstract Variational quantum models combine data encodings, trainable unitaries, and terminal measurements. We introduce a coherent quantum feedforward neural network (QFNN) with an explicit neural-style activation: trainable projector overlaps act as pre-activations, and alternating reflections implement exact Chebyshev responses. For a state |ψ⟩, an orthogonal projector Q, and p = ⟨ψ|Q|ψ⟩, d reflection pairs produce Td(2p − 1). For pure-state inputs, we characterize the complete idealized degree-at-most-D one-layer hypothesis space as ρ ↦→ Tr(Aρ⊗D) with A ∈ Herm(SymD H). This yields its exact dimension, polynomialfidelity kernel, strict degree hierarchy, domain-restricted capacity, interpolation properties, and finite separation from single-copy observable models. We derive Chebyshev approximation bounds, prove Universal Approximation Theorems for scalar and finite-dimensional continuous targets, and obtain norm- and margin-sensitive generalization bounds. The construction extends to fully coherent depth without intermediate measurement. The exact activation law and idealized Universal Approximation Theorem persist at every prescribed finite depth, while a process-feature representation and an exact Chebyshev-composition subarchitecture clarify the expressive and resource roles of recursion. Six-seed simulations on eight-class MNIST, Fashion-MNIST, and KMNIST obtain exact test accuracies of 90.57 ± 0.20%, 74.55 ± 0.51%, and 67.61 ± 0.65%, respectively. Together, these results provide an explicit mathematical framework for neural-style polynomial responses in coherent variational quantum architectures.
•Sep 14, 2026•0 citations