A Hybrid Quantum Neural Network for Closing-Stock Price Forecasting: Variational Circuits, Temporal Memory, and Attention with a Theoretical Account of Expressivity, Trainability, and Scaling

Abstract

Predicting the closing price of an equity remains stubbornly hard: price series are noisy, non-stationary, and shaped by feedback that breaks the assumptions of linear models. We revisit the problem with a hybrid quantum neural network (HQNN) that places a small variational quantum circuit (VQC) in front of a recurrent–attention forecaster, and — going beyond an empirical demonstration — we supply a theoretical account of why such a model is expressive, trainable, and computationally lean. The architecture works in two coupled stages. A four-qubit VQC first maps an amplitude-encoded feature vector through parameterised / rotations and CNOT–CZ entangling gates, with exact gradients supplied by the parameter-shift rule; the resulting expectation values form a compact quantum latent code. A stacked Long Short-Term Memory (LSTM) network with eight-head self-attention then resolves the temporal structure of that code over a sixty-day look-back window. On the evaluation data the model reaches a root-mean-square error (RMSE) of 1.74 USD, a mean absolute error (MAE) of 1.42 USD, and a coefficient of determination , ahead of a standalone LSTM (RMSE 3.45), a DCQ-GA Elman network (4.82), a blind-quantum-computing QNN (5.13), and a variational QNN–LSTM (3.97). An ablation isolates each component and shows that removing the quantum stage is by far the most damaging single change, cutting accuracy from 98.26% to 93.41%. Our theoretical contribution interprets the VQC as a truncated Fourier model whose accessible spectrum is set by the encoding depth, shows that the shallow four-qubit design sits well clear of the barren-plateau regime, recasts the circuit’s KL penalty as a quantum information bottleneck, and derives the sub-linear training-time scaling — about against the LSTM’s near-quadratic growth — that we observe empirically up to 100,000 samples. Together these results position the HQNN as a compact and theoretically motivated forecaster for non-stationary financial series, with downstream value for risk control and capital allocation.