Abstract:Variational quantum circuits are increasingly studied as continuous-function approximators, but quantum regression remains difficult to train when global losses, finite-shot stochasticity, and circuit-depth growth combine to produce weak or ill-conditioned gradient signals. We study this trainability problem in a controlled hybrid quantum--classical regression design. The central ingredient is a capacity-controlled classical embedding that acts as a learnable geometric preconditioner: it reshapes the input distribution seen by a data-reuploading variational circuit while preserving a low-dimensional quantum bottleneck. We pair this representation design with a curriculum protocol that grows circuit depth progressively and switches from SPSA-based stochastic exploration to Adam-based analytic-gradient fine-tuning. We formalize the mechanism through a local quantum-tangent contraction statement: in the linearized quantum-parameter dynamics, the embedding changes the empirical Gram matrix that controls residual contraction and one-step loss decrease. Across finite-size statevector audits on PDE-informed regression benchmarks and small-data tabular tasks, the Hybrid QNN lowers error relative to Pure QNN baselines under matched quantum-model budgets. Strong classical references remain competitive, and in several cases are better in absolute error; the evidence therefore supports a trainability claim for the hybrid QNN design rather than a claim of classical or hardware quantum advantage.
| Subjects: | Machine Learning (cs.LG); Quantum Physics (quant-ph) |
| Cite as: | arXiv:2601.11942 [cs.LG] |
| (or arXiv:2601.11942v3 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2601.11942 arXiv-issued DOI via DataCite |
Submission history
From: Yangshuai Wang [view email]
[v1]
Sat, 17 Jan 2026 07:32:18 UTC (1,428 KB)
[v2]
Thu, 29 Jan 2026 05:49:33 UTC (1,428 KB)
[v3]
Wed, 13 May 2026 00:50:12 UTC (1,102 KB)