Representation Learning with Quantum Signal Processing

• URL: http://arxiv.org/abs/2608.28828v1
• Date: Fri, 28 Aug 2026 19:57:26 GMT
• Title: Representation Learning with Quantum Signal Processing
• Abstract summary: We establish quantum signal processing as a solvable quantum model of the representation-learning regime.<n>We prove a sparse-data guarantee for the full nonlinear flow without freezing or ensemble-averaging the kernel.<n>These results give a controlled theory of learned quantum data geometry with provable training dynamics beyond the frozen limit.
• Score: 14.534656001951594
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Representation learning begins when training changes the features that define similarity between data. A frozen-kernel model only reweights a fixed geometry. We establish quantum signal processing (QSP) as a solvable quantum model of the representation-learning regime. At arbitrary depth, we compute the exact mean and variance of its quantum neural tangent kernel, revealing an input-dependent angular geometry whose diagonal remains non-self-averaging even when the underlying unitary approaches Haar randomness. We also prove a sparse-data guarantee for the full nonlinear gradient flow without freezing or ensemble-averaging the kernel: the realized dynamics converges to an integrable scalar flow with a time-dependent kernel closure and explicit convergence times. A finite-depth speed limit holds for every data set and trajectory. At higher data density, numerical results show coupled evolution beyond both the scalar and frozen-kernel descriptions. These results give a controlled theory of learned quantum data geometry with provable training dynamics beyond the frozen limit.

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