Representation Learning with Quantum Signal Processing
Artificial IntelligenceMachine Learning
Summary
The authors study how quantum models learn to represent data by changing the way similarities between inputs are measured. They use a particular type of quantum model called quantum signal processing (QSP) to exactly analyze how the model's features evolve during training. Their results show that the geometry of the data representation changes in a complex but predictable way, even when the model behaves nearly randomly. They also prove that the learning dynamics converge in a controlled manner for sparse data and identify limits on how fast training can proceed. At higher data densities, training behaves more intricately, going beyond simpler approximations.
Authors
Junqi Wang, Junyu Liu
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.