Representation Learning with Quantum Signal Processing

Junqi Wang; Junyu Liu

quant-phAug 28, 2026

Representation Learning with Quantum Signal Processing

Authors: Junqi Wang, Junyu Liu

Organizations: Department of Computer Science, University of Pittsburgh, Pittsburgh, PA 15260, USA

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.

Figures & tables

Figure 1: Non-self-averaging initialization and angular locality of the mean QNTK. (a) Distribution of the normalized diagonal QNTK

z=3K_{d}(x,x)/M

x=0.6

. The distributions approach the

\operatorname{Beta}(3/2,1/2)

law (black). The finite-depth weight above

z=1

reflects the residual anisotropy of the frame tensor. (b) Exact coefficient of variation

\sqrt{\operatorname{Var}K_{d}(x,x)}/\mathbb{E}K_{d}(x,x)

for three fixed interior inputs. All three curves approach the nonzero asymptotic value

1/3

(dashed). (c) Along

\alpha+\beta=\pi/2

, the normalized mean kernel

4\mathbb{E}K_{d}(x,y)/M=\cos^{d}(\alpha-\beta)

approaches

e^{-u^{2}/2}

under the rescaling

u=\sqrt{d}\,|\alpha-\beta|

. The inset shows the corresponding narrowing in the unscaled angular coordinate.

Figure 2: Learning with isolated and overlapping quantum data neighborhoods. QSP loss for a representative initialization (blue) and the interquartile band over

30

initializations, compared with the realization-matched scalar flow (red dotted), the realized kernel frozen at

K(0)

(yellow dashed), and the continuous-flow speed-limit envelope

e^{-2Mt}

(gray). (a) For

(d,n,w)=(1024,4,0.125)

, both reduced descriptions track the QSP descent, with median ratios

t_{\mathrm{sc}}/t_{\mathrm{QSP}}=0.97

and

t_{K(0)}/t_{\mathrm{QSP}}=1.04

. (b) For

(d,n,w)=(128,64,5.66)

, overlapping tangent neighborhoods drive the QSP trajectory away from both predictions. The numerical protocol is given in the Supplemental Material.

Representation Learning

Representation learning develops data transformations that expose useful structure for prediction, transfer, generation, and control.

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Towards unsupervised representation learning for quantum data: quantum models with inference and generation

Robin Lorenz, Eric Brunner, Marcello Benedetti

Quantinuum, London, United Kingdom

Latent Variable Models Quantum Generative Modeling Quantum Machine Learning Unsupervised Representation Learning

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With quantum sensors, simulators and networks emerging, a future of quantum technology may produce quantum states as data---that is, coherently rather than as classical measurement records---thus motivating the study of suitable quantum generalisations of modern machine learning, including the automated, unsupervised extraction of useful representations. Two ingredients are central to the latter: inference, mapping observations to latent representations, and generation, mapping latent states back to synthetic data. Both are related to each other and to joint distributions for training models by the chain-rule of classical probability theory. The fact that quantum states however lack such universal, standard factorisation property thus poses a challenge. Here we develop a conceptual and mathematical framework for unsupervised representation learning from quantum data. Models are joint quantum states over visible and latent systems; state-over-time maps provide a notion of factorisation into a marginal state and inference (generation) channel; models with inference (generation) are ambiguous states---states for which such factorisation obtains---subject to a further consistency condition on extended inference maps as data extension. These stipulations are restrictive: we show that non-trivial models must feature non-linear such maps to the extended space. For three representative state-over-time maps, we completely characterise the ambiguous states, uncovering a hierarchy tied to the positive-partial-transpose (PPT) criterion from entanglement theory. Notably, the Leifer-Spekkens construction supports inference and generation exactly for model classes of PPT states, thus allowing genuinely quantum visible-latent correlations. We also formulate quantum counterparts of exact and approximate inference training, explore weaker notions of data extension and sketch a future research programme.

Abstract

Representation Learning

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