Equivalence between exponential concentration in quantum machine learning kernels and barren plateaus in variational algorithms

Sep 17, 2026

Uploaded by Noah

We formalize a rigorous connection between barren plateaus (BPs) in variational quantum algorithms and exponential concentration (EC) of quantum kernels used for machine learning. The results indicate that circuit-engineering strategies proposed to build BP-free quantum circuits can produce useful quantum kernels without exponential concentration. A numerical example demonstrates a provably BP-free quantum neural network constructing kernel matrices for classification datasets of increasing dimensionality while avoiding exponential concentration.

LanguageEnglish

CategoryResearch & Report

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Pages18

Reading Time45 min

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I. INTRODUCTION

II. DEFINITIONS

arXiv :2501 .07433v3 [ quant-ph] 15 Jan 2025 Equivalence between exponential concentration in quantum machine learning kernels and barren plateaus in variational algorithms Pranav Kairon, 1, 2 Jonas Jäger,3, 2 and Roman V. Krems2, 4 1 Department of Physics and Astronomy, University of British Columbia, Vancouver, B. C. , Canada a 2 Stewart Blusson Quantum Matter Institute, Vancouver, B. C. , Canada 3 Department of Computer Science and Institute of Applied Mathematics, University of British Columbia, Vancouver, B. C., Canada 4 Department of Chemistry, University of British Columbia, Vancouver, B. C. , Canada† (Dated: January 16, 2025) We formalize a rigorous connection between barren plateaus (BP) in variational quantum algorithms and exponential concentration of quantum kernels for machine learning. Our results imply that recently proposed strategies to build BP-free quantum circuits can be utilized to construct useful quantum kernels for machine learning. This is illustrated by a numerical example employing a provably BP-free quantum neural network to construct kernel matrices for classification datasets of increasing dimensionality without exponential concentration. I. INTRODUCTION Machine learning with quantum kernels [1–5] and variational quantum algorithms (VQAs) [6] are two distinct paradigms currently explored for computation with noisy quantum devices [7–11] . VQAs are restricted by barren plateaus (BPs) [12 , 13] . BPs are regions in the variational parameter space where gradients of the cost function vanish exponentially with the number of qubits, leading to increasingly narrow gorges of optimal solutions [14] . This implies that an exponentially large number of measurements is required to estimate cost functions for VQAs with BP, as these estimates are otherwise dominated by the statistical measurement error (i.e., shot noise) . The origin of BPs can be traced to the vanishing of inner products between quantum states in exponentially large Hilbert spaces. Exponential vanishing of inner products also hinders applications of quantum computing for estimating kernels for machine learning models [5 , 15] . In kernel methods of machine learning, models are based on kernel matrices [15], which in quantum machine learning (QML), are estimated by measurements on a quantum computer [1–3 , 16] . Refs. [17 , 18] show that both the expected value a Currently at Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA † [rkrems@chem.ubc.ca](rkrems@chem.ubc.ca) 2 and variance of the off-diagonal elements of quantum kernel matrices decrease exponentially as the number of qubits used for building quantum kernels grows. This leads to exponential concentration (EC) of the kernel matrix around a diagonal matrix, implying that quantum kernels reduce to δ-functions in a large Hilbert space and make quantum machine learning impractical. Since BPs preclude the quantum advantage of VQAs [12], a significant focus of recent work has been on how to engineer quantum circuits for VQAs without BPs. It has been shown that BPs can be avoided by restricting the depth of quantum circuits [19], by building equivariance into the quantum ansatz [20], as well as by employing specific initialization strategies for variational optimization [21–25] . Previous work on how to prevent EC of quantum kernels is much more scarce. The proposed approaches include covariant quantum kernels [26], quantum kernel bandwidth methods [27], quantum Fisher kernels [28], and projected quantum kernel methods [29] . All these methods encode inductive bias into the quantum models, thereby restricting learning to a particular subspace of the Hilbert space. A rigorously proven relation between BPs in VQAs and concentration of kernel matrices is required to leverage and transfer results from the BP literature to quantum kernels. It has been assumed that BPs and EC of quantum kernels are different manifestations of the

FAQs

What is the main connection the document establishes?

It connects barren plateaus in variational quantum algorithms with exponential concentration of quantum kernels used in machine learning.

How do barren plateaus affect variational quantum algorithms?

Barren plateaus cause gradients of the cost function to vanish exponentially with the number of qubits, leading to measurement requirements that scale poorly due to shot noise.

What does the paper show about BP-free circuits and quantum kernels?

If an ansatz yields BP-free variational problems with a global observable, the paper shows there exists a corresponding quantum kernel that does not suffer from exponential concentration.

Sep 17, 2026

Uploaded by Noah