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.
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.
Abstract In recent years, variational quantum circuits (VQCs) have been widely explored to advance quantum circuits against classic models on various domains, such as quantum chemistry and quantum machine learning. Similar to classic machine-learning models, VQCs can be trained through various optimization approaches, such as gradient-based or gradient-free methods. However, when employing gradient-based methods, the gradient variance of VQCs may dramatically vanish as the number of qubits or layers increases. This issue, a.k.a. barren plateaus (BPs), seriously hinders the scaling of VQCs on large datasets. To mitigate the barren plateaus, extensive efforts have been devoted to tackling this issue through diverse strategies. In this survey, we conduct a systematic literature review of recent works from both investigation and mitigation perspectives. Furthermore, we propose a new taxonomy to categorize most existing mitigation strategies into five groups and introduce them in detail. Also, we compare the concurrent survey papers about BPs. Finally, we provide insightful discussion on future directions for BPs.
While quantum computers can give exponential advantages over classical computers, the difficulties in constructing quantum computers have led to small and erroneous devices which are unable to realize the theoretical advantage for useful problems. However, there are some results which show that current quantum computers can outperform classical devices on some toy problems. This has led to the search of an algorithm that could give exponential advantages on small and erroneous near-term devices. Variational quantum algorithms (VQA) and quantum kernel methods (QKM) have been considered as potential candidates for near-term devices. Unfortunately, these methods often suffer from a barren plateau (BP) problem, where the values, which we evaluate using quantum computers, get exponentially smaller as qubits are added. This is an issue, because quantum mechanics limit us to only have access to samples, which we use to approximate the values of interest. When approximating a value with samples, we need to take more samples when we want to increase accuracy. As we need to keep the error, that comes from sampling, smaller than the value of interest, the BP problem leads to the need to take exponentially many samples. This diminishes the advantage that we gain from using quantum computers, because taking exponentially many shots leads to exponential scaling. In this work, we go over research about the BP problem and how we can potentially get quantum advantage while avoiding it, which is not easy because the avoidance of the BP problem often leads to us being able to solve the problem in polynomial time without needing a quantum computer. We will show that the avoidance of the BP problem and polynomial classical computability greatly alters the way that VQAs and QKMs operate. Because of this we will also look at some results about VQAs and QKMs being viable for near-term devices. We find out that these results do not lead to current approaches of VQAs and QKMs being inherently suitable for near-term devices.
Quantum kernel methods (QKMs) have emerged as a prominent framework for supervised quantum machine learning. Unlike variational quantum algorithms, which rely on gradient-based optimisation and may suffer from issues such as barren plateaus, non-variational QKMs employ fixed quantum feature maps, with model selection performed classically via convex optimisation and cross-validation. This separation of quantum feature embedding from classical training ensures stable optimisation while leveraging quantum circuits to encode data in high-dimensional Hilbert spaces. In this review, we provide a thorough analysis of non-variational supervised QKMs, covering their foundations in classical kernel theory, constructions of fidelity and projected quantum kernels, and methods for their estimation in practice. We examine frameworks for assessing quantum advantage, including generalisation bounds and necessary conditions for separation from classical models, and analyse key challenges such as exponential concentration, dequantisation via tensor-network methods, and the spectral properties of kernel integral operators. We further discuss structured problem classes that may enable advantage, and synthesise insights from comparative and hardware studies. Overall, this review aims to clarify the regimes in which QKMs may offer genuine advantages, and to delineate the conceptual, methodological, and technical obstacles that must be overcome for practical quantum-enhanced learning.
Abstract—Quantum Machine Learning (QML) operating on Noisy Intermediate-Scale Quantum (NISQ) hardware represents an emerging computational paradigm to address high-dimensional optimization problems that challenge classical Von Neumann supercomputing architectures. By mapping complex feature vectors into exponentially large 2^N Hilbert state spaces via Parameterized Quantum Circuits (PQCs) and Variational Quantum Algorithms (VQAs), QML targets computational advantages in combinatorial optimization, high-dimensional classification, and quantum-enhanced molecular modeling. However, the theoretical promise of NISQ-era variational computing is severely constrained by three fundamental bottlenecks: (1) Barren Plateaus, characterized by the exponential concentration of measure and vanishing variance of cost function gradients; (2) Hardware Decoherece Noise and readout error rates that corrupt quantum state fidelity; and (3) Classical-to-Quantum State Preparation bottlenecks that can erase theoretical computational speedups. This paper delivers an exhaustive, mathematically formal technical investigation and empirical benchmark of QML architectures for high-dimensional optimization. We formalize the Haar-distributed geometric mechanics of barren plateaus under 2-design unitary assumptions, evaluate barren plateau mitigation through shallow local cost operators and layer-by-layer parameter initialization, and contrast Variational Quantum Eigensolvers (VQE) with Quantum Kernel Support Vector Machines (Q-SVM). Furthermore, we construct an empirical benchmarking testbed executing across both state-vector simulators and superconducting quantum backends (IBM Quantum Heron / Eagle architectures), evaluating 127-qubit Hamiltonian optimization and high-dimensional financial portfolio allocation. Our findings demonstrate that while global cost functions suffer exponential gradient decay Var[grad] ~ O(2^-N) across circuit depth d >= 4, local Hamiltonian decomposition maintains trainable variance, achieving a 98.2% ground-state fidelity under zero-noise extrapolation (ZNE) error mitigation. Finally, we formulate open challenges in quantum error mitigation scaling, non-perturbative barren plateau escapes, and quantum data loading efficiency. Index Terms—Quantum Machine Learning (QML), Parameterized Quantum Circuits (PQCs), Variational Quantum Eigensolver (VQE), Barren Plateaus, Quantum Kernel Methods, NISQ Computing, Error Mitigation, Hilbert Space.
Abstract We define laziness to describe a large suppression of variational parameter updates for neural networks, classical or quantum. In the quantum case, the suppression is exponential in the number of qubits for randomized variational quantum circuits. We discuss the difference between laziness and barren plateau in quantum machine learning created by quantum physicists in McClean et al (2018 Nat. Commun. 9 1–6) for the flatness of the loss function landscape during gradient descent. We address a novel theoretical understanding of those two phenomena in light of the theory of neural tangent kernels. For noiseless quantum circuits, without the measurement noise, the loss function landscape is complicated in the overparametrized regime with a large number of trainable variational angles. Instead, around a random starting point in optimization, there are large numbers of local minima that are good enough and could minimize the mean square loss function, where we still have quantum laziness, but we do not have barren plateaus. However, the complicated landscape is not visible within a limited number of iterations, and low precision in quantum control and quantum sensing. Moreover, we look at the effect of noises during optimization by assuming intuitive noise models, and show that variational quantum algorithms are noise-resilient in the overparametrization regime. Our work precisely reformulates the quantum barren plateau statement towards a precision statement and justifies the statement in certain noise models, injects new hope toward near-term variational quantum algorithms, and provides theoretical connections toward classical machine learning. Our paper provides conceptual perspectives about quantum barren plateaus, together with discussions about the gradient descent dynamics in Liu et al (2023 Phys. Rev. Lett. 130 150601).
Quantum kernel methods promise enhanced expressivity for learning structured data, but their usefulness has been limited by kernel concentration and barren plateaus. Both effects are mathematically equivalent and suppress trainability. We analytically prove that covariant quantum kernels tailored to datasets with group symmetries avoid exponential concentration, ensuring stable variance and guaranteed trainability independent of system size. Our results extend beyond prior two-coset constructions to arbitrary coset families, broadening the scope of problems where quantum kernels can achieve advantage. We further derive explicit bounds under coherent noise models - including unitary errors in fiducial state preparation, imperfect unitary representations, and perturbations in group element selection - and show through numerical simulations that the kernel variance remains finite and robust, even under substantial noise. These findings establish a family of quantum learning models that are simultaneously trainable, resilient to coherent noise, and linked to classically hard problems, positioning group-symmetric quantum kernels as a promising foundation for near-term and scalable quantum machine learning.