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.
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.
Variational quantum algorithms (VQAs) promise efficient use of near-term quantum computers. However, training VQAs often requires an extensive amount of time and suffers from the barren plateau problem where the magnitude of the gradients vanishes with increasing number of qubits. Here, we show how to optimally train VQAs for learning quantum states. Parameterized quantum circuits can form Gaussian kernels, which we use to derive adaptive learning rates for gradient ascent. We introduce the generalized quantum natural gradient that features stability and optimized movement in parameter space. Both methods together outperform other optimization routines in training VQAs. Our methods also excel at numerically optimizing driving protocols for quantum control problems. The gradients of the VQA do not vanish when the fidelity between the initial state and the state to be learned is bounded from below. We identify a VQA for quantum simulation with such a constraint that thus can be trained free of barren plateaus. Finally, we propose the application of Gaussian kernels for quantum machine learning.
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—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.
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 Variational quantum algorithms (VQAs) are widely applied in the noisy intermediate-scale quantum era and are expected to demonstrate quantum advantage. However, training VQAs faces difficulties, one of which is the so-called barren plateaus (BPs) phenomenon, where gradients of cost functions vanish exponentially with the number of qubits. In this paper, inspired by transfer learning, where knowledge of pre-solved tasks could be further used in a different but related work with training efficiency improved, we report a parameter initialization method to mitigate BP. In the method, a small-sized task is solved with a VQA. Then the ansatz and its optimum parameters are transferred to tasks with larger sizes. Numerical simulations show that this method could mitigate BP and improve training efficiency. A brief discussion on how this method can work well is also provided. This work provides a reference for mitigating BP, and therefore, VQAs could be applied to more practical problems.
Variational Quantum Algorithms (VQAs) have gained prominence as a viable framework for exploiting near-term quantum devices in applications ranging from optimization and chemistry simulation to machine learning. However, the effectiveness of VQAs is often constrained by the so-called barren plateau problem, wherein gradients diminish exponentially as system size or circuit depth increases, thereby hindering training. In this work, we propose a reinforcement learning (RL)-based initialization strategy to alleviate the barren plateau issue by reshaping the initial parameter landscape to avoid regions prone to vanishing gradients. In particular, we explore several RL algorithms (Deterministic Policy Gradient, Soft ActorCritic, and Proximal Policy Optimization, etc.) to generate the circuit parameters (treated as “actions”) that minimize the VQAs cost function before standard gradient-based optimization. By pre-training with RL in this manner, subsequent optimization using methods such as gradient descent or Adam proceeds from a more favorable initial state. Extensive numerical experiments under various noise conditions and tasks consistently demonstrate that the RL-based initialization method significantly enhances both convergence speed and final solution quality. Moreover, comparisons among different RL algorithms highlight that multiple approaches can achieve comparable performance gains, underscoring the flexibility and robustness of our method. These findings shed light on a promising avenue for integrating machine learning techniques into quantum algorithm design, offering insights into how RL-driven parameter initialization can accelerate the scalability and practical deployment of VQAs. Opening up a promising path for the research community in machine learning for quantum, especially barren plateau problems in VQAs.