A key challenge in classical machine learning is to mitigate overparameterization by selecting sparse solutions. We translate this concept to the quantum domain, introducing quantum sparsity as a principle based on minimizing quantum information shared across multiple parties. This allows us to address fundamental issues in quantum data processing and convergence issues such as the barren plateau problem in Variational Quantum Algorithm (VQA). We propose a practical implementation of this principle using the topological Entanglement Entropy (TEE) as a cost function regularizer. A non-negative TEE is associated with states with a sparse structure in a suitable basis, while a negative TEE signals untrainable chaos. The regularizer, therefore, guides the optimization along the critical edge of chaos that separates these regimes. We link the TEE to structural complexity by analyzing quantum states encoding functions of tunable smoothness, deriving a quantum Nyquist-Shannon sampling theorem that bounds the resource requirements and error propagation in VQA. Numerically, our TEE regularizer demonstrates significantly improved convergence and precision for complex data encoding and ground-state search tasks. This work establishes quantum sparsity as a design principle for robust and efficient VQAs.
A key challenge in classical machine learning is to mitigate overparameterization by selecting sparse solutions. We translate this concept to the quantum domain, introducing quantum sparsity as a principle based on minimizing quantum information shared across multiple parties. This allows us to address fundamental issues in quantum data processing and convergence issues such as the barren plateau problem in Variational Quantum Algorithm (VQA). We propose a practical implementation of this principle using the topological Entanglement Entropy (TEE) as a cost function regularizer. A non-negative TEE is associated with states with a sparse structure in a suitable basis, while a negative TEE signals untrainable chaos. The regularizer, therefore, guides the optimization along the critical edge of chaos that separates these regimes. We link the TEE to structural complexity by analyzing quantum states encoding functions of tunable smoothness, deriving a quantum Nyquist-Shannon sampling theorem that bounds the resource requirements and error propagation in VQA. Numerically, our TEE regularizer demonstrates significantly improved convergence and precision for complex data encoding and ground-state search tasks. This work establishes quantum sparsity as a design principle for robust and efficient VQAs.
The curse of dimensionality in high-dimensional optimization: It is inherently seen that the dimensionality of feature space presents a fundamental challenge to high-dimensional optimization: as dimensionality increases, feature space becomes computationally inefficient, overfitted and model interpretability declines, thus requiring highly robust sparsity-based solutions. To find a solution to these problems, a new quantum-inspired framework of sparse optimization is proposed in this study, which combines probabilistic Q-bit representation and L1-regularized optimization to efficiently consider large solution spaces but with a sparsity constraint. The suggested method builds on quantum-inspired search algorithms such as initiation by superposition and rotation gate updates to the search to improved exploration and exploitation ratio, leading to a better convergence process. Experimental analyses show that the framework produces better optimization accuracy, indicated by much lower Mean Squared Error (MSE) and convergence rates than classical and met heuristic baselines, with the framework also having a higher level of sparsity in the solution space. These results highlight the effectiveness of the proposed method in handling high-dimensional problems, making it particularly suitable for applications in signal processing, machine learning-based feature selection, and large-scale scientific computing tasks.
Quantum machine learning (QML) models operating on classical data require an encoding step to map information into the Hilbert space. However, this imposes a computational bottleneck absent in classical frameworks, necessitating rigorous theoretical complexity analysis and algorithmic mitigations. We provide theoretical foundations of different encoding methods that prepare quantum states for both simulators and hardware, analyzing normal and sparse high energy physics data. In addition, we propose a quantum data augmentation method based on randomly sampled circuits to improve sample efficiency and mitigate the data scarcity bottleneck. We demonstrate the effectiveness of this method on a generative quantum diffusionlike model to produce synthetic samples of classical images. Our results demonstrate that choosing domain optimal encoding and applying the novel quantum data augmentation leads to faster convergence and improved model performance, offering a highly sample-efficient and scalable approach to encoding and augmenting sparse classical data for QML.
Variational quantum algorithms (VQAs) have established themselves as a central computational paradigm in the Noisy Intermediate-Scale Quantum (NISQ) era. By coupling parameterized quantum circuits (PQCs) with classical optimization, they operate effectively under strict hardware limitations. However, as quantum architectures transition toward early fault-tolerant (EFT) and ultimate fault-tolerant (FT) regimes, the foundational principles and long-term viability of VQAs require systematic reassessment. This review offers an insightful analysis of VQAs and their progression toward the fault-tolerant regime. We deconstruct the core algorithmic framework by examining ansatz design and classical optimization strategies, including cost function formulation, gradient computation, and optimizer selection. Concurrently, we evaluate critical training bottlenecks, notably barren plateaus (BPs), alongside established mitigation strategies. The discussion then explores the EFT phase, detailing how the integration of quantum error mitigation and partial error correction can sustain algorithmic performance. Addressing the FT phase, we analyze the inherent challenges confronting current hybrid VQA models. Furthermore, we synthesize recent VQA applications across diverse domains, including many-body physics, quantum chemistry, machine learning, and mathematical optimization. Ultimately, this review outlines a theoretical roadmap for adapting quantum algorithms to future hardware generations, elucidating how variational principles can be systematically refined to maintain their relevance and efficiency within an error-corrected computational environment.
In recent years, as datasets have grown larger, quantum computing has shown potential in addressing the rapidly increasing costs of data processing due to its superposition and entanglement capabilities. Limited by quantum hardware and the number of controllable qubits, variational quantum algorithms (VQAs) are algorithms suitable for NISQ devices, capable of computation on medium-scale noisy devices. VQAs are hybrid classical-quantum algorithms that use parameterized quantum circuits to prepare parameterized quantum states and employ classical optimizers to minimize objective functions, enabling efficient solutions for complex problems. Classical optimizers reduce the stringent requirements on quantum hardware and can achieve quantum advantage on current devices. This paper systematically elaborates on the core foundational theories of quantum computing and neural networks, with a focus on reviewing the research and application status of VQAs in three main scenarios: machine learning (supervised, unsupervised, and reinforcement learning), solving systems of equations (linear and nonlinear), and network attack detection. It analyzes the current technical bottlenecks in this field and provides prospects for future development, offering a reference for further research on variational quantum algorithms.
This paper introduces a new technique for relaxing constraints in sparse linear systems, leveraging quantum-inspired algorithms to enhance computational efficiency. Sparse linear systems are prevalent in various fields, presenting significant challenges due to their inherent sparsity and computational complexity. Traditional methods often struggle with these issues, necessitating innovative approaches. This work proposes a novel framework that employs quantum-like principles to identify and exploit sparse patterns within the system, ultimately reducing computational cost and improving solution quality. The technique is designed to be effective across a range of sparse linear systems, addressing key challenges associated with these data. The core mechanism involves a dynamic adjustment of constraint bounds, guided by an iterative process that mimics quantum phenomena. This approach offers a promising pathway to overcome limitations of current methods and facilitate more efficient solutions. The paper details the algorithm's implementation, discusses its performance evaluation, and concludes with insights into future research directions.
Abstract Gaussian processes are well-established Bayesian machine learning algorithms with significant merits, despite a strong limitation: lack of scalability. Clever solutions address this issue by inducing sparsity through low-rank approximations, often based on the Nystrom method. Here, we propose a different method to achieve better scalability and higher accuracy using quantum computing, outperforming classical Bayesian neural networks for large datasets significantly. Unlike other approaches to quantum machine learning, the computationally expensive linear algebra operations are not just replaced with their quantum counterparts. Instead, we start from a recent study that proposed a quantum circuit for implementing quantum Gaussian processes and then we use quantum phase estimation to induce a low-rank approximation analogous to that in classical sparse Gaussian processes. We provide evidence through numerical tests, mathematical error bound estimation, and complexity analysis that the method can address the “curse of dimensionality,” where each additional input parameter no longer leads to an exponential growth of the computational cost. This is also demonstrated by applying the algorithm in a practical setting and using it in the data-driven design of a recently proposed metamaterial. The algorithm, however, requires significant quantum computing hardware improvements before quantum advantage can be achieved.