Variational Quantum Circuits (VQCs) have emerged as a cornerstone of hybrid quantum–classical algorithms designed to harness the computational potential of near-term quantum devices. By combining parameterized quantum gates with classical optimization, VQCs provide a flexible framework for tackling machine learning, chemistry, and optimization problems intractable for classical methods. This review comprehensively overviews VQC design principles, ansatz structures, optimization strategies, and real-world applications. Furthermore, we discuss fundamental challenges such as barren plateaus, the expressibility–trainability trade-off, and current noisy intermediate-scale quantum (NISQ) hardware limitations. Finally, we highlight emerging directions that could enable scalable, noise-resilient, and physically interpretable variational quantum models for future quantum computing applications
Variational Quantum Circuits (VQCs) have emerged as a cornerstone of hybrid quantum–classical algorithms designed to harness the computational potential of near-term quantum devices. By combining parameterized quantum gates with classical optimization, VQCs provide a flexible framework for tackling machine learning, chemistry, and optimization problems intractable for classical methods. This review comprehensively overviews VQC design principles, ansatz structures, optimization strategies, and real-world applications. Furthermore, we discuss fundamental challenges such as barren plateaus, the expressibility–trainability trade-off, and current noisy intermediate-scale quantum (NISQ) hardware limitations. Finally, we highlight emerging directions that could enable scalable, noise-resilient, and physically interpretable variational quantum models for future quantum computing applications
Variational Quantum Algorithms are a major method of harnessing Noisy Intermediate-Scale Quantum devices to solve classically intractable problems in physics, chemistry, optimization, and machine learning. The current review offers a critical overview of VQAs, with special emphasis on their theoretical backgrounds, algorithmic structures, and application performance within realistic models of noise. Several key variants, such as the Variational Quantum Eigensolver, Quantum Approximate Optimization Algorithm, and Quantum Neural Networks, are compared in terms of noise robustness, ansatz archi- tecture, and hybrid quantum-classical methods of optimization. Benchmarks in fields such as quantum chemistry, finance modeling, and particles and nuclei physics reveal the poten- tial of VQAs in achieving practical quantum advantage. Fundamentally limiting factors 1such as barren plateaus, optimization overhead, and hardware constraints are considered in addition to mitigation strategies, including adaptive construction of ansatz, informed training in the presence of noise, and quantum error correction using QVECTOR. The paper is concluded with an overview of the scalability and future of VQAs in the direction of transitioning toward fault-tolerant quantum computing.
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.
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 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 a comprehensive analysis of VQAs and their progression toward the FT 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 (QEM) 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.
This chapter introduces variational quantum algorithms, a class of hybrid methods that leverage parametrized quantum circuits and classical optimization to solve problems in chemistry, optimization, and machine learning. Focusing on the variational quantum eigensolver and the quantum approximate optimization algorithm, we explore their algorithmic structure, theoretical motivation, and practical implementation on near-term devices. We discuss key challenges such as barren plateaus, ansatz design, and measurement overhead, as well as opportunities for adaptive circuits, noise-aware optimization, and emerging applications in quantum machine learning. The chapter lays the foundation for understanding variational classifiers and generative models in subsequent chapters.
Variational quantum algorithms (VQAs) are a leading framework for realizing quantum advantages in the Noisy Intermediate-Scale Quantum (NISQ) era, with applications spanning discrete combinatorial problems and continuous optimization. While the topologies of parameterized quantum circuits (ansatzes) fundamentally govern both expressibility and trainability in continuous landscapes, existing reviews predominantly focus on static algorithmic classifications or discrete settings, leaving the structural evolution and practical limitations of ansatz architectures insufficiently explored. To address this gap, this review presents a systematic analysis of variational ansatz architectures, tracing their progression from static, pre-defined topologies to adaptive growth mechanisms. Beyond traditional gradient-driven and architecture-search paradigms, we evaluate supplementary strategies such as layerwise training and noise-adaptive construction, revealing inherent vulnerabilities such as local minima entrapment and the compilation overhead induced by calibration drift. The mathematical foundations of VQAs are outlined, and representative fixed ansatz architectures, including hardware-efficient, physics-inspired, and problem-specific designs, are characterized within continuous-domain mappings. Intrinsic limitations arising from barren plateaus (BPs) and noise-induced barren plateaus (NIBPs) are analyzed, revealing the fundamental coupling between circuit depth, parameter scaling, and trainability degradation. Furthermore, adaptive construction strategies and recent advances in automated variational quantum architecture search (VQAS) are examined. Through the synthesis of intrinsic limitations (BPs, NIBPs, and hardware-algorithm coupling) and the evaluation of standardized benchmarking protocols, this review rigorously assesses the resource trade-offs of current VQA frameworks. Ultimately, next-generation ansatz design will adopt hardware–software co-design principles grounded in physical constraints, enabling scalable and noise-resilient quantum optimization.
Variational quantum algorithms (VQAs) have emerged as the leading strategy for using these noisy devices to achieve an advantage over their classical counterparts.VQAs use a hybrid quantum-classical framework, in which one employs the quantum computer to prepare a parameterized quantum state and outsource the optimization to a classical subroutine. VQAs have been proposed for applications in various fields, including physics, quantum chemistry, machine learning, and optimization. Nevertheless, many challenges must be addressed before we can use them for practical tasks. In this dissertation, we propose strategies to overcome some challenges in implementing VQAs. We first look at the issue of trainability for VQAs and propose three methods for optimizing them.The first method is based on natural evolutionary strategies (NES) and can alleviate the issue of vanishing gradients for randomly initialized parameterized quantum circuits (PQCs). The second method is based on the notion of information flow in PQCs and uses a mutual information-based distance metric to define paths for stochastic optimization of VQAs. Finally, we develop gradient evaluation procedures for optimizing objectives with the unitary coupled-cluster ansatz. Next, we study the issue of expressibility of PQCs used in variational algorithms.First, using the theory of quantum optimal control we define a notion of reachability of general ans\"atze used in VQAs. Next, using a variational autoencoder (VAE) coupled with a regression model, we try to gain a better understanding of the parameterization of PQCs. Finally, we attempt to remedy the inefficient usage of quantum resources for storing quantum data using evolutionary strategies to design quantum circuits for compressing quantum data. We present numerical evidence that the work presented here can increase trainability, efficiency, and understanding of VQAs.We believe our work is of interest to the quantum computing community as we continue to push the limits of NISQ devices for solving practical problems.