The variational quantum algorithm (VQA) is designed to simulate an $$N$$ -particle wave function of an electronic system of molecules using programmable quantum computers. This brief review, presented at the International Workshop on Mathematical Problems of Quantum Information Technologies (May 2024, Dubna), traces the evolution of the algorithm from a version originating from the unitary coupled cluster method, UCCSD, in computational quantum chemistry to a new version, ADAPT-VQA, adapted for practical implementation on noisy intermediate-scale quantum (NISQ) computing devices.
The variational quantum algorithm (VQA) is designed to simulate an N -particle wave function of an electronic system of molecules using programmable quantum computers. This brief review, presented at the International Workshop on Mathematical Problems of Quantum Information Technologies (May 2024, Dubna), traces the evolution of the algorithm from a version originating from the unitary coupled cluster method, UCCSD, in computational quantum chemistry to a new version, ADAPT-VQA, adapted for practical implementation on noisy intermediate-scale quantum (NISQ) computing devices.
The variational quantum algorithm, VQA, is designed to simulate the N-particle wave function of an electron system of molecules using programmable quantum computers. This brief review, presented at the International Workshop on Mathematical Problems of Quantum Information Technologies (May 2024, Dubna), traces the evolution of the algorithm from a version originating from the unitarily coupled cluster method, UCCSD, in computational quantum chemistry to a new version, ADAPT-VQA, adapted for practical implementation on noisy intermediate-scale quantum, NISQ, computing devices.
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 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) are the leading candidates for practical applications of noisy intermediate-scale quantum (NISQ) devices. Recently, pulse-based variational quantum algorithms (p-VQAs) have emerged as a compelling alternative to traditional gate-based VQAs. By optimizing continuous control fields directly on hardware-native Hamiltonians, pulse-based methods are easy to implement and can significantly reduce quantum resource overhead, thereby enhancing the overall performance of quantum algorithms on NISQ devices. In this review, we present a comprehensive survey over recent advances in p-VQAs, covering theory, algorithms, and hardware deployments across various quantum computing platforms. We also review applications of p-VQAs to variational quantum eigensolver in quantum chemistry, quantum approximate optimization algorithm for combinatorial optimization, and quantum machine learning tasks.
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.
Abstract This practical guide explores several instances and extensions of variational quantum algorithms (VQAs), based on IBM Q learning resources. The exercises were completed by the author on May 9, 2024, using IBM Q cloud resources. This lesson introduces advanced applications of VQAs and demonstrates how different algorithmic designs can be adapted for specific computational tasks. Participants study multiple quantum variational algorithms, including: Variational Quantum Eigensolver (VQE) Subspace-Search VQE (SSVQE) Variational Quantum Deflation (VQD) Quantum Subspace Regression (QSR) These algorithms highlight various design concepts, such as the use of weighted cost functions, penalty terms, resampling strategies, and mitigation of undersampling effects. By exploring these approaches, participants gain insight into customizing variational algorithms for specific problems in quantum chemistry, optimization, and machine learning. The Variational Quantum Eigensolver (VQE) is emphasized as a foundational algorithm that serves as a template for many other variational techniques. Exercises focus on constructing parameterized ansätze, defining appropriate cost functions, and integrating extensions that improve convergence, accuracy, or computational efficiency. Participants are encouraged to experiment with these designs and share results within the quantum computing community. Hands-on exercises in Qiskit provide practical experience in applying these advanced algorithms on noisy intermediate-scale quantum (NISQ) devices, bridging theoretical concepts with real-world quantum computation.