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.
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.
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.
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.
In order to answer the problem of Quantum Phase Estimation Algorithm been not suitable for NISQ devices, and allows one to outperform classical computers, Variational Quantum Algorithms (VQAs) were designed. Our subject of interest is the so-called Variational Quantum Eigensolver (VQE) algorithm and was originally designed to simulate electronic structures and to compute the ground state of a given molecule. VQE is made of two main components : an ansatz and a classical optimizer. The ansatz runs on the quantum device and aims to simulate the wavefunction, the parameters of the ansatz will be optimized until the expectation value is minimum. The very first ansatz that has originally been used is called UCCSD and it is based on Coupled Cluster Theory. The main issue considering UCCSD is the large amount of parameters to optimize and this leads us to the introduction of Adaptive Derivative-Assembled Pseudo-Trotter ansatz VQE (ADAPT-VQE) which determines a quasi-optimal ansatz with a minimal number of parameters. The key point of ADAPT-VQE is to grow the ansatz at every step, by adding operators chosen from a pre-determined pool of operators one-at-a-time, assuring that the maximal amount of correlation energy is recovered at each step. There exists different kind of ADAPT-VQE depending on the starting pool of operators as the fermionic-ADAPT, the qubit-ADAPT or even the qubit excitation based (QEB). Our goal is to implement the different types of ADAPT-VQE mentioned before. After a quick review of the theoretical background under all of these concepts, we will implement each algorithm using quiskit. We will also compare all of these algorithms on different criterions such as the number of parameters, the accuracy or the number of CNOT gate used on H2 and LiH molecules. Then we will have a small discussion about the results we obtained.
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.