Abstract Variational quantum‐classical hybrid algorithms are emerging as important tools for simulating quantum chemistry with quantum devices. These algorithms can be applied to evaluate various molecular properties, including potential energy surfaces. Here in, recent progresses on the development of the so‐called variational quantum eigensolver (VQE) are surveyed. The eigensolver aims at reducing the consumption of quantum resources as much as possible. The key feature of VQE is that variation quantum states are optimized by a feedback process, where the measurement of the Hamiltonian is implemented term by term. This approach avoids the need of encoding all of the information about the molecular Hamiltonian in a quantum circuit. The VQE method is also compatible with classical methods in quantum chemistry, such as unitary coupled‐cluster ansatz. Furthermore, basic elements of VQE are covered, such as qubit encoding, mapping rules of the fermionic operators, ansatz preparation, together with several techniques for improving the performance, including constraining, and error mitigation.
Abstract Variational quantum‐classical hybrid algorithms are emerging as important tools for simulating quantum chemistry with quantum devices. These algorithms can be applied to evaluate various molecular properties, including potential energy surfaces. Here in, recent progresses on the development of the so‐called variational quantum eigensolver (VQE) are surveyed. The eigensolver aims at reducing the consumption of quantum resources as much as possible. The key feature of VQE is that variation quantum states are optimized by a feedback process, where the measurement of the Hamiltonian is implemented term by term. This approach avoids the need of encoding all of the information about the molecular Hamiltonian in a quantum circuit. The VQE method is also compatible with classical methods in quantum chemistry, such as unitary coupled‐cluster ansatz. Furthermore, basic elements of VQE are covered, such as qubit encoding, mapping rules of the fermionic operators, ansatz preparation, together with several techniques for improving the performance, including constraining, and error mitigation.
Abstract The variational quantum eigensolver (VQE) is a method that uses a hybrid quantum-classical computational approach to find eigenvalues of a Hamiltonian. VQE has been proposed as an alternative to fully quantum algorithms such as quantum phase estimation (QPE) because fully quantum algorithms require quantum hardware that will not be accessible in the near future. VQE has been successfully applied to solve the electronic Schrödinger equation for a variety of small molecules. However, the scalability of this method is limited by two factors: the complexity of the quantum circuits and the complexity of the classical optimization problem. Both of these factors are affected by the choice of the variational ansatz used to represent the trial wave function. Hence, the construction of an efficient ansatz is an active area of research. Put another way, modern quantum computers are not capable of executing deep quantum circuits produced by using currently available ansatzes for problems that map onto more than several qubits. In this review, we present recent developments in the field of designing efficient ansatzes that fall into two categories—chemistry–inspired and hardware–efficient—that produce quantum circuits that are easier to run on modern hardware. We discuss the shortfalls of ansatzes originally formulated for VQE simulations, how they are addressed in more sophisticated methods, and the potential ways for further improvements.
New approaches into computational quantum chemistry can be developed through the use of quantum computing. While universal, fault-tolerant quantum computers are still not available, and we want to utilize today's noisy quantum processors. One of their flagship applications is the variational quantum eigensolver (VQE)-an algorithm for calculating the minimum energy of a physical Hamiltonian. In this study, we investigate how various types of errors affect the VQE and how to efficiently use the available resources to produce precise computational results. We utilize a simulator of a noisy quantum device, an exact statevector simulator, and physical quantum hardware to study the VQE algorithm for molecular hydrogen. We find that the optimal method of running the hybrid classical-quantum optimization is to: (i) allow some noise in intermediate energy evaluations, using fewer shots per step and fewer optimization iterations, but ensure a high final readout precision; (ii) emphasize efficient problem encoding and ansatz parametrization; and (iii) run all experiments within a short time-frame, avoiding parameter drift with time. Nevertheless, current publicly available quantum resources are still very noisy and scarce/expensive, and even when using them efficiently, it is quite difficult to perform trustworthy calculations of molecular energies.
The Variational Quantum Eigensolver (VQE) has emerged as a transformative approach in quantum computing, providing a practical means to tackle complex quantum systems. This chapter delves into the essential components and strategies for enhancing VQE performance, focusing on key areas such as hybrid optimization techniques, quantum circuit design, and measurement efficiency. By integrating classical and quantum resources, hybrid optimization methods significantly improve convergence rates while reducing the computational burden on quantum devices. The chapter explores the design of quantum circuits, emphasizing the importance of selecting appropriate variational ansätze to achieve optimal results. Challenges related to Hamiltonian simulation and measurement are critically examined, alongside innovative solutions to mitigate noise and enhance measurement fidelity. Through these comprehensive insights, this chapter aims to illuminate the potential of VQE in advancing quantum machine learning applications across diverse scientific domains.
The Variational Quantum Eigensolver (VQE) has become a central hybrid quantum-classical method for addressing electronic structure and materials science problems on near-term quantum hardware. As researchers across physics, chemistry, and quantum information engage with VQE, they encounter a wide range of algorithmic choices, including problem encoding, ansatz construction, classical optimization, and measurement strategies to choose from. This article provides a structured overview of these components and discusses practical factors that influence VQE performance in real settings, such as noise characteristics, available error-mitigation techniques, and hardware-dependent circuit considerations. Drawing partly on our early experiences, we highlight aspects that may be especially relevant for newcomers and outline common challenges observed in practical workflows. Our aim is to offer an accessible entry point for researchers approaching VQE-based quantum simulation and to support informed decision-making as the method and underlying hardware continue to evolve, which we support by discussion of examples from physics and quantum chemistry.
Abstract Quantum chemistry (QC) is one of the most promising applications of quantum computing. However, present quantum processing units (QPUs) are still subject to large errors. Therefore, noisy intermediate‐scale quantum (NISQ) hardware is limited in terms of qubit counts/circuit depths. Variational quantum eigensolver (VQE) algorithms can potentially overcome such issues. Here, we introduce the OpenVQE open‐source QC package. It provides tools for using and developing chemically‐inspired adaptive methods derived from unitary coupled cluster (UCC). It facilitates the development and testing of VQE algorithms and is able to use the Atos Quantum Learning Machine (QLM), a general quantum programming framework enabling to write/optimize/simulate quantum computing programs. We present a specific, freely available QLM open‐source module, myQLM‐fermion. We review its key tools for facilitating QC computations (fermionic second quantization, fermion‐spin transforms, etc.). OpenVQE largely extends the QLM's QC capabilities by providing: (i) the functions to generate the different types of excitations beyond the commonly used UCCSD ansatz; (ii) a new Python implementation of the “adaptive derivative assembled pseudo‐Trotter method” (ADAPT‐VQE). Interoperability with other major quantum programming frameworks is ensured thanks to the myQLM‐interop package, which allows users to build their own code and easily execute it on existing QPUs. The combined OpenVQE/myQLM‐fermion libraries facilitate the implementation, testing and development of variational quantum algorithms, while offering access to large molecules as the noiseless Schrödinger‐style dense simulator can reach up to 41 qubits for any circuit. Extensive benchmarks are provided for molecules associated to qubit counts ranging from 4 to 24. We focus on reaching chemical accuracy, reducing the number of circuit gates and optimizing parameters and operators between “fixed‐length” UCC and ADAPT‐VQE ansätze. This article is categorized under: Software > Quantum Chemistry Quantum Computing > Algorithms
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) hardware to estimate eigenvalues of many-body Hamiltonians. Unlike fully quantum approaches such as quantum phase estimation (QPE), VQE trades deep coherent circuits for repeated state preparation, measurement, and classical optimization, making it more compatible with limited qubit counts and finite coherence times. Recent developments have focused on reducing quantum resource requirements while retaining chemically meaningful wavefunction structure. In this paper, we examine recent progress in VQE methods for molecular electronic structure with an emphasis on three themes: (i) strategies for circuit and ansatz complexity reduction, including adaptive and selectively screened approaches, (ii) chemically motivated workflows that combine VQE with orbital optimization, fragmentation, and localized active-space ideas to better address strong correlation, and (iii) extensions of VQE to excited-state calculations. Throughout, we emphasize the tradeoffs among parameter count, gate depth, symmetry preservation, measurement overhead, and classical preprocessing, and discuss where these approaches may become most useful for chemically challenging active spaces. We also highlight benchmarking considerations for assessing both accuracy and resource requirements, and conclude with a perspective on regimes in which VQE may offer the greatest long-term value, particularly multireference active spaces and low-lying excited-state manifolds.