Exploring many-body chemical systems on classical computers often involves solving the Schrödinger equation. However, this approach is frequently limited by the exponential increase in the dimensionality of the Hamiltonian as the number of degrees of freedom increases. In contrast, quantum computing, specifically through the variational quantum eigensolver (VQE) framework, shows promise in overcoming this exponential cost. VQE can utilize the collective properties of quantum states to model the wavefunction in polynomial time. Despite the current limitations of quantum hardware, significant advances have been made in the development of VQE-based algorithms. In this review, we provide an overview of emerging protocols, focusing on their applications in simulating the ground state, excited state, and vibrational properties of chemical systems. By examining notable algorithmic advancements and applications, this review aims to shed light on the challenges and potential of VQE-based algorithms in addressing relevant chemical problems.
Exploring many-body chemical systems on classical computers often involves solving the Schrödinger equation. However, this approach is frequently limited by the exponential increase in the dimensionality of the Hamiltonian as the number of degrees of freedom increases. In contrast, quantum computing, specifically through the variational quantum eigensolver (VQE) framework, shows promise in overcoming this exponential cost. VQE can utilize the collective properties of quantum states to model the wavefunction in polynomial time. Despite the current limitations of quantum hardware, significant advances have been made in the development of VQE-based algorithms. In this review, we provide an overview of emerging protocols, focusing on their applications in simulating the ground state, excited state, and vibrational properties of chemical systems. By examining notable algorithmic advancements and applications, this review aims to shed light on the challenges and potential of VQE-based algorithms in addressing relevant chemical problems.
The variational quantum eigensolver (or VQE), first developed by Peruzzo et al. (2014), has received significant attention from the research community in recent years. It uses the variational principle to compute the ground state energy of a Hamiltonian, a problem that is central to quantum chemistry and condensed matter physics. Conventional computing methods are constrained in their accuracy due to the computational limits facing exact modeling of the exponentially growing electronic wavefunction for these many-electron systems. The VQE may be used to model these complex wavefunctions in polynomial time, making it one of the most promising near-term applications for quantum computing. One important advantage is that variational algorithms have been shown to present some degree of resilience to the noise in the quantum hardware. Finding a path to navigate the relevant literature has rapidly become an overwhelming task, with many methods promising to improve different parts of the algorithm, but without clear descriptions of how the diverse parts fit together. The potential practical advantages of the algorithm are also widely discussed in the literature, but with varying conclusions. Despite strong theoretical underpinnings suggesting excellent scaling of individual VQE components, studies have pointed out that their various pre-factors could be too large to reach a quantum computing advantage over conventional methods. This review aims at disentangling the relevant literature to provide a comprehensive overview of the progress that has been made on the different parts of the algorithm, and to discuss future areas of research that are fundamental for the VQE to deliver on its promises. All the different components of the algorithm are reviewed in detail. These include the representation of Hamiltonians and wavefunctions on a quantum computer, the optimization process to find ground state energies, the post processing mitigation of quantum errors, and suggested best practices. We identify four main areas of future research: (1) optimal measurement schemes for reduction of circuit repetitions required; (2) large scale parallelization across many quantum computers; (3) ways to overcome the potential appearance of vanishing gradients in the optimization process for large systems, and how the number of iterations required for the optimization scales with system size; (4) the extent to which VQE suffers for quantum noise, and whether this noise can be mitigated in a tractable manner. The answers to these open research questions will determine the routes for the VQE to achieve quantum advantage as the quantum computing hardware scales up and as the noise levels are reduced.
Quantum mechanics has introduced a new theoretical framework for the study of molecules, enabling the prediction of properties and dynamics through the solution of the Schrödinger equation applied to these systems. However, solving this equation is computationally expensive, which has led to the development of various mathematical frameworks and computational methods designed to balance the available resources with the desired level of accuracy. In particular, quantum computers have emerged as a promising technology with the potential to address these problems more efficiently in the coming decades, whether through reductions in memory, time, and energy consumption, i. e., reductions in computational complexity or by enhancing precision. This research field is known as Quantum Simulation. Given the current technological limitations of quantum computers, Variational Quantum Algorithms (VQAs), especially the Variational Quantum Eigensolver (VQE), stand out as a feasible approach to demonstrating advantages over classical methods in the near term. This feasibility arises from their lower demand for quantum gates and the reduced depth of the circuits required for their implementation. In this work, we present the application of quantum mechanics to the study of molecules, provide an introduction to the fundamentals of quantum computing, and explore the integration of these fields by employing the VQE in molecular simulations. Finally, we discuss the spatial and temporal complexity associated with the algorithm, highlighting its implications and challenges.
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.
The advancements in classical computational components such as CPUs and GPUs have had a reverberating impact on the democratization of computational power available, with the field of computational chemistry reaping significant benefits from the propulsion of the limits of systems studied. The accurate prediction of chemical properties and reaction dynamics through computational simulation is now a centerpiece of modern chemical research, significantly accelerating discoveries research areas of materials science and drug development, as well as industrially relevant processes such as the emission of aldehydes from soft foams, flame retardants and pyrolysis and the carbonylation of epoxides. These achievements of computational chemistry still are overshadowed in the face of fundamental bottlenecks where classical methods remain constrained by the exponential scaling when treating large and/or strongly correlated quantum many-body systems. This innate computational complexity often constitutes chemically accurate calculations based on first principles impractical, or plain impossible, for systems beyond modest sizes. Quantum computing offers a potential pathway around these limitations, originally formally formulated by David Deutsch as a tool for simulating quantum systems, with the first quantum algorithms indicating speed-ups compared to classical counterparts arising from Peter Shor and Lov Grover for the prime factorization of an integer number and unstructured search respectively. Relying on fundamental theorems by Claude Shannon, it has been demonstrated that through the use of error-correcting codes fault-tolerant quantum computers are able to safeguard information against noise and decoherence, strongly indicating towards exponential speedups in solving central computational chemistry problems, such as determining ground state energies or reaction pathways of chemical systems. Substantial strides in experimental quantum computing over the past decade, including proof-of-principle demonstrations of quantum error correction, mean to bring full-scale fault-tolerant quantum computing within the grasp of humanity. The breakthroughs needed to attain scalable, error-corrected quantum computation with capabilities exceeding those of classical methods in real-world scenarios is likely still some years away. While striving to address the engineering challenges standing in the way of achieving fault tolerant quantum computing, substantial effort has been focused toward developing and benchmarking hybrid algorithms tailored for the Noisy Intermediate- Scale Quantum (NISQ) devices, available now to researchers. These hybrid quantum-classical algorithms, with the most notable being the Variational Quantum Eigensolver (VQE), allow the probing of quantum computing’s utility in computational chemistry even in the face of significant hardware noise and limited coherence times. The work presented in this thesis introduces several key additions the VQE framework, geared especially towards to NISQ hardware, where the mindful use of available resources is imperative. First, the implementation of the Conjugate Model Search optimizer is described, an optimization method that integrates simple models that are surrogate to the original function, combined with conjugate gradient optimization to navigate effectively parameter landscapes typical of variational algorithms, prone to noise. Second, this thesis contains the proposal and extensive study of a hyperparameter tuning procedure employing genetic algorithms, allowing for comprehensive simulation benchmarks that augment the robustness and efficiency of VQE implementations with respect to convergence time through real wall-time models of quantum processors. Continuing, a bridge is presented between quantum simulation and quantum chemistry, through qubit-to-fermion mappings of existing quantum gates, further enabling the direct and accurate modeling of chemical systems on trapped atom quantum hardware with the added arsenal provided by techniques designed for qubit architectures. Additionally, experimental data contained in this thesis include detailed benchmarking data collected from state-of-the-art trapped ion quantum processors available at the Johannes Gutenberg Universität Mainz, demonstrating the actual viability and performance of various methodologies for extracting results. Lastly, the state of VQE is examined, as it served as a springboard for a plethora of advancements in the development of quantum computing hardware, as well as the direction the field of simulation of computational quantum chemistry via quantum devices is headed towards in the coming years.
Quantum computers have the potential to deliver speed-ups for solving certain important problems that are intractable for classical counterparts, making them a promising avenue for advancing modern computation. However, many quantum algorithms require deep quantum circuits, which are challenging to implement on current noisy devices. To address this limitation, variational quantum algorithms (VQAs) have been actively developed, enabling practical quantum computing in the noisy intermediate-scale quantum (NISQ) era. Among them, the variational quantum eigensolver (VQE) stands out as a leading approach for solving problems in quantum chemistry, many-body physics, and even integer factorization. The VQE algorithm can be implemented on various quantum hardware platforms, including photonic systems, quantum dots, trapped ions, neutral atoms, and superconducting circuits. In particular, photonic platforms offer several advantages: they operate at room temperature, exhibit low decoherence, and support multiple degrees of freedom, making them suitable for scalable, high-dimensional quantum computation. Here we present methodologies for realizing VQE on photonic systems, highlighting their potential for practical quantum computing. We first provide a theoretical overview of the VQE framework, focusing on the procedure for variationally estimating ground state energies. We then explore how photonic systems can implement these processes, showing that a wide variety of problems can be addressed using either multiple qubit states or a single qudit state.
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.