This document is a pdf version of the series of blogposts about variational quantum algorithms (VQA) I originally posted on my blog Musty Thoughts. It provides an explanation of the basic variational algorithms, such as Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA), as well as a more general framework for VQAs. It also describes some more advanced techniques that can be used to make these algorithms more efficient, as well as the challenges associated with using them.
This document is a pdf version of the series of blogposts about variational quantum algorithms (VQA) I originally posted on my blog Musty Thoughts. It provides an explanation of the basic variational algorithms, such as Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA), as well as a more general framework for VQAs. It also describes some more advanced techniques that can be used to make these algorithms more efficient, as well as the challenges associated with using them.
The rapid evolution of quantum computing has introduced groundbreaking approaches to solving complex optimization problems, traditionally limited by classical methods. This talk explores the principles and applications of quantum optimization, with a focus on Variational Quantum Algorithms (VQAs). These algorithms, including the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA), leverage hybrid quantum-classical systems to address challenges in diverse fields such as combinatorial optimization, quantum chemistry, and material design. Attendees will gain insights into the theoretical foundations of quantum computing, the operational mechanics of quantum circuits, and the integration of quantum programming tools like Qiskit. Practical examples and demonstrations will highlight how VQAs outperform classical methods in specific scenarios, offering a transformative perspective on optimization in the quantum era.
Noisy intermediate-scale quantum computers (NISQ computers) are now readily available, motivating many researchers to experiment with Variational Quantum Algorithms (VQAs). Among them, the Quantum Approximate Optimization Algorithm (QAOA) is one of the most popular one studied by the combinatorial optimization community. In this tutorial, we provide a mathematical description of the class of Variational Quantum Algorithms, assuming no previous knowledge of quantum physics from the readers. We introduce precisely the key aspects of these hybrid algorithms on the quantum side (parametrized quantum circuit) and the classical side (guiding function, optimizer). We devote a particular attention to QAOA, detailing the quantum circuits involved in that algorithm, as well as the properties satisfied by its possible guiding functions. Finally, we discuss the recent literature on QAOA, highlighting several research trends.
In this chapter, we will take a look at the various optimization techniques that use quantum computing in their formulation. A couple of such algorithms that we are going to work through in great detail are the variational quantum eigensolver, popularly known as VQE, and the quantum approximate optimization algorithm, also known as QAOA. The central idea in both methods is to define cost objectives as an expectation of appropriate Hamiltonians pertaining to quantum systems. Based on the maximization or minimization problem, we look to derive the maximum or minimum eigenvalue state through these optimization techniques. Both of these methods are variational in that they combine both quantum and classical methods for the optimization. Since the topic is centered around Hamiltonians, we will study the popular Isling Hamiltonian model as well in this chapter. Similarly, the QAOA technique is based on the adiabatic evolution of a quantum system, and hence we will study its underlying math in great detail in this chapter.
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.
The exponential runtime of quantum simulators on classical machines, along with long queue times and high costs associated with real quantum devices, presents significant challenges for the efficient optimization of Variational Quantum Algorithms (VQAs) such as the Variational Quantum Eigensolver (VQE), Quantum Approximate Optimization Algorithm (QAOA), and Quantum Neural Networks (QNNs). To address these limitations, we propose a novel approach, DyPP (Dynamic Parameter Prediction), which accelerates the convergence of VQAs by leveraging regular trends in the parameter weights to update parameters efficiently. We introduce two techniques for optimal prediction performance: Naive Prediction (NaP) and Adaptive Prediction (AdaP). Through extensive experimentation and training of multiple QNN models on various datasets, we demonstrate that DyPP achieves a speedup of approximately 2.25×compared to standard training methods, while also providing improved accuracy (up to 2.3% higher) and lower loss (up to 6.1% reduction) with minimal storage and computational overhead. We further evaluate DyPP’s effectiveness in VQE for molecular ground-state energy estimation and in QAOA for graph MaxCut. Our results show that, on average, DyPP leads to a speedup of up to 3.1× for VQE and 2.91× for QAOA compared to traditional optimization, while requiring up to 3.3× fewer shots (i.e., repeated circuit executions). Even in the presence of hardware noise, DyPP outperforms existing optimization techniques, delivering up to 3.33× speedup and requiring 2.5× fewer shots, thereby enhancing the efficiency of VQAs.