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Quantum Variational Optimization and Adiabatic Methods

Santanu Pattanayak

Published Jan 1, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.1007/978-1-4842-6522-2_7

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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.
Quantum computing is quickly becoming a field that can change the game. It can completely change how businesses solve optimisation problems. We will be looking at three different quantum algorithms in great detail: the Quantum Approximate Optimisation Algorithm (QAOA), the Variational Quantum Eigensolver (VQE), and Grover's Algorithm. We look into how these algorithms work on the inside, how they compare to more traditional methods, and how they might be used in areas like energy, banking, and logistics. The piece then talks about current research projects that are trying to fix the technical issues and hardware limits of quantum technology. In the end, we look ahead to possible future developments that might help solve optimisation problems, such as better quantum gear and more complex quantum algorithms. By combining what has already been written with what is new, this study aims to shed light on how quantum computing could help solve tough optimisation problems and spark new ideas.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm for finding the minimum eigenvalue of a Hamiltonian that involves the optimization of a parameterized quantum circuit. Since the resulting optimization problem is in general nonconvex, the method can converge to suboptimal parameter values that do not yield the minimum eigenvalue. In this work, we address this shortcoming by adopting the concept of variational adiabatic quantum computing (VAQC) as a procedure to improve VQE. In VAQC, the ground state of a continuously parameterized Hamiltonian is approximated via a parameterized quantum circuit. We discuss some basic theory of VAQC to motivate the development of a hybrid quantum-classical homotopy continuation method. The proposed method has parallels with a predictor-corrector method for numerical integration of differential equations. While there are theoretical limitations to the procedure, we see in practice that VAQC can successfully find good initial circuit parameters to initialize VQE. We demonstrate this with two examples from quantum chemistry. Through these examples, we provide empirical evidence that VAQC, combined with other techniques (an adaptive termination criteria for the classical optimizer and a variance-based resampling method for the expectation evaluation), can provide more accurate solutions than “plain” VQE, for the same amount of effort.
ABSTRACT: Even if the quantum computing popularity has recently grown, the currently-available technology is not ready for a fundamental revolution of informatics. In this transition phase, hybrid quantum-classical algorithms can help in understanding the real power that this novel paradigm can reach in the future. In this work, we tested the capability of two such algorithms to solve combinatorial optimization problems: the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA). We used them to solve max-cut problems, consisting of separating the vertices of an undirected graph in two sub-groups so that the number of edges connecting vertices of different sub-groups is maximal. To do that, we mapped the problem into a quantum system, using the Ising formalism. In particular, we investigated the effect of changing the number of measurements of the quantum circuit associated with the system on the algorithms convergence. A large number of measurements (or shots) allows a better knowledge of the quantum state, but on the other hand, increases the convergence time, making the algorithms less competitive compared to classical techniques. The results showed that VQE needs a mínimum number of measurements to start converging towards the optimal problem solution. Once this threshold is overcome, the solution improves as 1/√shots. QAOA, on the other hand, does not seem to converge, independently of the number of shots.
This chapter investigates the application of quantum computing to optimization, a field traditionally dominated by classical methods but increasingly limited by computational demands in tackling complex, large-scale problems. The discussion focuses on quantum optimization techniques, particularly variational quantum algorithms like the Quantum Approximate Optimization Algorithm (QAOA) and the variational quantum eigensolver (VQE), along with quantum annealing (QA), a process inspired by physical annealing that leverages quantum tunnelling and superposition. The chapter also explores the integration of quantum approaches within machine learning frameworks, where quantum algorithms can potentially accelerate model training and enhance capabilities. Additionally, it presents practical applications and challenges associated with these quantum algorithms, covering their effectiveness on noisy intermediate-scale quantum (NISQ) devices and the evolving landscape of quantum hardware. This chapter concludes with insights into quantum optimization's future potential to address optimization and machine learning challenges that are currently intractable for classical methods, marking a significant shift in computational science.
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.