Quantum Optimization Algorithms

Quantum optimization allows for up to exponential quantum speedups for specific, possibly industrially relevant problems. As the key algorithm in this field, we motivate and discuss the Quantum Approximate Optimization Algorithm (QAOA), which can be understood as a slightly generalized version of Quantum Annealing for gate-based quantum computers. We delve into the quantum circuit implementation of the QAOA, including Hamiltonian simulation techniques for higher-order Ising models, and discuss parameter training using the parameter shift rule. An example implementation with Pennylane source code demonstrates practical application for the Maximum Cut problem. Further, we show how constraints can be incorporated into the QAOA using Grover mixers, allowing to restrict the search space to strictly valid solutions for specific problems. Finally, we outline the Variational Quantum Eigensolver (VQE) as a generalization of the QAOA, highlighting its potential in the NISQ era and addressing challenges such as barren plateaus and ansatz design. Combinatorial optimization problems, such as determining the most efficient routing of delivery trucks or finding the optimal schedule for a set of tasks, involve searching through a vast number of possible solutions to find the best one. For a small subset of potentially practically relevant problems ranging from the most basic maximum cut problem [1] to the optimization of higher order objective functions [2, 3]), there already exists promising evidence that the standard quantum optimization algorithm for gate model quantum computer, called Quantum Approximate Optimization Algorithm (QAOA), can outperform classical state-of-the-art solvers. Quantum computing hence offers promising approaches to tackle these complex problems by exploiting quantum mechanical principles.