Quantum Optimization Algorithms

TL;DR

This paper analyzes quantum optimization on gate-based devices by connecting adiabatic quantum computing to the Quantum Approximate Optimization Algorithm (QAOA) and framing it as a discretized, trainable evolution under cost and mixer Hamiltonians. It provides detailed circuit-level methods for Hamiltonian simulation, higher-order Ising terms, and parameter-shift training, and demonstrates a MaxCut example using Pennylane, including constraint handling with Grover mixers. It also outlines a real-world deployment guide from problem formulation to hardware mapping and benchmarking, and situates QAOA within the broader Variational Quantum Eigensolver (VQE) framework as a generalization with considerations for ansatz design and NISQ-era challenges. Together, the work offers a practical roadmap for applying quantum optimization to industrial problems, while highlighting limitations such as barren plateaus and the critical role of problem-specific ansätze.

Abstract

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.

Figures (11)

• Figure 1: Standard grid-based form of a classical Ising model with spins pointing up or down.
• Figure 2: Energy spectra for an (a) native and (b) discretized, i.e., approximated, continuous-time Hamiltonian. The differently colored lines display the different energy levels of the Hamiltonian.
• Figure 3: The time evolution speed in adiabatic quantum computing can be optimized based on the momentary spectral gap of the time-dependent Hamiltonian.
• Figure 4: Hamiltonian simulation of the components in the cost Hamlitonian $H_C$.
• Figure 5: Quantum circuit implementation of the QAOA starting in the initial state $\ket{+}^{\otimes n}$.