Quantum Optimization Algorithms

2025-11-15

AI Summary

This work addresses industrial-scale combinatorial optimization problems in the NISQ era, focusing on enhancing the practicality and constraint handling of the Quantum Approximate Optimization Algorithm (QAOA). Method: For NP-hard problems such as Max-Cut, we propose a Grover-mixer-based constraint-encoding scheme that rigorously embeds the feasible solution space into the QAOA variational circuit, eliminating sampling of infeasible solutions. We further generalize QAOA to a constrained Variational Quantum Eigensolver (c-VQE), supporting high-order Ising Hamiltonian modeling and analytical gradient computation via the parameter-shift rule. Contribution/Results: Implemented end-to-end on PennyLane, our approach demonstrates significant improvements in solution quality and convergence speed over unconstrained baselines. Systematic evaluation confirms c-VQE’s robustness under realistic noise models and its scalability to larger problem instances, establishing a foundation for near-term quantum optimization with hard constraints.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Quantum Machine Learning

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Developing quantum optimization algorithms for exponential speedups
• Implementing QAOA for gate-based quantum computers
• Addressing constraints and challenges in NISQ-era quantum optimization

Innovation

• Quantum Approximate Optimization Algorithm for gate-based computers
• Incorporates constraints using Grover mixers for valid solutions
• Generalizes to Variational Quantum Eigensolver for NISQ applications