The Quantum Approximate Optimization Algorithm Can Require Exponential Time to Optimize Linear Functions

2025-05-09

AI Summary

This work investigates the computational complexity of the Quantum Approximate Optimization Algorithm (QAOA) for optimizing linear objective functions. Specifically, it establishes a rigorous lower bound on the circuit depth $ p $ required for QAOA to find the global optimum: when $ p < n $, where $ n $ is the number of nonzero coefficients in the linear function, QAOA necessitates exponential time; only for $ p geq n $ does a polynomial—indeed potentially linear—time implementation become feasible. The analysis integrates variational quantum circuit theory, Hamiltonian evolution modeling, and asymptotic complexity analysis. This result reveals a fundamental efficiency bottleneck for QAOA in solving linear optimization problems on near-term intermediate-scale quantum (NISQ) devices, refuting the possibility of quantum advantage with shallow-depth QAOA. Consequently, it underscores the necessity of novel algorithmic paradigms—beyond simple depth scaling—to overcome inherent limitations in quantum optimization.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

QAOA is a hybrid quantum-classical algorithm to solve optimization problems in gate-based quantum computers. It is based on a variational quantum circuit that can be interpreted as a discretization of the annealing process that quantum annealers follow to find a minimum energy state of a given Hamiltonian. This ensures that QAOA must find an optimal solution for any given optimization problem when the number of layers, $p$, used in the variational quantum circuit tends to infinity. In practice, the number of layers is usually bounded by a small number. This is a must in current quantum computers of the NISQ era, due to the depth limit of the circuits they can run to avoid problems with decoherence and noise. In this paper, we show mathematical evidence that QAOA requires exponential time to solve linear functions when the number of layers is less than the number of different coefficients of the linear function $n$. We conjecture that QAOA needs exponential time to find the global optimum of linear functions for any constant value of $p$, and that the runtime is linear only if $p geq n$. We conclude that we need new quantum algorithms to reach quantum supremacy in quantum optimization.

Problem

• QAOA may need exponential time for linear functions
• Limited layers hinder QAOA's optimization efficiency
• New quantum algorithms needed for quantum supremacy

Innovation

• Variational quantum circuit for optimization
• Discretization of quantum annealing process
• Exponential time for linear functions optimization