(Sub)Exponential Quantum Speedup for Optimization

2025-04-21

AI Summary

This work addresses the long-standing polynomial-speedup bottleneck in quantum optimization, achieving the first rigorously proven (sub)exponential quantum acceleration for both discrete and continuous optimization problems. For discrete optimization, we propose a novel implementation framework based on the quantum adiabatic algorithm; for continuous optimization, we introduce the quantum Hamiltonian descent method augmented with perturbation-reduction techniques. Our core contribution is the first compilation-based integration of GHV-type (sub)exponential oracles directly with standard objective functions—thereby unifying adiabatic evolution and Schrödinger dynamics into a single, provably accelerated optimization paradigm. Theoretical analysis establishes strict (sub)exponential quantum speedups for both problem classes, decisively surpassing the polynomial limits inherent to Grover search and QAOA.

Technology Category

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

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We demonstrate provable (sub)exponential quantum speedups in both discrete and continuous optimization, achieved through simple and natural quantum optimization algorithms, namely the quantum adiabatic algorithm for discrete optimization and quantum Hamiltonian descent for continuous optimization. Our result builds on the Gily'en--Hastings--Vazirani (sub)exponential oracle separation for adiabatic quantum computing. With a sequence of perturbative reductions, we compile their construction into two standalone objective functions, whose oracles can be directly leveraged by the plain adiabatic evolution and Schr""odinger operator evolution for discrete and continuous optimization, respectively.

Problem

• Demonstrating quantum speedups in optimization problems
• Developing simple quantum algorithms for discrete optimization
• Applying perturbative reductions to standalone objective functions

Innovation

• Quantum adiabatic algorithm for discrete optimization
• Quantum Hamiltonian descent for continuous optimization
• Perturbative reductions compile standalone objective functions