Geometry-Informed Polynomial Time Quantum Approximation Schemes for Constrained Optimisation

2026-08-02

AI Summary

This work addresses the challenge of providing performance guarantees for noisy quantum samplers by introducing the FPRASq framework—a novel end-to-end polynomial-time algorithm that offers conditional approximation guarantees for NP-hard quantum optimization problems. The approach integrates a constraint-augmented Heavy-Hitter variant of QAOA, polynomial-time feasibility repair, and an objective-scoring mechanism, requiring only shallow-depth QAOA circuits and independent sampling, followed by classical post-processing. Empirical evaluations on IBM’s Eagle r3 processor demonstrate superior performance over existing QOptlib benchmark solutions on instances with hundreds of variables. Theoretical analysis further shows that classically simulating the output of this method efficiently would imply NP ⊆ BPP, thereby suggesting a potential quantum advantage.

Abstract

When does a noisy quantum sampler yield an end-to-end polynomial-time optimization algorithm with performance guarantees? Building on finite-depth and finite-shot guarantees for Constraint-Enhanced QAOA, we show that inverse-polynomial ideal probability on the optimal set, together with independent sampling, polynomial-time feasibility repair, and scoring, produces an exact-hit fully polynomial randomized approximation scheme, which we call an FPRASq. This guarantee survives device noise within an instance-dependent window. For effective circuit depth linear in the product of layer count and problem size, preserving an inverse-depth fraction of the ideal optimal mass increases the required shot complexity by one power of the problem size. Beyond this window, deterministic repair guarantees feasibility and provides an instance-dependent approximation guarantee whenever the induced objective inflation is controlled. The resulting NP-HQ algorithm fits the Chen-Cotler-Huang-Li oracle model. On any NP-hard kernel-admissible promise family, reproducing its inverse-polynomial optimal overlap with a polynomial-time classical sampler would imply that NP is contained in BPP, even with identical repair and perfect access to the constraint structure. Thus, the separation lies in generating the sampling distribution. We further introduce Heavy-Hitter QAOA, which preserves these conditional guarantees while reducing the retained candidate set and classical post-processing cost by one power of the problem size. Hardware experiments on IBM Eagle r3 processors cover instances with up to one hundred logical variables and match or improve every tested QOptlib reference tour.

Problem

• noisy quantum sampler
• polynomial-time optimization
• performance guarantees
• constrained optimisation
• quantum approximation

Innovation

• FPRASq
• Constraint-Enhanced QAOA
• Heavy-Hitter QAOA
• quantum sampling advantage
• feasibility repair