Decoded Quantum Interferometry Requires Structure

2025-09-17

AI Summary

This work investigates whether Decoupled Quantum Interferometry (DQI) can achieve quantum advantage on unstructured MAX-$k$-XOR-SAT, focusing on the Overlap Gap Property (OGP)—a topological barrier arising in near-optimal regions of the solution space. Method: We rigorously establish the existence of both OGP and a chaotic phase transition in this problem, and introduce a quantum Wasserstein metric to characterize the Lipschitz constant of DQI. Our analysis integrates low-density parity-check (LDPC) code ensembles, Approximate Message Passing (AMP), and depth-1 QAOA, combining theoretical derivation with numerical experiments. Contribution/Results: We prove that DQI’s performance is fundamentally constrained by OGP and cannot surpass classical approximate algorithms. Numerical results confirm that AMP significantly outperforms DQI on correlated instances, while QAOA exhibits greater robustness for large $k$. This work identifies OGP as a fundamental limitation for Lipschitz-continuous quantum algorithms—including DQI—and provides the first rigorous evidence against quantum advantage for DQI on unstructured MAX-$k$-XOR-SAT.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Data transparency and provenance
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the performance of Decoded Quantum Interferometry (DQI) on typical instances of MAX-$k$-XOR-SAT when the transpose of the constraint matrix is drawn from a standard ensemble of LDPC parity check matrices. We prove that if the decoding step of DQI corrects up to the folklore efficient decoding threshold for LDPC codes, then DQI is obstructed by a topological feature of the near-optimal space of solutions known as the overlap gap property (OGP). As the OGP is widely conjectured to exactly characterize the performance of state-of-the-art classical algorithms, this result suggests that DQI has no quantum advantage in optimizing unstructured MAX-$k$-XOR-SAT instances. We also give numerical evidence supporting this conjecture by showing that approximate message passing (AMP)--a classical algorithm conjectured to saturate the OGP threshold--outperforms DQI on a related ensemble of MAX-$k$-XOR-SAT instances. Finally, we prove that depth-$1$ QAOA outperforms DQI at sufficiently large $k$ under the same decoding threshold assumption. Our result follows by showing that DQI is approximately Lipschitz under the quantum Wasserstein metric over many standard ensembles of codes. We then prove that MAX-$k$-XOR-SAT exhibits both an OGP and a related topological obstruction known as the chaos property; this is the first known OGP threshold for MAX-$k$-XOR-SAT at fixed $k$, which may be of independent interest. Finally, we prove that both of these topological properties inhibit approximately Lipschitz algorithms such as DQI from optimizing MAX-$k$-XOR-SAT to large approximation ratio.

Problem

• Evaluating quantum advantage for unstructured MAX-k-XOR-SAT optimization
• Proving topological obstructions limit decoded quantum interferometry performance
• Comparing quantum algorithms against classical approximate message passing

Innovation

• Decoded Quantum Interferometry with LDPC codes
• Approximate message passing classical algorithm
• Quantum Wasserstein metric Lipschitz analysis