Decoded Quantum Interferometry Under Noise

TL;DR

Decoded Quantum Interferometry (DQI) is a promising quantum optimization method that faces challenges under noise, with performance depending on instance sparsity and noise levels, as shown through theoretical analysis and simulations.

Contribution

This paper provides the first rigorous analysis of DQI's robustness to noise, introducing a framework to understand how noise impacts its performance and potential quantum advantage.

Findings

Performance decays exponentially with decreasing sparsity under noise.

Solution quality is governed by a noise-weighted sparsity parameter.

Numerical simulations confirm the theoretical decay in specific problems.

Abstract

Decoded Quantum Interferometry (DQI) is a recently proposed quantum optimization algorithm that exploits sparsity in the Fourier spectrum of objective functions, with the potential for exponential speedups over classical algorithms on suitably structured problems. While highly promising in idealized settings, its resilience to noise has until now been largely unexplored. To address this, we conduct a rigorous analysis of DQI under noise, focusing on local depolarizing noise. For the maximum linear satisfiability problem, we prove that, in the presence of noise, performance is governed by a noise-weighted sparsity parameter of the instance matrix, with solution quality decaying exponentially as sparsity decreases. We demonstrate this decay through numerical simulations on two special cases: the Optimal Polynomial Intersection problem and the Maximum XOR Satisfiability problem. The Fourier-analytic methods we develop can be readily adapted to other classes of random Pauli noise, making our framework applicable to a broad range of noisy quantum settings and offering guidance on preserving DQI's potential quantum advantage under realistic noise.