Kernelized Decoded Quantum Interferometry

TL;DR

Kernelized Decoded Quantum Interferometry (k-DQI) enhances quantum optimization by integrating spectral engineering into quantum circuits, improving robustness against noise and enabling practical near-term quantum applications.

Contribution

The paper introduces a unified framework, k-DQI, that incorporates spectral kernels into quantum circuits to improve noise resilience and decoding success in quantum interferometry.

Findings

Kernel tuning acts as a spectral lens to recover signals lost to noise.

Explicit circuit implementations with Chirp and LCT kernels boost signal-to-noise ratio.

Theoretical guarantees link kernel optimization to increased decoding success under noise.

Abstract

Decoded Quantum Interferometry (DQI) promises superpolynomial speedups for structured optimization; however, its practical realization is often hindered by significant sensitivity to hardware noise and spectral dispersion. To bridge this gap, we introduce Kernelized Decoded Quantum Interferometry (k-DQI), a unified framework that integrates spectral engineering directly into the quantum circuit architecture. By inserting a unitary kernel prior to the interference step, k-DQI actively reshapes the problem's energy landscape, concentrating the solution mass into a ``decoder-friendly'' low-frequency head. We formalize this advantage through a novel robustness metric, the noise-weighted head mass $\Sigma_K$, and prove a Monotonic Improvement Theorem, which establishes that maximizing $\Sigma_K$ guarantees higher decoding success rates under local depolarizing noise. We substantiate these theoretical gains in Optimal Polynomial Interpolation (OPI) and LDPC-like problems, demonstrating that kernel tuning functions as a ``spectral lens'' to recover signal otherwise lost to isotropic noise. Crucially, we provide explicit, efficient circuit realizations using Chirp and Linear Canonical Transform (LCT) kernels that achieve significant boosts in effective signal-to-noise ratio with negligible depth overhead ($\tilde{O}(n)$ to $\tilde{O}(n^2)$). Collectively, these results reframe DQI from a static algorithm into a tunable, noise-aware protocol suited for near-term error-corrected environments.