Kernelized Decoded Quantum Interferometry

TL;DR

Kernelized Decoded Quantum Interferometry (k-DQI) reframes decoded quantum interferometry by inserting a spectral kernel before the interference stage to concentrate solution mass into decoder-friendly low-frequency modes, thereby improving robustness to hardware noise and spectral dispersion. The framework centers on the noise-weighted head mass $Σ_K$ and proves the Monotonic Improvement Theorem, which guarantees that larger head mass yields stronger decoding guarantees under local depolarizing noise. It validates structure-adaptive gains on Optimal Polynomial Interpolation and sparse Max-XORSAT, using Chirp and Linear Canonical Transform kernels and block-local designs, with explicit circuit constructions that incur only $ ilde{O}(n)$ to $ ilde{O}(n^2)$ depth overhead. The work further develops a practical resource model and scalable kernel-parameter search (analytic for OPI, variational for latent structure), yielding a realistic, near-term path to improved quantum-classical decoding performance through spectral preconditioning. Together, these results turn DQI into a tunable, noise-aware protocol that leverages spectral structure and decoding theory to achieve provable gains in structured optimization tasks.

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 $Σ_K$, and prove a Monotonic Improvement Theorem, which establishes that maximizing $Σ_K$ 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.

Figures (9)

• Figure 1: Pipeline of kernelized decoded quantum interferometry (k-DQI). A degree-$\ell$ polynomial $P$ first shapes the objective $f$, a unitary kernel $K$ reshapes the resulting spectrum before noise, and a decoder $\mathrm{Dec}$ maps measurement outcomes to a candidate solution $x^\star$. The kernel is designed to increase a noise-weighted head-spectrum mass $\Sigma_K$ that controls the resulting guarantees.
• Figure 2: Mechanism of threshold improvement via block-local alignment (BLA). The figure illustrates the behavior of the density evolution surrogate for a regular $(d_l,d_r)=(3,6)$ LDPC ensemble over the Binary Erasure Channel (BEC). a, The recursive DE map $y = \phi_\epsilon(x)$. The black point marks the critical "tangency condition" at the asymptotic BEC threshold ($\epsilon^*_{\mathrm{BEC}} \approx 0.429$), where the recursion gets stuck. The colored curves demonstrate the dose-response effect of the kernel structure: increasing the local alignment $\Delta\Sigma_{\mathrm{loc}}$ from $0$ (dark orange) to $0.06$ (yellow) lowers the effective noise parameter $\epsilon' = \epsilon(1 - \kappa \Delta\Sigma_{\mathrm{loc}})$. This shift pushes the curve strictly below the diagonal $y=x$ (red dashed line), opening a "tunnel" for successful decoding convergence. The gray band denotes the visual tolerance margin $|y-x| \le 0.02$. b, Contraction stability analysis via the derivative $\phi'_\epsilon(x)$. Reliable belief propagation requires the slope to remain below unity (dotted line at 1.0). The plot confirms that the BLA-induced shift dampens the derivative peak, restoring the strict contraction property ($\phi' < 1$) required to bypass the bottleneck.
• Figure 3: Noise-weighted head-spectrum mass $\Sigma_K(\ell,\eta;d_\star)$ for a representative OPI instance as a function of the chirp/LCT kernel parameter $\theta$. The curve exhibits a sharp peak around an optimal parameter $\theta_\star$, where the quadratic kernel cancels the OPI polynomial phase and concentrates almost all spectral mass in the head. The horizontal dashed line marks a KV/BM soft-decoding threshold: only a narrow neighbourhood of $\theta_\star$ pushes $\Sigma_K$ above this level, illustrating the tunable head-mass boost
• Figure 4: Finite-length validation of the regular (3,6) LDPC ensemble performance.a, Frame Error Rate (FER) versus channel crossover probability $\epsilon$ for the Binary Symmetric Channel (BSC). The vertical dashed line indicates the asymptotic Belief Propagation (BP) threshold $\epsilon^* \approx 0.064$. b, FER versus Signal-to-Noise Ratio ($E_b/N_0$) for the Additive White Gaussian Noise (AWGN) channel, with the threshold at $E_b/N_0^* \approx 1.02$ dB. In both panels, markers represent finite-length simulations (Finite-Length Sim), while solid curves denote the smooth response function derived from the global kernel fit (k-DQI theory). The strict alignment between simulation data and the theoretical waterfall region confirms the validity of the density evolution surrogate.
• Figure 5: Degradation of BP thresholds across regular LDPC ensembles. The plots compare the asymptotic noise tolerance for ensembles with varying degrees $(d_v, d_c) \in \{(3,6), (4,8), (5,10)\}$. a, Threshold trends for the BSC channel. Lower degree ensembles (e.g., (3,6)) exhibit higher error tolerance (larger $\epsilon^*$). b, Threshold requirements for the AWGN channel. Higher density codes require higher signal power (larger $E_b/N_0^*$) to achieve reliable decoding. This comparison illustrates the trade-off between code rate and channel robustness, quantifying the baseline requirements for block-local alignment (BLA) in the high-noise regime.

Theorems & Definitions (31)

• Theorem 1: Monotonic Improvement via Head Mass
• Lemma 3.1: Synthesis of quadratic chirp kernels
• proof
• Lemma 1.1: Per‑mode contraction under local noise
• proof
• Lemma 1.2: Head–tail coherence bound
• proof
• Proposition 2: Decoder threshold coupling
• proof : Proof of \ref{['thm:monotonicity']}
• Lemma 2.1: Per-mode contraction