Hidden Quantum Advantage near the Decoding Threshold of Decoded Quantum Interferometry

Abstract: Where is the true boundary of the quantum advantage region of decoded quantum interferometry (DQI)? The best existing answer is provided by Theorem 10.1 of Jordan et al., yet we show that this answer systematically underestimates the extent of quantum advantage. On the standard partial-win LDPC benchmark instance, there exist 26 consecutive parameter points (l in [642, 667]) at which Jordan's analysis declares no quantum advantage (<s>/m < 0.5), while quantum advantage is in fact present with an approximation ratio reaching 0.66. The root cause is that Jordan's bound penalizes the entire system with the worst-case Hamming-layer decoding failure rate epsilon = max_k epsilon_k, discarding the spectral structure of the DQI tridiagonal matrix. Exploiting the concentration of the Perron eigenvector, we replace the uniform penalty with the weighted average epsilon_bar = sum_k epsilon_k w_k^2 and establish a unified lower bound (Master Theorem) valid over arbitrary finite fields F_q, proving that it strictly improves upon the original bound from three independent sources.