Pairwise-Independent Dithering for Single-Stage Hadamard Quantization

Quantizing high-dimensional vectors is fundamental to similarity search, distributed learning, and model compression. Feng, Indyk, Kapralov, Krachun, and Prokhorov established sharp guarantees for an unbiased dithered quantizer based on a randomized Hadamard transform. Their $1/d$-scale inner-product estimator, however, uses a second randomized transform and residual quantization, increasing both communication and the leading constant in the proved bound. We show that this extra stage is unnecessary: pairwise-independent dithers across Hadamard coordinates suffice. The resulting unbiased single-stage estimator uses $b$ bits per coordinate and achieves $\mathrm{E}\left[{\left|⟨y,\hat{x}-x⟩\right|}^2\right]\le \left(\frac{3\pi \sqrt{3}}{2}+o(1)\right)\frac{\Vert y{\Vert }_2^2}{d{4}^b},$ as $b\to \infty$, with a dimension-free $o(1)$ term uniform over unit inputs and fixed queries. Compared with the two-stage construction of Feng et al., it eliminates the residual-stage $O(d)$-bit payload and reduces the leading upper-bound constant by a factor of approximately $5.93$.