QAM-W: Joint 2D Codebook Quantization for LLM Weights via Hadamard Rotation and Activation-Aware Scaling

Preetam Sharma; Kacper Dobek

cs.LGMay 25, 2026

QAM-W: Joint 2D Codebook Quantization for LLM Weights via Hadamard Rotation and Activation-Aware Scaling

Authors: Preetam Sharma, Kacper Dobek

Abstract

Scalar post-training quantizers discard pairwise coordinate structure within weight rows. We introduce QAM-W (Quadrature Amplitude Modulation for Weights), a codec that recovers this structure: each row is L2-normalized, block-Hadamard rotated, paired into 2D coordinates, and quantized against a single Lloyd-Max codebook trained on the unit circular Gaussian, with activation-aware per-channel scaling. In a cross-model study spanning five LLMs from four families (1.1B--13B parameters) and eight quantized configurations, the activation-aware variant at $\approx 5.5$ bpw stays within $\pm 0.4\%$ of BF16 WikiText-2 perplexity on every model, matching the SmoothQuant W8A8 quality envelope at $32\%$ fewer weight bits. Joint 2D coding outperforms polar (amplitude $\times$ phase) coding by 2--15pp $Δ$ PPL at equal bitrate, and paired KL against BF16 tracks $Δ$ PPL% at Spearman $ρ= 0.99$ across 37 (method, model) rows, consistent with a monotone composite bound from codec distortion to KL divergence. A 3.5bpw variant is competitive on quantization-tolerant architectures. At strict 4~~bpw, the rotated-codebook frontier method QTIP outperforms QAM-W; the contribution is the quality-preserving 5--6~~bpw band.

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Low-bit weight quantization saves storage but leaves errors that degrade LLM quality. We introduce activation-weighted seeded residual coding (AWSRC), a compact repair codec for an existing quantization backbone. Given a reconstructed weight

W_0

, AWSRC encodes the residual

W-W_0

using deterministic seed-generated bases. The sidecar stores seed selectors, low-bit coefficients, and scales rather than an explicit codebook. Two variants combine activation weighting with per-module byte quotas (

\mathrm{AWSRC\text{-}U}

), or blended activation/Fisher weighting with globally ranked progressive prefixes (

\mathrm{AWSRC\text{-}P}_{F}

) that support multiple byte budgets without refitting. On Qwen2.5-3B-Instruct, adding

0.162

scope-bits/weight to an RTN-INT4 baseline closes

88.2\%

78.9\%

, and

71.3\%

of the PPL, KL, and 11-task mean-accuracy gaps to BF16, respectively. AWSRC achieves the highest mean downstream accuracy in byte-matched residual-codec ablations and improves all metrics across model families with up to 32B parameters.

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