Qift: Shift-Friendly No-Zero W2 Post-Training Quantization for Rotated W2A4/KV4 LLM Inference

We apply the influence-adaptive Walsh geometry of a companion theory paper (arXiv:2605.01637) to extreme low-bit weight-only LLM quantization. The recipe is one math-invariant transformation: WHT-rotate each linear layer's weight matrix and rescale its columns by per-coordinate Walsh-basis activation energy before handing off to a reconstruction-error quantizer (Intel auto-round). This biases per-group integer rounding toward high-spectral-energy channels. On four pretrained decoder-only models from 135M to 1.5B parameters, BBT-spectral reduces wikitext-2 perplexity by 15-58% relative to vanilla auto-round at W2A16; we also report a TinyLlama-1.1B auxiliary data point. Three extensions transfer the recipe to families it failed on: a per-head PCA matrix-Gamma replacement of q_norm/k_norm for Qwen3 attention (PPL 136.76 -> 88.99 on Qwen3-0.6B); an SO(2) per-pair rotation that commutes with RoPE (PPL 36.93 -> 21.84 on Qwen2.5-1.5B); and an MoE-aware input-side absorption fix identified by architectural fuzzing of Laguna-style fused-expert layouts. A W2-vs-W4 ablation gives a deliberate negative control: the redistribution payoff falls within the +/-0.5 PPL noise floor at W4, consistent with the Schur-convexity intuition that the cost of unconcentrated influence vanishes as the noise budget shrinks. All quantized weights export to OpenVINO IR and run on Intel NPU + Arc dGPU + CPU with PPL invariant to device within +/-0.1. We do not claim a formal Boolean-to-real-valued transfer of the theory paper's majorization argument: the WHT activation energy used here is not the Boolean influence of the theory paper, the link is intuitive, and the contribution is engineering value rather than a transferred theorem. Head-to-head benchmarks against SpinQuant, QuaRot, QuIP-sharp, AQLM, OmniQuant, and ButterflyQuant at matched calibration are the main future-work item.

We revisit Kashin-decomposition-based weight quantization for large language models and propose an improved algorithm with stronger convergence properties and structured, efficient orthogonal transforms. The method retains the core factorization of each weight into two components -- one with bounded infinity norm and the other with bounded infinity norm after an orthogonal transformation -- but replaces the dense random orthogonal matrix with a sign-randomized Discrete Cosine Transform (DCT), reducing the per-iteration cost from $\mathcal{O}(N^2)$ to $\mathcal{O}(N \log N)$. The proposed greedy algorithm with alternating updates guarantees the four-peak distribution required for stable 2-bit clustering of each factor and admits closed-form initialization of cluster centers, removing the multi-restart k-means bottleneck of prior work. Composed with OPTQ-style sequential error compensation and QuIP-style incoherence preprocessing, the resulting JAX pipeline is competitive with OPTQ, QuIP, QuIP-RG and a fine-tuning- and vector-quantization-free variant of QuIP# at 4-bit per channel on OPT, Llama-2 and Pythia, with favorable wall-clock scaling. The bounded-$\ell_\infty$ factorization is also notably robust: on stress configurations where QuIP variants diverge to four-digit perplexity (Pythia-6.9B) or abort with NaNs in LDL back-substitution (Mistral-7B), Kashin-DCT remains numerically stable and stays close to FP16 baseline. At inference time, each weight decomposes into two 2-bit factor codes per channel that are structurally suited to native-2-bit hardware.

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 4bpw, the rotated-codebook frontier method QTIP outperforms QAM-W; the contribution is the quality-preserving 5--6bpw band.