# KBBQ: A Predictive Noise Law and the Limits of Spectrum Flattening in FP4 Quantization

arXiv: https://arxiv.org/abs/2609.08135 · HTML: https://mlsanity.com/paper/2609.08135/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 34 references that are in the map.
- First posted on arXiv 8 Sep 2026.

## Builds on (most cited first, 25 of 25)

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- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
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- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [QuIP#: Even Better LLM Quantization with Hadamard Incoherence and Lattice Codebooks](https://mlsanity.com/paper/2402.04396/) (2024)
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- [Pretraining Large Language Models with NVFP4](https://mlsanity.com/paper/2509.25149/) (2025)
- [FP4 All the Way: Fully Quantized Training of LLMs](https://mlsanity.com/paper/2505.19115/) (2025)
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- [HALO: Hadamard-Assisted Lower-Precision Optimization for LLMs](https://mlsanity.com/paper/2501.02625/) (2025)
- [WUSH: Near-Optimal Adaptive Transforms for LLM Quantization](https://mlsanity.com/paper/2512.00956/) (2025)
- [Full-Stack FP4: Stable LLM Pretraining with Quantized Projections, Optimizers, and Attention](https://mlsanity.com/paper/2607.04422/) (2026)
- [Dissecting Quantization Error: A Concentration-Alignment Perspective](https://mlsanity.com/paper/2603.04359/) (2026)
- [HARP: Hadamard-Preconditioned Adaptive Rotation Processor for Extreme LLM Quantization](https://mlsanity.com/paper/2605.29843/) (2026)
- [TORQ: Two-Level Orthogonal Rotation for MXFP4 Quantization](https://mlsanity.com/paper/2605.19561/) (2026)
- [KronQ: LLM Quantization via Kronecker-Factored Hessian](https://mlsanity.com/paper/2607.07964/) (2026)
- [ConQuR: Corner Aligned Activation Quantization via Optimized Rotations for LLMs](https://mlsanity.com/paper/2605.10793/) (2026)

## Abstract (arXiv)

We develop a second-order theory of quantization noise in matrix multiplication in which the quantization format is characterized by the variance it assigns to each element. The constant variance profile of integer quantization recovers existing integer-noise theory, while the multiplicative profile of floating-point rounding reduces the data dependence to a scalar, the participation factor kappa, yielding a closed-form signal-to-noise-ratio law. The resulting functional also admits a closed-form upper bound kappa* that no function-preserving linear transform can exceed and that is attained by a recent state-of-the-art method. Building on this analysis, we introduce KBBQ (Kappa-Braked Blockwise Quantization), which parameterizes the extent to which a transform approaches this ceiling. At W4A4, across four base models and two FP4 formats, KBBQ outperforms the prior state of the art without additional deployment-time computation.