# LoaQ: Layer-wise Output Approximation Quantization

arXiv: https://arxiv.org/abs/2509.06297 · HTML: https://mlsanity.com/paper/2509.06297/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 26 references that are in the map.
- First posted on arXiv 8 Sep 2025.

## Builds on (most cited first, 18 of 18)

- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference](https://mlsanity.com/paper/1712.05877/) (2017)
- [AWQ: Activation-aware Weight Quantization for LLM Compression and Acceleration](https://mlsanity.com/paper/2306.00978/) (2023)
- [QuaRot: Outlier-Free 4-Bit Inference in Rotated LLMs](https://mlsanity.com/paper/2404.00456/) (2024)
- [Quantizing deep convolutional networks for efficient inference: A whitepaper](https://mlsanity.com/paper/1806.08342/) (2018)
- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [GPTAQ: Efficient Finetuning-Free Quantization for Asymmetric Calibration](https://mlsanity.com/paper/2504.02692/) (2025)
- [Quantization Error Propagation: Revisiting Layer-Wise Post-Training Quantization](https://mlsanity.com/paper/2504.09629/) (2025)
- [MagR: Weight Magnitude Reduction for Enhancing Post-Training Quantization](https://mlsanity.com/paper/2406.00800/) (2024)
- [CDQuant: Greedy Coordinate Descent for Accurate LLM Quantization](https://mlsanity.com/paper/2406.17542/) (2024)
- [NeUQI: Near-Optimal Uniform Quantization Parameter Initialization for Low-Bit LLMs](https://mlsanity.com/paper/2505.17595/) (2025)

## Built on by (most cited first, 5 of 5)

- [LPCD: Unified Framework from Layer-Wise to Submodule Quantization](https://mlsanity.com/paper/2512.01546/) (2025)
- [OneComp: One-Line Revolution for Generative AI Model Compression](https://mlsanity.com/paper/2603.28845/) (2026)
- [One Qubit Can Beat One Bit: Quantum Advantage for Post-Training Quantization](https://mlsanity.com/paper/2608.05240/) (2026)
- [From Sweep to Seam: Interleaved Cross-Block Post-Training Quantization](https://mlsanity.com/paper/2608.09595/) (2026)
- [SchurQuant: Groupwise Discrete Optimization for Layer-Wise LLM Quantization](https://mlsanity.com/paper/2608.15567/) (2026)

## Abstract (arXiv)

A natural and intuitive idea in model quantization is to approximate each component's quantized output to match its original. Motivated by this idea, most layer-wise post-training quantization (PTQ) methods focus on weight approximation at the linear-layer level. As a result, this local objective often yields insufficient approximations and practical deviations from the guiding intuition. Recent work has improved the approximation of linear-layer outputs within the layer-wise PTQ framework, but such refinements remain inadequate for achieving alignment with the full-model output. Based on a deeper understanding of the structure of mainstream LLMs, we propose LoaQ, which incorporates output-matching factors when quantizing linear layers within the layer-wise PTQ framework. It better aligns with this intuition and can feature a simple closed-form solution, making it orthogonal to existing techniques and readily integrable into existing quantization pipelines. Experiments on the LLaMA and Qwen model families demonstrate that LoaQ performs effectively in both weight-only and weight-activation quantization. By integrating seamlessly with existing quantization strategies, it further enhances overall quantization quality and shows strong potential to advance the frontier of post-training quantization.