# ButterflyQuant: Ultra-low-bit LLM Quantization through Learnable Orthogonal Butterfly Transforms

arXiv: https://arxiv.org/abs/2509.09679 · HTML: https://mlsanity.com/paper/2509.09679/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 59 references that are in the map.
- First posted on arXiv 11 Sep 2025.

## Builds on (most cited first, 30 of 43)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [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)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [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)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [LLM.int8(): 8-bit Matrix Multiplication for Transformers at Scale](https://mlsanity.com/paper/2208.07339/) (2022)
- [QuaRot: Outlier-Free 4-Bit Inference in Rotated LLMs](https://mlsanity.com/paper/2404.00456/) (2024)
- [SpinQuant: LLM quantization with learned rotations](https://mlsanity.com/paper/2405.16406/) (2024)
- [OmniQuant: Omnidirectionally Calibrated Quantization for Large Language Models](https://mlsanity.com/paper/2308.13137/) (2023)
- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [Hyena Hierarchy: Towards Larger Convolutional Language Models](https://mlsanity.com/paper/2302.10866/) (2023)
- [QuIP#: Even Better LLM Quantization with Hadamard Incoherence and Lattice Codebooks](https://mlsanity.com/paper/2402.04396/) (2024)
- [SpQR: A Sparse-Quantized Representation for Near-Lossless LLM Weight Compression](https://mlsanity.com/paper/2306.03078/) (2023)
- [Massive Activations in Large Language Models](https://mlsanity.com/paper/2402.17762/) (2024)
- [Extreme Compression of Large Language Models via Additive Quantization](https://mlsanity.com/paper/2401.06118/) (2024)
- [Outlier Suppression: Pushing the Limit of Low-bit Transformer Language Models](https://mlsanity.com/paper/2209.13325/) (2022)
- [DuQuant: Distributing Outliers via Dual Transformation Makes Stronger Quantized LLMs](https://mlsanity.com/paper/2406.01721/) (2024)
- [LLM Inference Unveiled: Survey and Roofline Model Insights](https://mlsanity.com/paper/2402.16363/) (2024)
- [Quantizable Transformers: Removing Outliers by Helping Attention Heads Do Nothing](https://mlsanity.com/paper/2306.12929/) (2023)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [FlatQuant: Flatness Matters for LLM Quantization](https://mlsanity.com/paper/2410.09426/) (2024)
- [EfficientQAT: Efficient Quantization-Aware Training for Large Language Models](https://mlsanity.com/paper/2407.11062/) (2024)

## Built on by (most cited first, 10 of 10)

- [Pushing the Limits of Block Rotations in Post-Training Quantization](https://mlsanity.com/paper/2601.22347/) (2026)
- [HARP: Hadamard-Preconditioned Adaptive Rotation Processor for Extreme LLM Quantization](https://mlsanity.com/paper/2605.29843/) (2026)
- [ReSpinQuant: Efficient Layer-Wise LLM Quantization via Subspace Residual Rotation Approximation](https://mlsanity.com/paper/2604.11080/) (2026)
- [OptRot: Mitigating Weight Outliers via Data-Free Rotations for Post-Training Quantization](https://mlsanity.com/paper/2512.24124/) (2025)
- [Bielik-Q2-Sharp: A Comparative Study of Extreme 2-bit Quantization Methods for a Polish 11B Language Model](https://mlsanity.com/paper/2603.04162/) (2026)
- [Influence-Inspired Spectral Rotations for Extreme Low-Bit LLM Quantization](https://mlsanity.com/paper/2605.25203/) (2026)
- [When Quantization Is Free: An int4 KV Cache That Outruns fp16 on Apple Silicon](https://mlsanity.com/paper/2605.05699/) (2026)
- [When Local Variance Optimality Is Not Enough: RoPE-Aligned Q/K Rotations for Dynamic 4-Bit Quantisation](https://mlsanity.com/paper/2608.13365/) (2026)
- [Transforms for LLM Quantization: The Great Inversion and Format Co-Design](https://mlsanity.com/paper/2608.25188/) (2026)
- [SpectrumQuant: Spectrum Alignment for LLM Quantization](https://mlsanity.com/paper/2610.20303/) (2026)

## Abstract (arXiv)

Large language models require massive memory footprints, severely limiting deployment on consumer hardware. Quantization reduces memory through lower numerical precision, but extreme 2-bit quantization suffers from catastrophic performance loss due to outliers in activations. Rotation-based methods such as QuIP and QuaRot apply orthogonal transforms to eliminate outliers before quantization, using computational invariance: y = Wx = (WQT)(Qx) for orthogonal Q. However, these methods use fixed transforms--Hadamard matrices achieving optimal worst-case coherence μ = 1/√n--that cannot adapt to specific weight distributions. We identify that different transformer layers exhibit distinct outlier patterns, motivating layer-adaptive rotations rather than one-size-fits-all approaches. In this work, we propose ButterflyQuant, which replaces Hadamard rotations with learnable butterfly transforms parameterized by continuous Givens rotation angles. Unlike Hadamard's discrete +1, -1 entries that are non-differentiable and thus prohibit gradient-based learning, butterfly transforms' continuous parameterization enables smooth optimization while guaranteeing orthogonality by construction. This orthogonal constraint ensures theoretical guarantees in outlier suppression while achieving O(n log n) computational complexity with only fracn log n2 learnable parameters. We further introduce a uniformity regularization on post-transformation activations to promote smoother distributions amenable to quantization. Learning requires only 128 calibration samples and converges in minutes on a single GPU.