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

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: $\mathbf{y} = \mathbf{Wx} = (\mathbf{WQ}^T)(\mathbf{Qx})$ for orthogonal $\mathbf{Q}$. However, these methods use fixed transforms--Hadamard matrices achieving optimal worst-case coherence $\mu = 1/\sqrt{n}$--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. 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 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 $\frac{n \log n}{2}$ 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--a negligible one-time cost. On LLaMA-2-7B with 2-bit quantization, ButterflyQuant achieves 15.4 perplexity versus 22.1 for QuaRot.