Fairy2i: Training Complex LLMs from Real LLMs with All Parameters in $\{\pm 1, \pm i\}$

• URL: http://arxiv.org/abs/2512.02901v2
• Date: Wed, 03 Dec 2025 14:15:05 GMT
• Title: Fairy2i: Training Complex LLMs from Real LLMs with All Parameters in $\{\pm 1, \pm i\}$
• Authors: Feiyu Wang, Xinyu Tan, Bokai Huang, Yihao Zhang, Guoan Wang, Peizhuang Cong, Tong Yang,
• Abstract summary: We present Fairy2i, a universal framework that transforms pre-trained real-valued layers into an equivalent widely-linear complex form.<n>We demonstrate that Fairy2i restores the performance of LLaMA-2 7B at an effective 2-bit precision to levels nearly comparable with full-precision baselines.
• Score: 11.09481608287424
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Large language models (LLMs) have revolutionized artificial intelligence, yet their massive memory and computational demands necessitate aggressive quantization, increasingly pushing representations toward the theoretical limit of a single bit. While complex-valued LLMs, such as iFairy, offer a superior chance for low-bit representation compared to real-valued counterparts, they require training from scratch, preventing the utilization of the vast ecosystem of pre-trained real-valued foundation models. Here we present Fairy2i, a universal framework that transforms pre-trained real-valued layers into an equivalent widely-linear complex form, enabling extremely low-bit quantization while reusing existing checkpoints. By proving a lossless mathematical equivalence between real and widely-linear maps, we convert standard Transformers into the complex domain and employ a phase-aware quantization scheme with a highly efficient codebook of fourth roots of unity. Furthermore, we introduce a recursive residual quantization mechanism that iteratively minimizes quantization error, allowing inference to proceed via efficient multiplication-free accumulation. We demonstrate that Fairy2i restores the performance of LLaMA-2 7B at an effective 2-bit precision to levels nearly comparable with full-precision baselines, significantly outperforming state-of-the-art real-valued binary and ternary quantization methods. This work bridges the gap between the representational efficiency of complex-valued arithmetic and the practical utility of pre-trained models, paving a new way for efficient inference on commodity hardware.

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