Fairy2i-W2

🔗 Links

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 ({±1, ±i}). 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-W2 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.

Method

Fairy2i-W2 consists of three key components:

Widely-Linear Transformation

We transform pre-trained real-valued linear layers into an equivalent widely-linear complex form without altering the model's behavior. Each real linear layer R (a real matrix of size 2n×2m) is reparameterized into two complex matrices U and W (each of size n×m) such that y = Ux + Wx̅, where x̅ denotes the complex conjugate of x. This transformation is lossless and unique, preserving the original forward computation before quantization.

Phase-Aware Complex Quantization

We quantize complex weights using a phase-based scheme with the codebook {±1, ±i} (fourth roots of unity). For each complex weight, we project it to the nearest codeword by angle and apply axis-wise scaling factors. During QAT training, we maintain full-precision master weights and use quantized copies in the forward pass with straight-through estimator (STE) gradients.

Recursive Residual Quantization

To further reduce quantization error, we recursively quantize the residual error. Each complex weight is represented as a sum of low-bit terms: W_q ≈ Σ W^(t) (sum over t from 0 to T-1), where each term is quantized using the same phase-aware mechanism. For Fairy2i-W2 (T=2), we use 2 recursive stages, achieving an effective 2 bits per real parameter.

Evaluation

Main Results on LLaMA-2 7B

Method	Bits	C4 PPL↓	ARC-e	ARC-c	HellaSwag	PIQA	Winogrande	Avg.
LLaMA-2 (FP16)	16	6.63	75.59	43.17	57.06	77.91	69.85	64.72
Fairy2i-W2	2	7.85	72.73	39.76	53.33	76.17	68.03	62.00
AQLM	2	8.54	63.68	32.76	49.55	74.76	65.67	57.28
QuIP#	2	11.01	55.56	28.84	42.94	71.38	62.43	52.23
Real-Ternary (QAT)	1.58	11.06	55.93	24.15	38.43	69.80	55.17	48.70
Fairy2i-W1	1	11.03	56.56	24.82	38.19	70.08	53.67	48.66
Real-Binary (QAT)	1	11.75	53.32	22.70	35.57	66.81	52.64	46.21
GPTQ	3	10.61	58.46	31.06	45.21	71.49	59.19	53.08

Key Results:

• Fairy2i-W2 (2-bit) achieves a perplexity of 7.85, closing the gap to FP16 (6.63) while outperforming all 2-bit PTQ methods
• Fairy2i-W2 achieves 62.00% average accuracy on zero-shot tasks, highly competitive with FP16 (64.72%)
• Fairy2i-W1 (1-bit) outperforms real-valued binary and ternary baselines at the same or lower bit budgets

Quick Start

Fairy2i-W2 is based on LLaMA-2 7B architecture, with only the linear layers replaced by complex-valued QAT layers. The model structure is otherwise identical to LLaMA-2.

Installation

pip install torch transformers safetensors huggingface_hub

Loading the Model

Please refer to load_model.py for detailed implementation. Basic usage:

from load_model import load_model

# Load Fairy2i-W2 model
model, tokenizer = load_model()

# The model is ready to use!
prompt = "Hello, how are you?"
inputs = tokenizer(prompt, return_tensors="pt").to(model.device)

with torch.no_grad():
    outputs = model.generate(
        **inputs,
        max_new_tokens=50,
        do_sample=True,
        temperature=0.7
    )

response = tokenizer.decode(outputs[0], skip_special_tokens=True)
print(response)

Model Details

• Base Model: LLaMA-2 7B
• Quantization Method: Complex-Phase V2 (2-step recursive residual quantization)
• Effective Bit Width: 2 bits per real parameter
• Codebook: {±1, ±i} (fourth roots of unity)
• Training: QAT (Quantization-Aware Training) on 30B tokens from RedPajama dataset

Files in Repository

• load_model.py: Model loading script
• qat_modules.py: QAT linear layer implementations
• quantization.py: Quantization functions (PhaseQuant, BitNet, etc.)
• config.json: Model configuration (identical to LLaMA-2 7B)
• model.safetensors.index.json: Weight file index
• model-0000X-of-00003.safetensors: Sharded model weights
• Tokenizer files: tokenizer.json, tokenizer_config.json, etc.

Citation

If you use Fairy2i-W2 in your research, please cite:

@article{wang2025fairy2i,
  title={Fairy2i: Training Complex LLMs from Real LLMs with All Parameters in {±1, ±i}},
  author={Wang, Feiyu and Tan, Xinyu and Huang, Bokai and Zhang, Yihao and Wang, Guoan and Cong, Peizhuang and Yang, Tong},
  journal={arXiv preprint},
  year={2025}
}

License

This model follows the same license as LLaMA-2. Please refer to the original LLaMA-2 license for details.

Contact

For questions or issues, please contact: [email protected]