---
license: llama2
base_model: meta-llama/Llama-2-7b-hf
tags:
  - llama-2
  - quantization
  - qat
  - complex-valued
  - 2-bit
  - text-generation
  - recursive
  - safetensors
language:
  - en
pipeline_tag: text-generation
---

# Fairy2i-W2

**🔗 Links**

[![Paper](https://img.shields.io/badge/Paper-arXiv-red?logo=arxiv&logoColor=white)](https://arxiv.org/abs/2512.02901) [![GitHub](https://img.shields.io/badge/GitHub-181717?logo=github&logoColor=white)](https://github.com/PKULab1806/Fairy2i-W2) [![ModelScope](https://img.shields.io/badge/ModelScope-624AFF?logoColor=white)](https://modelscope.cn/models/PKULab1806/Fairy2i-W2)

## 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

```bash
pip install torch transformers safetensors huggingface_hub
```

### Loading the Model

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

```python
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:

```bibtex
@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: tanxinyu330@gmail.com