Abstract: Post-training quantization of Large Language Models (LLMs) is challenging. In this work, we introduce **L**ow-rank **Q**uantization **E**rror **R**eduction (LQER), which combines quantization and low-rank approximation to recover the model capability. LQER leverages an activation-induced scale matrix to drive the singular value distribution of quantization error towards a desirable distribution, which enables nearly-lossless W4A8 quantization on various LLMs and downstream tasks without the need for knowledge distillation, grid search, or gradient-based iterative optimization. Unlike existing methods, the computation pattern of LQER eliminates the need for specialized Scatter and Gather processes to collect high-precision weights from irregular memory locations. Our W4A8 LLMs achieve near-lossless performance on six popular downstream tasks, while using $1.36 \times$ fewer hardware resources than the leading state-of-the-art method. We will open-source our framework at [https://github.com/ChengZhang-98/lqer](https://github.com/ChengZhang-98/lqer)