Training large language models (LLMs) is a computationally and memory-intensive task, often requiring vast resources. Imagine training a powerful 7-billion parameter model on a single consumer-grade graphics card like the NVIDIA RTX 4060 Ti, which has only 16GB of memory. This seemingly impossible feat has now become a reality thanks to a groundbreaking new technique called Q-GaLore. Traditional methods like full-parameter training or even existing low-rank approaches like GaLore struggle with the immense memory demands of LLMs. Q-GaLore tackles this challenge head-on by combining quantization and low-rank gradient projection. Quantization cleverly reduces the precision of model weights and other components without significantly sacrificing performance. Meanwhile, low-rank projection leverages the inherent redundancy in LLM gradients, optimizing in a smaller, more memory-efficient subspace. Q-GaLore builds upon GaLore by intelligently adapting the frequency of subspace updates. It recognizes that some layers in an LLM converge quickly, while others evolve more dynamically. By selectively updating subspaces based on their convergence behavior, Q-GaLore further reduces computational overhead and latency. The results are impressive. Q-GaLore enables full-parameter training of a LLaMA-7B model from scratch on a single RTX 4060 Ti, matching the performance of its full-rank counterpart. Not just limited to pre-training, Q-GaLore shines in fine-tuning scenarios as well, outperforming other popular methods like LoRA and QLoRA while using considerably less memory. This breakthrough opens doors for researchers and developers with limited resources, democratizing access to large-scale language model training and potentially accelerating innovation in the field.