Abstract: Current post-training quantization methods for LLMs compress the weights down to 4-bits, with moderate to low degradation in accuracy. However, further reducing the number of bits or accelerating the network while avoiding large accuracy drops, especially for smaller, sub 7B models, remains an actively researched and open problem. To address this, in this work, we introduce Quantization with Binary Bases (QBB), a new approach for low-bit quantization that effectively removes (nearly) all multiplications, reducing the implementation to summations. Our novel approach works by decomposing the original weights into a set of binary (1-bit) matrices using an iterative process. For a given layer, starting from a weight matrix, we first construct an initial approximation using an analytical solution, where each new binary matrix, paired with a scaling vector, approximates the residual error of the previous estimation. Secondly, using gradient descent and a progressive learning curriculum, we find the optimal set of binary matrices and scaling vectors that minimize the $\ell_2$ distance between the produced approximation and original weights. Thirdly, as previous steps are input agnostic, we holistically optimize the scaling vectors alone, calibrating them in student-teacher fashion, with the teacher providing both the data, by autoregressive generation starting from a random token, and the target logits. When evaluated across multiple LLM families, our approach matches and outperforms all prior works, setting a new state-of-the-art result using a summation-only based approach.