The Math Behind E8 Lattice Quantization (with Code)

The Math Behind E8 Lattice Quantization (with Code) Standard scalar quantization — what every LLM quantizer from GPTQ to AWQ does — rounds each number independently to the nearest representable value. E8 lattice quantization rounds groups of 8 numbers jointly to the nearest point on a mathematical lattice. The difference sounds subtle. It isn't. This post is a complete walkthrough of how E8 quantization works, why it beats scalar quantization by ~30% in distortion, and exactly what the algorithm does line by line. Why Lattices? The core problem in quantization is sphere packing. You want to cover n-dimensional space with the fewest representable points, such that any real vector is "close" to at least one codebook entry. For 1D scalar quantization, you're placing points on a number line. Easy — evenly space them. For 8D vector quantization, you want to pack 8D balls as densely as possible. The densest known packing in 8 dimensions is the E8 root lattice, proven optimal by Maryna Viazov