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arXiv:2605.10655

BCJR-QAT: A Differentiable Relaxation of Trellis-Coded Weight Quantization

Venugopalan Iyengar

Published May 11, 2026• 0 citations
arXivHTMLPDF

Abstract

Trellis-coded quantization sets the current 2-bit post-training frontier for LLMs (QTIP), but pushing below the PTQ ceiling requires quantization-aware training, and QAT on a trellis is obstructed by the non-differentiable Viterbi argmax. We introduce BCJR-QAT, a relaxation that replaces the argmax with the BCJR forward-backward sum-product algorithm at temperature $T$, producing a soft codeword equal to the Boltzmann expectation over trellis paths, exactly differentiable, recovering the hard QTIP code as $T \to 0$, and mathematically identical to the transfer-matrix computation for a 1D Ising-like spin chain. We contribute (i) a fused Triton kernel making BCJR tractable on a single consumer GPU ($6.57\times$ speedup, fp32 parity); (ii) a quantitative drift-budget theory of when BCJR-QAT can escape the QTIP-PTQ Voronoi basin, verified across four experiments; and (iii) a positive empirical result on Llama-3.2-1B at 2 bpw under end-to-end forward-KL distillation: with the right schedule (skip the high-$T$ phase to avoid an overshoot we diagnose), single-layer BCJR-QAT beats QTIP-PTQ by $\mathbf{-0.084}$ PPL on WikiText-2, and multi-layer compounding is super-additive.

DOI
10.48550/arxiv.2605.10655

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Trellis-coded quantization sets the current 2-bit post-training frontier for LLMs (QTIP), but pushing below the PTQ ceiling requires quantization-aware training, and QAT on a trellis is obstructed by the non-differentiable Viterbi argmax. We introduce BCJR-QAT, a relaxation that replaces the argmax with the BCJR forward-backward sum-product algorithm at temperature TTT, producing a soft codeword equal to the Boltzmann expectation over trellis paths, exactly differentiable, recovering the hard QTIP code as T→0T \to 0T→0, and mathematically identical to the transfer-matrix computation for a 1D Ising-like spin chain. We contribute (i) a fused Triton kernel making BCJR tractable on a single consumer GPU (6.57×6.57\times6.57× speedup, fp32 parity); (ii) a quantitative drift-budget theory of when BCJR-QAT can escape the QTIP-PTQ Voronoi basin, verified across four experiments; and (iii) a positive empirical result on Llama-3.2-1B at 2 bpw under end-to-end forward-KL distillation: with the right schedule (skip the high-TTT phase to avoid an overshoot we diagnose), single-layer BCJR-QAT beats QTIP-PTQ by −0.084\mathbf{-0.084}−0.084 PPL on WikiText-2, and multi-layer compounding is super-additive.
The best two-bit weight quantizers for large language models, such as QTIP and Proteus, rotate each weight matrix by a random orthogonal transform, which must be undone at every decoding step, then encode it with a trellis or lattice code under a Euclidean search; the layer Hessian enters only through error feedback between coding blocks. We show that this leaves part of the Hessian unused. Error feedback turns the loss into a weighted sum of per-coordinate rounding errors whose weights, the diagonal of the Hessian's LDL factorization, existing quantizers compute but never read. We put these weights into the Viterbi branch metric, so the search follows the curvature within each coding block. This also explains the rotation: it removes this within-block variation, so weighting in the native basis and rotating are substitutes. On three models the weighted native search matches a full-dimension randomized Hadamard to within about one point of downstream accuracy, and weighting after the rotation gains little. Around this search we build CurveTQ, a trellis codec with no rotation, which handles the weights' amplitude and marginal shape with a factored scale field and a closed-form quantile table, and stores a start state per coding block so the trellis can adapt to the residual that error feedback carries into it. At two bits CurveTQ is 1-3 points higher in mean downstream accuracy than QTIP and Proteus on three 4-8B Instruct models, even after both are given our start state, which alone lifts either baseline by 1-3 points. It also leads on a 35B mixture of experts, to our knowledge the first trellis-coded result on such a model. With no rotation to undo, our decoder is the fastest of the three at all tested batch sizes and bit widths.
Low-bit quantization reduces the memory footprint of large language models, but preserving accuracy with efficient training remains challenging. Scalar quantization encodes weights independently, while explicit-codebook vector quantization introduces costly search during training. We propose E8-QAT, a 2-bit quantization-aware training framework that uses the E8E_8E8​ lattice as an implicit eight-dimensional codebook. It jointly quantizes weight groups through arithmetic projection without storing or searching an explicit codebook during the training forward pass. A 2-bit core encoding and sparse extension preserve the selected lattice points exactly, adding less than 0.1\% to core index storage. On Qwen3.6-35B-A3B, E8-QAT matches the performance of the full-precision model in terms of the average score across the four tasks. E8-QAT improves over the state-of-the-art baseline by 2.03\% on IFEval and by 1.51\% on LiveCodeBench, under the same data, schedule, and bit width. The 2-bit representation reduces peak memory by 80\% and supports a 4×4\times4× larger batch. At each representation's largest feasible batch, E8-QAT reaches 1.97×1.97\times1.97× the decoding throughput of BF16.
One-bit post-training quantization represents each weight using only its sign, requiring all deployment contexts to share the same binary weight matrix even when their activation statistics favor different sign patterns. We study this shared-sign constraint and introduce Quantum Random Access Quantization (QRAQ). This framework encodes context-dependent signs in a quantum random-access code and retrieves them via context-matched Pauli measurements. Under an explicit fresh-copy logical readout model, QRAQ produces an unbiased, context-specific binary surrogate with a tractable shot-noise penalty. We prove a row-wise separation from shared-sign one-bit PTQ with signed per-row scales. When the optimal context-wise signs are incompatible, QRAQ achieves a strictly lower ideal reconstruction risk. We also derive finite-shot and calibrated-noise conditions under which this separation is retained. Fixed-readout quantum schemes are classically simulable, so the relevant resource in this model is measurement incompatibility rather than quantization alone. Finally, we characterize the role of scale granularity, provide finite-sample certificates, and evaluate the predicted ideal, finite-shot, noisy, and multi-context regimes in simulator experiments.
Quantizing the weights of large language models (LLMs) from 16-bit to lower bitwidth is the de facto approach to deploy massive transformers onto more affordable accelerators. While GPTQ emerged as one of the standard methods for one-shot post-training quantization at LLM scale, its inner workings are described as a sequence of algebraic updates that obscure geometric meaning or worst-case guarantees. In this work, we show that, when executed back-to-front (from the last to first dimension) for a linear layer, GPTQ is mathematically identical to Babai's nearest plane algorithm for the classical closest vector problem (CVP) on a lattice defined by the Hessian matrix of the layer's inputs. This equivalence is based on a sophisticated mathematical argument, and has two analytical consequences: first, the GPTQ error propagation step gains an intuitive geometric interpretation; second, GPTQ inherits the error upper bound of Babai's algorithm under the assumption that no weights are clipped. Leveraging this bound, we design post-training quantization methods that avoid clipping, and outperform the original GPTQ. In addition, we provide efficient GPU inference kernels for the resulting representation. Taken together, these results place GPTQ on a firm theoretical footing and open the door to importing decades of progress in lattice algorithms towards the design of future quantization algorithms for billion-parameter models. Source code is available at https://github.com/IST-DASLab/GPTQ-Babai.
Leech-lattice quantization gives good quality at two bits per weight, but its codebooks hold more than 10^14 points, too many for a lookup table. Our earlier kernel expanded the codes at load time and read 4.804 bits per weight from GPU memory for 2 bits of code. We present Tetra, a new codebook on the same lattice. A 24-weight block still takes 48 bits, most of which index a 64-state trellis of the Golay code and one shared 16 KiB table. The kernel decodes a block with six table loads and two small lookups inside the matrix-vector product, and reads 2.148 bits per weight. For full models, we retrain one scale per matrix row, store the matrices that lose the most as 4-bit integers, and pay for them with 4-bit embedding tables. Our Qwen3-4B, 8B and 14B files hold 2.73, 2.70 and 2.73 bits per parameter over the whole model. They score 63.37, 69.58 and 75.66 on the full MMLU test set, 4.76, 4.21 and 2.46 points below 4-bit AWQ at 5.3 to 6.0 bits per parameter. They generate 113.8, 95.0 and 57.2 tokens per second in our engine. On GSM8K, through the served kernel, they lose 9.63, 4.62 and 3.26 points to FP16. At 4B our file scores 23.6 points above llama.cpp's IQ2_XXS (2.48 bits per parameter). Every number we measured for a table or figure comes from one NVIDIA L40S GPU. We preregistered the main experiments.
The growing need for privacy protection, instant response, and offline access has pushed large language model inference from the cloud to local edge devices. However, decoder-only LLMs remain hard to run on smartphones and embedded systems, as their performance is limited not just by parameter count, but also by memory bandwidth, hardware operator compatibility, and the rising cost of long-context generation. Post-training quantization (PTQ) is the most viable approach today, as it shrinks model size and speeds up inference without full retraining. Yet most PTQ methods are tested on cloud GPUs, and their claimed improvements rarely hold up on real edge hardware. This survey reviews on-device PTQ for decoder-only LLMs from a hardware-focused angle, covering static and adaptive schemes, key hardware bottlenecks, evaluation flaws, and KV cache scaling issues. It concludes that model size alone is a poor measure of edge efficiency, and future work must better align quantization algorithms, runtime backends, and on-device evaluation standards.