Weight-only post-training quantization (PTQ) pipelines such as AWQ ultimately commit to a deterministic rounding decision for every scalar weight, and recent discrete coordinate-descent quantizers refine those decisions one weight at a time. We show that both are blocked by the same obstacle: correlated activation channels create multi-spin energy barriers in the discrete rounding landscape — configurations in which no single rounding flip helps, yet a small group of coordinated flips strictly reduces the layer reconstruction error. We characterize these barriers exactly under the empirical activation metric and prove that they arise only from frustration between local quantization fields and cross-channel couplings. Building on this analysis we propose Transverse-Field Ising Correction (TFIC), a barrier-aware discrete optimizer that post-processes any AWQ-style pipeline. TFIC casts fixed-grid rounding as a classical Ising ground-state problem, selects the spins near a rounding boundary as a mobile pool, drives that pool to a single-flip local minimum by exact coordinate descent, and then crosses barriers with cluster moves that flip small groups of coupled spins together. Every accepted move is scored against the exact reconstruction energy, and a robust per-layer noise-floor acceptance rule prunes flips whose apparent gain is within calibration-estimation noise; together these guarantee that each layer’s calibration reconstruction error is monotonically non-increasing. TFIC requires no gradients and no architectural changes. On Mistral-7B and Llama-3-8B, applying TFIC on top of a 4-bit AWQ baseline consistently reduces WikiText-2 and C4 perplexity, at a per-layer cost of tens of seconds and with no inference-time overhead.
Weight-only post-training quantization (PTQ) pipelines such as AWQ ultimately commit to a deterministic rounding decision for every scalar weight, and recent discrete coordinate-descent quantizers refine those decisions one weight at a time. We show that both are blocked by the same obstacle: correlated activation channels create multi-spin energy barriers in the discrete rounding landscape — configurations in which no single rounding flip helps, yet a small group of coordinated flips strictly reduces the layer reconstruction error. We characterize these barriers exactly under the empirical activation metric and prove that they arise only from frustration between local quantization fields and cross-channel couplings. Building on this analysis we propose Transverse-Field Ising Correction (TFIC), a barrier-aware discrete optimizer that post-processes any AWQ-style pipeline. TFIC casts fixed-grid rounding as a classical Ising ground-state problem, selects the spins near a rounding boundary as a mobile pool, drives that pool to a single-flip local minimum by exact coordinate descent, and then crosses barriers with cluster moves that flip small groups of coupled spins together. Every accepted move is scored against the exact reconstruction energy, and a robust per-layer noise-floor acceptance rule prunes flips whose apparent gain is within calibration-estimation noise; together these guarantee that each layer’s calibration reconstruction error is monotonically non-increasing. TFIC requires no gradients and no architectural changes. On Mistral-7B and Llama-3-8B, applying TFIC on top of a 4-bit AWQ baseline consistently reduces WikiText-2 and C4 perplexity, at a per-layer cost of tens of seconds and with no inference-time overhead.
Post-training quantization (PTQ) is widely used to compress large language models without retraining. However, many existing weight-only methods rely on heuristic objectives and greedy rounding strategies, leading to noticeable degradation under low-bit quantization. We propose OJBKQ, a layer-wise PTQ method that formulates weight quantization as a joint optimization problem over activations and weights, yielding a multiple-right-hand-side box-constrained integer least squares (BILS) problem per layer. To solve this NP-hard problem, we extend Klein’s randomized variant of Babai’s nearest-plane algorithm to handle box constraints and run the extended algorithm K times to generate K box-constrained Klein points. The minimum-residual candidate among these points and the box-constrained Babai point is selected as a suboptimal solution. Experiments on large language models show that OJBKQ achieves lower perplexity at 3–4 bits than existing PTQ methods, while maintaining comparable computational cost.
Recent post-training quantization (PTQ) methods have adopted block rotations to diffuse outliers prior to rounding. While this reduces the overhead of online full-vector rotations, the effect of block structure on outlier suppression remains poorly understood. To fill this gap, we present the first systematic, non-asymptotic analysis of outlier suppression for block Hadamard rotations. Our analysis reveals that outlier suppression is fundamentally limited by the geometry of the input vector. In particular, in the deterministic worst case, post-rotation outliers are minimized when the pre-rotation ℓ1 norm mass is evenly distributed across blocks. Guided by these insights, we introduce PeRQ (Permute, Rotate, then Quantize), a PTQ framework that redistributes activation mass via permutations prior to rotation. We propose a greedy mass diffusion algorithm to calibrate permutations by equalizing the expected blockwise ℓ1 norms. To avoid adding inference overhead, we identify permutation-equivariant regions in transformer architectures to merge these permutations into model weights before deployment. Experiments show that PeRQ consistently improves accuracy across all block sizes, recovering up to 90% of the full-vector rotation perplexity when quantizing Llama3 1B to INT4 with block size 16, compared to 46% without permutations.
Post-training quantization (PTQ) is a primary approach for deploying large language models without fine-tuning, and the quantized performance is often strongly affected by the calibration in PTQ. By contrast, in vision-language models (VLMs), substantial differences between visual and text tokens in their activation distributions and sensitivities to quantization error pose significant challenges for effective calibration during PTQ. In this work, we rethink what PTQ calibration should align with in VLMs and propose the Token-level Importance-aware Layer-wise Quantization framework (TLQ). Guided by gradient information, we design a token-level importance integration mechanism for quantization error, and use it to construct a token-level calibration set, enabling a more fine-grained calibration strategy. Furthermore, TLQ introduces a multi-GPU, quantization-exposed layer-wise calibration scheme. This scheme keeps the layer-wise calibration procedure consistent with the true quantized inference path and distributes the complex layer-wise calibration workload across multiple RTX3090 GPUs, thereby reducing reliance on the large memory of A100 GPUs. TLQ is evaluated across two models, three model scales, and two quantization settings, consistently achieving performance improvements across all settings, indicating its strong quantization stability. The code will be released publicly.
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.
The training loss of binary and ternary neural networks, expanded as a multilinear polynomial over the discrete weights, is structurally equivalent to the energy of a hierarchical high-order Ising model. We exploit this equivalence to apply Ising solvers to discrete-weight network training, while accounting for a fundamental difference in objectives: an Ising solver seeks the ground state of a fixed energy function, whereas training seeks solutions that generalize to unseen data. Our key contribution is ΣΔ-Sampler, an optimizer that reconciles these objectives by embedding a stochastic sampling-and-annealing step as a noisy quantizer inside a first-order ΣΔ noise-shaping loop. The loop pushes noise from weight quantization, high-order frustration, and mini-batch sampling out of the low-frequency adaptation band, so that discrete weight updates follow the smooth descent trajectories embedded in the rugged loss landscape. We validate the framework on a planted quadratic unconstrained binary optimization (QUBO) benchmark with a known minimizer; on CIFAR-10, where a 13M-parameter ternary model comes within 1% of full-precision accuracy; on ImageNet-1k, where a ternary ResNet-18 comes within 0.4% Top-5 accuracy of a ternary baseline; on class-incremental continual learning, where prior-task accuracy is preserved; and on MatMul-free language models up to 2.7B parameters, where it outperforms a straight-through-estimator baseline under matched continued pre-training. Compared with standard quantization-aware training, ΣΔ-Sampler reduces the training memory footprint by over 50%, delivers an order-of-magnitude speed-up, and reduces weight-update memory traffic by four orders of magnitude, making it attractive for on-device training and fine-tuning of large binary and ternary models.
Maximum-likelihood decoding (MLD) achieves the minimum logical error rate of stabilizer codes under known i.i.d. Pauli noise, but its exact evaluation is \#P-hard. Practical pipelines therefore approximate MLD by minimum-weight decoding (MWD), retaining only the lowest-weight recovery per syndrome and discarding the coset degeneracy. The minimum-weight search is in turn implemented by stochastic solvers. We introduce approximate maximum-likelihood decoding (AMLD), a black-box framework that recycles the candidate samples discarded by stochastic inner decoders into a per-class truncated free-energy estimator. For every logical class represented in the candidate pool, the estimator is provably bounded below by the exact free energy and above by the empirical minimum weight. AMLD returns the logical class minimizing the estimated free energy with linear classical overhead. In SA-based Ising-decoder benchmarks, AMLD closes up to 83% of the MWD--MLD threshold gap across the toric and color codes under bit-flip and depolarizing noise. The largest threshold improvement, from 17.28% to 18.62%, occurs on the 6.6.6 color code under depolarizing noise. We further demonstrate AMLD on the [[144,12,12]] bivariate-bicycle code, whose bit-flip decoding problem has a hypergraph structure. This application requires neither matching-based enumeration nor code-specific tensor-network contraction. At p=0.05, AMLD reduces the logical error rate by 13% relative to MWD evaluated on the same BP-OSD candidate pool.