One Qubit Can Beat One Bit: Quantum Advantage for Post-Training Quantization

Post-training quantization (PTQ) compresses large language models by mapping weights to low-bit representations. The scaling factor that defines the quantization grid is typically chosen using simple, data-free heuristics. In this work, we present PiSO (Piecewise Scale Optimization), an algorithm that leverages calibration data to compute the optimal channel-wise weight scales exactly and efficiently under round-to-nearest quantization. PiSO partitions the scale search space into finitely many intervals on which the objective admits a closed-form minimizer. We extend PiSO to group-wise quantization via principled heuristics and propose effective strategies for interleaving scale optimization with error correction. Experiments on Llama and Qwen models across multiple model sizes and target weight bit-widths demonstrate consistent improvements in perplexity and downstream zero-shot accuracy, both standalone and combined with error correction. In particular, we observe increased benefits as the target bit-width narrows and quantization becomes more challenging.

Post-training quantization (PTQ) converts a trained full-precision model into low-bit weights without task-level retraining, while quantization-aware training (QAT) incorporates quantization into the training loop. Although PTQ is efficient and often accurate at moderate bitwidths, it can fail sharply at aggressive bitwidths; QAT is more expensive but can often recover the lost accuracy. We propose a unified geometric framework that explains both PTQ failure and QAT recovery. We model full-precision training as following a low-loss \emph{river} inside a wider \emph{valley}: a normal neighborhood of the river forms a nearly flat \emph{basin}, while leaving this basin incurs a sharp loss increase. When the quantization grid is comparable to the basin width, local PTQ objectives, including rounding and Hessian-based second-order reconstruction, can select a high-loss deployed quantized point outside the basin even when nearby low-loss quantized points exist. In this regime, straight-through-estimator-based QAT has a useful bias: it evaluates gradients at the deployed quantized weights while updating latent full-precision weights, causing the gradient to sense the valley wall and acquire an inward component that steers subsequent quantized iterates back into the basin. We formalize this mechanism through a local landscape model, construct a geometric PTQ failure mode, and prove finite-time QAT recovery under local quantizer-compatibility assumptions. Experiments across vision and language models under multiple neural-network quantization schemes corroborate the predicted basin-crossing failure of PTQ and the corresponding recovery mechanism of QAT.

Quantization can preserve predictive accuracy under low-bit deployment while silently breaking algorithmic recourse: an actionable change that flips a decision before quantization may fail after quantization, or become substantially more costly. We formalize counterfactual sensitivity under quantization through validity, cost, and direction stability, and introduce two metrics: Validity Drop (VD) and Counterfactual Recourse Gap (CRG) that reveal recourse failures invisible to accuracy. We propose Counterfactual-Faithful Quantization (CFQ), which trains quantizer parameters and mixed-precision bit allocation to preserve counterfactual behavior by enforcing the target outcome at teacher recourse points under a global bit budget. A margin-based analysis gives a sufficient condition for recourse transfer under bounded quantization perturbations. Experiments on Adult, German Credit, and COMPAS show that accuracy-matched baselines can significantly degrade recourse stability, while CFQ maintains accuracy and substantially improves VD and CRG across bit budgets.