Qronos: Correcting the Past by Shaping the Future... in Post-Training Quantization
Authors: Shihao Zhang
Haoyu Zhang
Ian Colbert
Rayan Saab
Venue: ICLR 2026 Poster (2026)
Keywords: Post Training Quantization
Large Language Models
Foundation Models
Efficient Machine Learning
Primary area: other topics in machine learning (i.e., none of the above)

Abstract: We introduce Qronos---a new post-training quantization algorithm that not only explicitly corrects errors due to both weight and activation quantization, but also corrects errors accumulated from previously quantized layers. Our iterative algorithm is based on an interpretable and disciplined optimization framework that surpasses existing data-driven approaches. At each step, Qronos alternates between error correction and diffusion via optimal update rules. Importantly, we prove that Qronos admits an equivalent formulation that significantly improves algorithmic efficiency; we use our discovery to reduce peak memory usage by 18\times on Llama3 8B, and our scaling analysis shows a speedup of up to  13.8\times for a single-layer microbenchmark. We demonstrate compatibility with existing transformation techniques such as Hadamard-based incoherence processing and weight-activation scaling equalization, among others. We evaluate Qronos using recent language models in the Llama3 and Qwen3 families; Qronos consistently outperforms previous state-of-the-art adaptive rounding methods when quantizing the weights, activations, and/or KV caches to 4 bits or fewer.

Published as a conference paper at ICLR 2026
QRONOS: CORRECTING THE PAST BY SHAPING THE FUTURE... IN POST-TRAINTS TRAINING QUANTIZATION
Shihao Zhang*
Haoyu Zhang
Department of Mathematics
Department of Mathematics University of California, San Diego
University of California, San Diego
102201@@ucs..o
ha2053 www.euccdeee.......
Ian Colbert
Rayan Saab
Software Architecture Advanced Micro Devices, Inc. icolbert amd. com
Department of Mathematics HDSI University of California, San Diego SeaabBuesd.. edu
ABSTRACT
We introduce Qronos, a new post-training quantization algorithm that not only explicitly corrects errors due to both weight and activation quantization, but also corrects errors accumulated from previously quantized layers. Our iterative al- gorithm is based on an interpretable and disciplined optimization framework that surpasses existing data-driven approaches. At each step, Qronos alternates be- tween error correction and diffusion via optimal update rules. Importantly, we prove that Qronos admits equivalent formulation that significantly improves algorithmic efficiency; we use our discovery to reduce peak memory usage by 18x Llama3 8B, and our scaling analysis shows a speedup of up to 13.8x for single-layer microbenchmark. We demonstrate compatibility with existing transformation techniques such as Hadamard- bassed incoherence processing and weight-activation scaling equalization, among others. We evaluate Qronos using recent language models in the Ilama3 and Qwen3 families: Qronos consistently outperforms previous state-of-le the-ttr-rrt adaptive rounding methods when quantizing the weights, activations, and/or KV caches to bits or fewer.
INTRODUCTION
Recent advances in post-training quantization (PTQ) have enabled the practical use of few-bit weights and activations for large language model (LLM) inference, typically by focusing on one or both aspects of the quantization pipeline, visualized in Figure [ The first aspect involves mod- ifying the weights and activations of a model to make them more amenable to quantization, often through transformations that exploit invariances within the compute graph. The second aspect more directly concerns the design of the quantization mapping itself. It involves using data to minimize quantization error by either calibrating the quantization grid, which is defined by bit width, scaling factor, and zero point, or adaptively rounding the (potentially transformed) weights.
The latest innovations in PTQ, including Ashkboos al. (2024); Liu al. (2025), among many others, are skewed towards proposing and improving transformations that address the quantization challenges exacerbated in LLMs. These studies often only consider round-to-neane OPTQ (Frantar et al. 2023), also known as GPTQ. Meanwhile, our work explicitly focuses on improving the rounding method while remaining compatible with these transformations.
(RTN) and
Contributions. We introduce Qronos as a new scalable algorithm that not only explicitly corrects quantization error in both the weights and activations, but also residual quantization error coming from previously quantized layers. In contrast, OPTQ can only correct weight quantization error. We derive Qronos in a well-disciplined and mathematically interpretable form, then rigorously derive an equivalent efficient implementation (see Theorem 3.1 that significantly improves algorithm scaling
"Equal contribution.

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Published as a conference paper at ICLR 2026
pre-trained weights and aclivations
transformed
quantized weights and output activations
weights and activations
10 X
Stage Transform
W x
Stage
q 10
X
(potentially quantized) Input activations from quantized model
Figure 1: The modern quantization pipeline is typically two-stage process consisting of (1) trans- formations that make weights and/or activations more amenable to quantization, followed by (2) rounding functions that map weights and/or activations onto quantization grid.
(see Remark 3.3 and Section 4.3) As a non-trivial by prooouuu,,, we address a theoretical blind spot of OPTQ by deriving novel interpretation (Corollary (.4), which shows that its seemingly local greedy update rules in fact correct the weight quantization error accumulated over all previous iter- ations. Our novel interpretation also offers clear geometric insights: each step, OPTQ performs an optimal grid selection followed by an orthogonal projection onto a lower dimensional hyperplane spanned by future columns of the data matrix. This is one of the first results on the geometry of LLM quantization, among a few concurrent works (Birnick 2025 Chen et al. 2025
We evaluate Qronos on the Llama3 (Grattafiori et al. 2024) and Qwen3 (Yang et al. 2025) model families, and compare against RTN, OPTQ. GPFQ (Lybrand and Saab (021] and GPTAQ (Li et al., 2025) while demonstrating compatibility with notable transformations for both weight-only quan- tization and weight-activatioon quantization. To our knowledge, this is the first work to isolate the impact of the rounding algorithm through a carefully designed experimental setup that fixes the quantization grid for each transformation method (or lack thereof). Our experiments show that Qronos consistently yields marked improvement over existing methods, as highlighted in Table [
Table 1: Weight-only quantization of Llama3 foundation models. We jointly apply Hadamard- based incoherence processing (Ashkboos et al. (2024) and MagR (Zhang et al. (0044) as quantization transforms (stage in Figure| and compare different rounding methods (stage 2).
WikiText22 (L)
0-shot (t)
3B
8B
IB
8B
BF16
8.9
7.1
5.9
59.4
67.5
74.4
RTN
3e3
5c3
3e3
32.4
32.2
33.0
OPTQ
24.6
13.2
10.4
39.3
47.3
55.2
2-bit
GPFQ
25.8
14.4
11.3
38.6
46.9
51.8
GPTAQ
22.0
12.2
9.6
39.8
49.2
54.8
Qronos
17.8
11.4
9.3
42.6
50.7
55.8
RTN
5e5
4e4
9e4
32.3
32,9
32.2
OPTQ
2c2
52.0
43.3
32.7
32.5
34.9
1,58-bit
GPFQ
1e2
51.3
35.8
32.4
32.6
33.4
GPTAQ
99.0
41.8
35.3
33.3
33.7
34.7
Qronos
39.3
22.8
18.0
34.8
36.5
37.8
BACKGROUND AND RELATED WORK
We first provide a short review of prior works that focus on the two key aspects of quantization we have mentioned: transformation techniques and rounding schemes. Figure illustrates how these two aspects interact within the quantization pipeline.
Methods based on transformations. Many recent works propose transformations of weights and/or activations to facilitate quantization. One line work, initially proposed for MobileNets (Nagel al. 2019), exploits scaling invariance in neural network compute graphs to equalize the range or precision weights and activations before quantization. Recent variants leverage scale invariance to redistribute quantization difficulty between weights and activations, with various proposals for
learning scales or ranges based on custom objective functions (Xiao et al., [023 Shao et al.) [024]
Lin et al. 2024 Another line of work rotations within compute graph normalize weight
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and activation distributions, initially leveraging random orthogonal rotations to promote weight in- coherence (Chee et al. 2023). Recent variants employ efficient Hadamard rotations (Tsnng et al.
2024 Ashkboos al.; (024). Stiefel manifold optimizations (Liu et al., 2025 Hu et al., 2025), ), and rotation expansion techniques (Adepu et al. 2024 Franco et al. (2025a). Finally, distinct from these invariance-based approaches, Magk (Zhang et al. 2024) directly minimizes the f norm of weights via proximal gradient descent to reduce dynamic range before quantization. While we do not intro- duce novel transformations of this type in this work, we demonstrate that existing transformations can be combined with our proposed method.
Methods based on rounding. The earliest line of work on rounding relies on continuous optimiza- tion strategies based on gradient descent (Nagel et al. (220). Although more recent methods exist (Hubara et al. 2021 Li et al., 202T), they had not been commonly evaluated on L.LMs due to their
computational cost until Cheng et al. (2024). Thus, early work on LLMs focused on grid scaling or shifting to reduce weight quantization error; for example, .nt.Mint8)) (Dettmers et al., (022) and ZeroQuant (Yao et al. 2022) directly round to nearest after heuristically selecting the quantization grid (i.e., bit width, scaling factors, and zero points). The most relevant line of work to ours adopts principled discrete optimization using greedy, gradient-free rounding strategies to select quantized weights to minimize the layer-wise wise reconstruction error, and includes OBQ (Frantar and Alistarh, 2022), OPTQ (Frantar al. 2023), GPFQ (Lybrand and Saab 2021 /hhng et 2023) and GPTAQ [Li et al. 2025
Qronos falls within this category.
Notation. Throughout the paper, the weight matrix of a layer is denoted by W E NxV', where each of the N' columns represents a N-dimensional channel. A denotes the discrete quantiza- tion grid (or alphabet) used for weight quantization, and Q denotes the corresponding RTN op- erator associated with AA. given by Q(W) == s (clip ([") clip(1; Omin, 0maxx min { max amiu), amax)) ). while the quantization step size (or scaling fac- tor) is denoted by s and the quantization grid is shifted by an offset denoted by 2, often referred to as a zero point. We specify our selection of 8, E R- for the various settings in Section 4 When quantizing W, we use X E RT*N to denote the input calibration dataset of m samples (e.g., tokens) for the layer, resulting from the original pre-trained model, and X E m to denote the input calibration dataset coming from the partially quantized model. Given a vector E R", we use o for its i-th entry, Uiÿ for the subvector (U....., v,,) and we define vzz analogously. ||v| is the Euclidean norm of Given a matrix A E Rx we use A, to denote its 2-th column. We use A2j to denote the submatrix (Aj..... A.). Similarly. A22, denotes the submatrix of A obtained by removing the first row and the first column. We use col(A) (A) denote the column space of A. PA is the orthogonal projection onto col( A), and PA the projection onto its orthogonal complement. Throughout this paper, all indices start from 1, following the standard mathematical convention.
x 2; min A, max -2) Here,
Layer-wise reconstruction and error correction. Data-driven weight quantization methods typi- cally aim approximately minimize the layer-wise reconstruction error given by
min QEAAXXN
|XW - -XQI%
At an arbitrary layer, the goal is to compute a quantized weight matrix Q E ANNNN that preserves the output activations XW under quantization. In practice, however, quantizing weights in earlier layers affects the input to subsequent layers. Let X E mR denote the activation matrix produced by a partially quantized model, where earlier layers have already been quantized. To account for the propagation of quantization error, we use a modified formulation, instead of Equation [] the mismatch between the original output XW and XQ by approximately solving
that targets
min QEAM*NN
|XWW- XQ
The type of mismatch in this formulation is typically not addressed in the literature but arises natu- rally in both onlg and weight-activation quantization settings. For instance, in weight-only quantization, X arises as the output of previously quantized layers, while in weight- quan- tization, one may encounter Q(X) rather than X if activations are quantized. Throughout this paper, we use the notation (X, X) to refer generically to mismatched input pairs.
'Equation is instance of integer leaat-squares problems, which are known to be NP-hard (asssii and Thus, the hest that one can hope for are approximate solutions.
Vikalo. 2002
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3 QRONOS
The iterations follow a disciplined and mathematically interpretable framework that alternates between error correction and diffusion using optimal update rules. We then prove that the explicit solutions to these minimization problems admit an efficient implementation. In Section Qronos in the context of previous stateoof-the-ar rounding algorithms, namely GPFQ and OPTQ. We also derive a novel interpretation of OPTQ (Corollary 3.4). which shows that it corrects the cumulative weight quantization error incurred over all the previous iterations. The proofs for all results in Section 3 are provided in the appendix.
We begin by describing the iterations associated with Qronos in Section 3.1
3.2 we provide deeper intuition behind
3.1 ALGORITHM AND EFFICIENT IMPLEMENTATION
Let us first note that Qronos can process each column us E RN of E RNN''' independently and in parallel to produce each column E A of Q E ANNN", Ideally, the goal is to find q that minimizes dll l2 Since this problem is NP-hard, we propose efficient sequential algorithm to approximate its solution. At each iteration, Qronos first selects the quantized weight that optimally corrects the current approximation error, holding the remaining weights fixed; Equation below. then updates the unquantized weights to optimally compensate for the rounding error, a process refer to error diffusion; see Equation 4
Let w. without superscripts or subscripts, denote the original unquantized weights. After deter- (t-1) mining Qt-1, let wh--") represent the updated unquantized weights corresponding to indices t through N. The full state of the algorithm after step t thus given by the vector ((t-1) At step t, the algorithm alternates between
(qst-1, w(--') with the initialization w()) lecting 9. through eπror correction and updating the remaining weights through error diffusion. The update rules are given by
t-1 Σ j=1
N Xg - -px- - Σ
q: = argmin
X
(3)
PEA
j=t+1
2241
(0)
argmin (Me) 1 "
X
Egix; Σ
j=t+1
These optimization problems admit the following closed-fome solutions (see Proposition E.1
(xx=
(Xu - ΣØ1 8;X;; Ejeet w 11-1) X,, X.
Q
(5)
12
w 2111 =
zt 21+1 (Xw - Xz19st)
(6)
While these expressions follow directly from the optimization problems, computing and w
WS++1
in this form is not computationally efficient and scales poorly, as we will show in Section 4.3 To address this, we present Theorem 3.1] which shows that for all t 2 2, 9. be computed via RTN, enabling a simpler implementation. In Lemma 3.2 we further show that the update for w also admits an efficient implementation using Cholesky decomposition to solve the associated least- squares problem. Together, these results yield a practical and scalable implementation of Qronos.
Theorem 3.1. Let (gt, w (/(-1))) be the iterates generated by Equation 3 and Equation 4 with ini- (0) tialization 21 w. Define alternative sequence (ğe, 1 (o((-1) using the same initialization
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(0) "22 w, by setting
NV.
(1 = arg min
2 Xw pxx -
Σ
Wy
(7)
PEA
j=2
N
1 22
arg min ( JRNN- 2
IXw - 91
Σ 0
(8)
5=2
and, = ,....,N, define Ô 2(1))-1)
N,
(9)
N
2022111 arg min
2 ((
X )
+
1 - )
(uy
(10)
(N MERNNN JERN-I
j=t+1
Then for t = I, the two procedures yield identical iterates: (qt, (u = (ge, 20
Starting from the second iteration, Theorem 3.T shows that the updates Equation and Equation 4 can be equivalently reformulated Equation 9 and Equation [0] respectively. This reformulation allows Oe to be obtained via RTN for 2 2. followed by an adjustment of the remaining weights us-
ing only the (potentially quantized) activation matrix X to compensate for the one-step quantization
error (q: - (t- (t-1)) Xx-
To further accelerate this adjustment step, we now present Lemma 3.2 which establishes the equiv- alence of the update in Equation 10 (for t 2) with Cholesky-based least-squares solution For notational simplicity, we slightly abuse the indexing by treating = 2 as a 'restart.'
Lemma 3.2 (Equivalence of Least-Squares Formulation and Cholesky Formulation). Assume that
H = X X is invertible, and let H LL T denote its Cholesky decomposition, with L lower triangular. Then, starting from (0) w. the update rules
= 2w
(11)
N Σ j=#+1
(NE 2 l1( - w/ (1-1))
arg min
)xxe
Xx, +
(v) w )
(12)
2141
equivalent to the Cholesky- based iterations
9t = 2(w! (111)
(13)
W 22+1 w 2(+1 + 1
(t) ((-1)
(14)
where (11) -(w((t)) qe) LoLL N-
Lee
Remark 3.3 (Memory Efficiency). At the first iteration, both qı and w depend X. X e RmxN requiring (mnN) peak memory, often where m >> N. For example, Llama3. 1-8B requires over 30 GB just to store 128 sumples of 2048-token sequences at float32. We optimize this first iteration to use only square matrices such that
G1.21--11.20 - -111220 (0)
9 = w (1) (H22,22) (G22,210 H H22,191).
(15)
111
(16)
where - XTX NN and H XT RNNN see Proposition E.2 for justification. Note that calculating G and H does not require storing X, X, as one can sequentially accumulate the outer products of each of the m samples. Thus, this square matrix formulation reduces peak memory requirements of Qronos from ((mN) to ((N2), yielding an 18x reduction the case of Llama3.1-88 We note that Colbert et al. (2024) similarly identify memory optimization for GPFQ. but use singular value decompositions that may not scale well with N.
We do claim that Lemma 3.2 novel, though we were unable find it stated explicitly in the literature.
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This completes reduction of the original updates [ to the equivalent implemen-
tation given by Equations 13 14 15| and [6 The pseudocode for this efficient version is provided in Appendix [ We further present runtime analysis comparing this efficient version with the base version (i.e., a direct evaluation of the closed-form solution) in Section 4.3
3.2 THEORETICAL INTERPRETATION AND INTUITION
Theorem and Lemma 3.2 connect the initial disciplined optimization formulation of Qronos to our efficient implementation. These results guarantee that Qronos is both interpretable and scalable, explicitly correcting emoor from the mismatched input pairs X and X. Here, we provide deeper intuition the context of previous state-of-ther the-t rounding algorithms, namely GPFQ and OPTQ.
When quantizing w, GPFQ (Lybrand and Saab 2021 Zhang al. 2023 Zhang and Saab, 2023) interprets Xw as the endpoint of the path Dj=1 W for t = 1,..., N, and handles mismatched
inputs by aiming to match Σj= J and Σ j=1 Øj- for all t. More precisely, q: is selected as arg min Σラ= my xyyx - -j - 1
Although path following handles the case when X X well, additional considerations are re- quired when X * X since, in such a case, the tails of the two paths generally do not align when E (X; - X2) # O. Qronos handles this drawback by adopting a natural remedy to replace the unquantized weights w; by auxiliary weights w for i 2 + 1, so that
9,
X
Xw
W,X,
1=++
OPTQ (Frantar et al., 2023) explores similar weight update idea, but only in the where X X, by modifying the remaining unquantized weights after qe is selected. The Cholesky refor- mulation used in Lemma 3.2 also resembles the key mechanism in OPTQ. In this way, the runtime of Qronos scales similarly to OPTQ while also explicitly addressing the mismatch between and X: see Section 4.3 for details. This unexpected connection of Qronos to OPTQ also allows us to derive novel interpretation of OPTQ, which now present.
Corollary 3.4. The OPTQ iterations, when applied to a single layer input X, are equivalent to
N
4t = arg min = PEA
1 X
Σ
X, - px, -
Σ
(1-1)
(17)
2
j=1
#=
(1) w 21+1
arg min
4i X,
Σ
0g 113
(18)
(x JRRN
/=1
j=t-1
with w (0)
In other words, the updated weights and quantized weights at every iteration that are produced by
OPTQ are identical those produced by Equations 17 and 18 In particular, Equation 18 shows
that, at each step the updated weights (!) 22++1 indeed optimally correct for the errors produced by
the hitherto quantized sequence 91...., via orthogonal projection onto col col(Xx++1) as further discussed in Appendix H
Noticeably, OPTQ suffers from systematic bias when the activation mismatch is non-negligible as, unlike Qronos, it does not explicitly minimize the true discrepancy minge,AN ||Xw Xyl12- Consequently, as discussed in Appendix[D] Qronos consistently reduces the relative error (measured in lg norm) of block outputs compared to OPTQ, as illustrated in Figure
4 EXPERIMENTS
The core contribution of this work is Qronos -orr principled data-driven rounding algorithm that al- ternates between (1) explicitly correcting quantization error due both the weights and activations,

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Published as a conference paper at ICLR 2026
Table 2-bit weight-only quantization of Qwen3 instruction fine-tuned models. We apply HIP and compare different rounding methods (stage 2).
(stage in Figure
WikiText2 ())
O-shot (1) 4B
0.6B
1.7B
4B
8B
14B
32B
0.6B
1.7B
8B
14B
32B
BFI6
18.6
15.2
12.2
8.6
7.6
6.8
51.1
61.4
68.9
72.4
75.4
77.2
RTN
7c5
8e6
4c5
4e4
3e5
lc5
32.1
31.9
32.4
31.8
32.9
32.8
OPTQ
le2
60.0
22.8
14.7
14.9
12.8
32.0
32.8
37.4
41.4
42.5
47.0
GPPQ
le2
45.3
25.4
17.1
15.6
13.4
33.0
32.4
35.9
39.4
40.4
46.0
GPTAQ
74.5
37.0
21.0
13.6
14.4
12.9
32.3
34.0
38.7
42.5
43.3
47.3
Qronos
46.0
23.5
17.8
12.9
13.4
12.0
35.0
36.7
41.5
44.7
45.2
48.0
Table 3: Weight-only quantization of Llama3 foundation models. We individually apply various quantization transforms (stage in Figure T to isolate the impact of different rounding functions (stage 2) when quantizing to 3 and bits, respectively denoted W3 and W4.
W3
W4
WikiText2 (1)
0-shot (1)
WikiText2 (1)
0-shot
Stage
Stage
IB
3B
8B
IB
3B
8B
IB
3B
8B
IB
3B
8B
BF16
8.9
7.1
5.9
59.4
67.5
74.4
8.9
7.1
5.9
59.4
67.5
74.4
RTN
2e4
le4
3e4
32.3
32.4
32.6
18.0
10.1
8.4
60.8
67.4
OPTQ
42.5
13.8
11.4
37.5
48.1
53.8
10.4
7.8
6.5
54.3
63.4
71.0
None
GPFQ
35.3
13.4
11.1
35.7
49.9
53.5
10.4
7.8
6.5
56.0
65.2
71.2
GPTAQ
28.4
12.6
10.3
39.3
49.6
57.1
10.3
7.8
6.5
56.3
63.3
71.0
Qronos
22.8
11.3
9.3
39.5
53.1
56.7
10.1
7.6
6.4
56.2
64.5
72.0
RTN
6e3
9e3
5e4
32.7
32.9
31.4
15.2
9.6
8.1
51.4
61.5
67.5
OPTQ
29.6
13.6
12.6
37.0
46.9
47.3
10.4
7.9
6.6
56.2
65.3
70.2
Smooth
GPFQ
30.1
14.7
36.5
44.8
45.4
10.8
7.9
$3.9
64.4
69.9
Quant
GPTAQ
25.0
12.9
11.4
37.9
46.8
49.1
10.4
7.9
6.6
55.2
63.1
71.2
Qronos
19.1
11.6
10.3
40.7
50.6
50.5
10.3
7.8
6.5
56.7
64.8
70.2
RTN
2c3
5e4
33.8
33.5
35.1
13.8
10.3
7.2
53.1
58.1
69.7
OPTQ
20.1
12.9
8.1
44.2
45.6
59.7
10.3
8.0
6.5
56.4
60.0
69.0
MagR
GPFQ
21.0
14.0
8.3
43.9
48.4
61.7
10.4
8.0
6.5
55.4
61.1
70.3
GPTAQ
18.0
12.4
8.0
46.8
51.2
60.7
10.3
8.0
6.4
56.2
60.0
70.3
Qronos
16.9
11.8
46.6
51.2
60.0
10.1
8.0
6.4
56.2
61.1
70.4
RTN
7e2
3e2
le2
34.2
33.3
36.3
13.8
8.8
7.2
$2.0
62.8
70.0
OPTQ
16.1
10.3
8.6
44.1
56.6
58.8
9.9
7.6
6.3
56.8
66.1
72.1
HIP
GPFQ
16.6
10.4
8.6
44.9
54.8
58.9
9.9
7.6
6.3
56.5
65.7
72.0
GPTAQ
14.7
9.9
8.3
46.5
56,9
59.3
9.8
7.5
6.3
57.8
66.0
72.4
Qronos
12.9
9.3
7.8
48.1
59.6
62.2
9.6
7.5
6.2
57.1
65.9
71.0
and (2) diffusing excess eтror into future weights yet to be quantized. Thus, our primary comparison metric preserving model quality in challenging quantization scenarios. We design our experiments to isolate the impact of the rounding function (stage 2 in Figure IT. while varying the quantization transforms (stage in Figure
T. as further discussed in Sections and 4.2
Models & Datasets. We conduct experiments on Llama3 Grattatiori et al. 2024) and Qwen3 (Yang al., 2025) models using WikiText2 (Merity et al., 2016) for evaluation. We use, without modifica- tion, the implementations made publicly available via Huggingface (Wolf et al. 2020). We provide additional results Appendix BB
We use LightFval (Fourrier et al.] (003) to evaluate generalization via 5 zero-shot reasoning tasks: ARC (challenge and easy) (Clark al. 2018). HellaSwag (Zellers) ctt al. 2019). PIQA (Bisk et al. 2020), and Winogrande (Sakaguchi et al. 2021), and report the normalized average accuracy.
Setup. We implement Qronos in PyTorch (Paszke et al. 2019) using the Brevitas quantization li- brary (Franco et al.) 2025b), and quantize all models using single AMD MI210 GPU with 64 GB of memory. Unless otherwise specified, we construct our calibration dataset using 128 random se- quences of 2048 tokens sampled from the WikiText2 dataset for all data-driven PTQ algorithms. We

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Published as a conference paper at ICLR 2026
Table 4: Weight-activation quantization of Llama3 foundation models. We individually apply
various transformations (stage 1) to isolate the impact of different rounding functions (stage 2).
W4A4KV16
W444KV4
WikiText2 (D)
O-shot (1)
WikiText2 ())
0-shot (1)
Stage
Stage
1B
3B
8B
IB
3B
8B
IB
3B
8B
IB
3B
8B
BF16
8.9
7.1
5.9
59.4
67.5
74.4
8.9
7.1
5.9
59.4
67.5
74.4
RTN
22.0
12.6
9.6
45.4
55.0
62.6
41.8
22.0
15.9
41.5
49.8
57.4
OPTQ
14.3
9.8
8.0
50.4
59.9
66.7
19.8
14.3
10.3
45.8
56.2
64.1
QuaRot
GPFQ
13.6
9.3
7.6
50.9
60.9
67.6
22.0
14.7
11.4
43.3
53.9
59.8
GPTAQ
13.4
9.2
7.4
51.2
61.4
68.1
18.0
12.2
9.3
46.6
57.3
64.8
Qronos
13.2
9.1
7.4
50.9
61.5
68.9
17.8
11.6
9.3
47.8
57.3
64.8
RTN
22.4
12.2
11.1
42.9
54.8
62.6
39.3
19.5
34.3
40.4
49.4
50.6
OPTQ
13.6
9.5
7.9
51.0
60.3
68.5
18.6
12.9
16.
45.9
55.9
59.1
RmoothRot
GPFQ
12.9
8.8
7.4
50.4
62.0
67.7
20.8
14.3
12.2
44.4
54,9
59.0
GPTAQ
12.6
8.9
7.3
51.1
61.4
68.8
16.6
11.6
10.8
48.6
57.8
63.7
Qronos
12.6
8.8
7.2
50.8
60.9
69.4
16.9
11.6
9.5
47.1
57.8
65.2
RTN
20.5
12.6
9.3
47.7
57.5
64.2
33.5
20.2
13.4
43.1
52.2
60.8
OPTQ
13,4
9.2
7.7
52.0
61.1
67.0
17.9
15.0
8.9
47.9
58.5
65.5
SpinQuant
GPFQ
13.5
9.2
7.5
51.2
61.2
67.0
21.1
14.3
10.9
45.3
53.6
60.9
GPTAQ
12.9
7.4
51.8
61.1
68.3
17.1
NaN
8.7
49.4
NaN
65.3
Qronos
12.3
7.2
52.8
62.1
68.4
16.4
11.1
8.7
48.2
58.2
65.8
compare Qronos against RTN and the unmodified Brevitas implementations of OPTQ and GPFQ. also leveraging the unmodified Brevitas implementations of the various quantization transforms. We provide quantization transform hyperparameter details in Appendix C as well as ablation studies.
Baselines. Our baselines are RTN, OPTQ, GPFQ and GPTAQ. For OPTQ, we use the standard dampened covariance matrix H = H + AI, where À is 1% the average diagonal of H. We similarly use dampened covariance matrix for Qronos, but choose to be based on the maximum singular value H such that À a- which limits the condition number of H to be less than -!. We select 1e-6 for weight-only quantization and 1e-3 for weight-activation quantization. Additionally, we apply GPFQ, GPTAQ, and Qronos block-by-block; this corresponds to resetting X = X at the beginning of each block. Finally, we quantize weights in descending order of the diagonals of H, as is now common practice (IST 2022 Franco et al. (025b).
4.1 WEIGHT-ONLY QUANTIZATION
We first present state-of-the-art 2-bit and 1.58-bit results for weight-only PTQ on Llama3 foundation models, controlling for the quantization transform and grid selection while varying the rounding function. We quantize weights using the standard asymmetric weight quantizer (Frantar et al.) (2023), where scaling factor and zero point
scaled min-max grid such that defined per-channel Following the analysis of Zhang 8 3 (max(w) - min( fain(u) (2' 1) and = = B moin(wo//s
al. (2024). we choose / 0.8 when quantizing to 2 bits fewer. We combine Hadamard-based incoherence processing (HIP) (Tseng et al. 2024| Ashkboos et al. 2024) with weight magnitude reduction (MagR) (Zhang et al. 2024) jointly act our quantization transform, they are both known to be effective at few-bit weight quantization (Chee et al. 2023 Adepu et al. 2024). We present our results in Table as well as the BF16 baselines, and highlight that Qronos consistently outperforms existing rounding methods. For example, when compared to OPTQ, Qronos provides a 1.4x reduction in WikiText2 perplexity and +3.3% increase average zero-shot accuracy for Llama3.2-1B at 2 bits, and massive improvement in perplexity (4.9x) at 1.58 bits. We provide additional 2-bit and 1.58-bit results with 3 in Appendix C.I
Next, we present state-of-the-art 2-bit weight-only PTQ results on Qwen3 instruction fine-tuned models. Here, we use HIP as our quantization transform then tune the grid to minimize the mean squared error loss between the transformed weights and their RTN-quantized counterparts via linear search over s and 2. Table provides the results from Qwen3 0.6B to 32B. Qronos again yields clear and consistent improvements for all models in this family.
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Table 5: Calibration Runtime Analysis. We report the end-to-end calibration time of OPTQ and Qronos for the Qwen3 model family, normalized to Qwen3-0.6B, as measured on an AMD MI325X.
0.6B
1.7B
4B
8B
14B
32B
OPTQ
1.0
1.4
2.8
3.7
5.6
10.6
Qronos
1.2
1.6
3.1
4.0
6.1
11.5
Overhead
19.7%
16.2%
11.1% %
9.0%
8.7%
8.7%
Finally, we present 3-bit and 4-bit weight-only PTQ results (denoted W3 and W4, respectively) on Llama3 foundation models while independently demonstrating compatibility with 3 notable quan- tization transforms: SmoothQuant (Xiao et al. 2023). MagR, and HIP. Table 3 shows the results Qronos consis-
across three models the Llama3 family. For both W3 and W4, use tently provides higher quality quantized models than RTN, OPTQ, GPFQ and GPTAQ, as measured in both WikiText2 perplexity and average zero-shot accuracy. Consistent with emerging work on rotation- based quantization transforms (Chee et al. 2023 Tseng et al.) 2024). incoherence pro- cessing outperforms other transforms, with HIP + Qronos providing the best overall results. Note that HIP OPTQ is similar in spirit to QulP by Theorem 6 in (Chee et al. 2023), which equates LDLQ to OPTQ, with notable difference that QulP proposed random orthogonal matrices instead of Hadamard matrices.
4.2 WEIGHT-ACTIVAL QUANTIZATION
We present 4-bit weight-activation quantization results with and without 4-bit KV cache quantiza- tion (denoted W4A4KV16 and W4A4KV4, respectively) while demonstrating compatibility with QuaRot (Ashkboos al., 2024), SmoothRot (Czakó et al. 2025), and SpinQuant (Liu al. 2025). We quantize weights using the standard symmetric weight quantizer with per-channel scaling factors optimized via linear search over the mean square error loss between the full-pree full-precision and quantized weights. We quantize activations using the standard asymmetric activation quantizer with dynamic per-token scaling factors and zero points defined on the min-max grid, as common practice (Liu et al. 2025). When quantizing KV caches, we similarly use per-token scaling and zero points.
Table the results across three foundation models the Llama3 family. Qronos again consis- tently outperforms RTN, OPTQ, GPFQ and GPTAQ as measured in both WikiText2 perplexity and average shot accuracy. Consistent with emerging work on learned rotations (Liu et al.. [2025] Hu
al. 2025 Franco et al. 2025a), SpinQuant outperforms QuaRot and SmoothRot, with SpinQuant + Qronos providing the best overall results with and without KV cache quantization. We remark that experiments use per-token quantization for both the activations and KV caches, while Ashkboos
et al.
2024) and Liu et al. (2025) both use per-group scaling for KV cache quantization.
Our experimental analysis reveals important pattern: Qronos provides larger improvements as quantization tasks become more challenging. Specifically, Qronos demonstrates larger relative im- provements over existing methods when transitioning from weight-only weight-activation quan- tization (i.e., W4 versus W4A4), and even more substantial gains when incorporating KV cache quantization (i.e., W4A4 versus W4A4KV4), We further validate this pattern with additional W3A3 results in Appendix B (Table which show larger improvements than both W4A4 and W3 weight- only quantization. These findings suggest that Qronos is particularly effective in scenarios where multiple sources of quantization eтror interact, making it especially valuable for aggressive quanti- zation settings where traditional methods struggle to maintain model quality.
4.3 HARDWARE EFFICIENCY AND RUNTIME ANALYSIS
The hardware efficiency benefits of quantization (i.e., improved throughput, memory, power, and area) are well-established (Jacob et al., 2018 Colbert et al. 2024). Since Qronos and other round- ing algorithms leave the compute graph unaltered, they capture these benefits without introducing inference overhead beyond the quantization transform. Prior works have already profiled inference
We observed instability with GPTAQ, as reflected by the NaN entries in Table 4 and similar issues have been reported by others attempting reproduce results from Li al. 2025) with their official repository.

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Published as a conference paper at ICLR 2026
800
50
Relative Algerthi Runtimo
GPFQ
Relative Overall Runtime
OPTQ
600
Qrones (Base Ver.]
Oronos (Emficiett Ver.)
30
400
20
200
400
850
1000
250
550
850
1000
Irput Features (N)
Input Features (N)
(a) Runtime of Rounding Algorithm
(b) Runtime of Quantization Pipeline
Figure 2: We compare the runtime of (a) the rounding algorithm and (b) the overall quantization pipeline scale the input features N, measured on an AMD MI210. We average all measure- ments over 3 seeds and normalize to the runtime of OPTQ where = 32.
speedups and overheads; for example, Ashkboos et al. (2024) report up to 2.16x speedup for W4A4 Llama2 7B over FP16, with Hadamard transforms adding most 7% overhead. Therefore, we focus our runtime analysis on the quantization pipeline itself.
Microbenchmark. We perform our initial runtime analysis using a single linear layer. We use calibration set of m = 10,000 random data sampled from K -ddimensional Gaussian distribution. [32, 1024] inputs with K/4 outputs. Figure 2 shows how the runtime
The linear layer has of OPTQ, GPFQ, and Qronos scale with K, where (a) isolates the algorithm runtime (i.e., without the added inference cost calculating H and G) and (b) aggregates the end-to-end runtime of calibration. To highlight the benefits of our equivalent formulation, implement base version of Qronos that uses the iterates for q and 22++1 from Equations 5 and 6 Note that via Theorem 3.1 and Lemma 3.2 we significantly improve the runtime scaling Qronos over the base version, with a 13.8x reduction in algorithm runtime and a 3.6x reduction in overall runtime when K = 1024.
Calibration Runtime. Compared with OPTQ, which only needs to collect X, GPFQ and Qronos re- quire collecting both X and X each layer, which requires two forward passes (with and without quantization) and increases the overall quantization pipeline runtime. To evaluate the overhead of two forward passes practice, we compare the calibration runtime of OPTQ and Qronos when quantizing the Qwen3 model family. Table 5 provides the runtimes for each model from 0.6B to 32B, normalized to the calibration runtime when using OPTQ to quantize Qwen3-0.6B on an AMD MI325X. We observe the overhead of Qronos decreases from 19.7% 8.7% as model size increases from 0.6B to 32B, indicating that algorithmic cost dominates the cost of executing inference twice and underscoring the importance of Theorem 3.1
5 CONCLUSIONS
We introduce Qronos-a new backpropagation-free rounding algorithm that alternates between cor- recting quantization error in both the weights and activations of previous layers and diffusing error into future weights within the current layer. Qronos is based on an interpretable and disciplined optimization framework, and demonstrably surpasses existing data-driven approaches. Our im- plementation exploits several optimizations that together yield orders of magnitude improvements memory and compute efficiency. Our experiments isolate the impact of the rounding function in the quantization pipeline while varying transformations on a scaled min-max grid. Our results show that Qronos consistently offers improvements over previous state-of-the-at methods when quantiz- ing weights, activations, and/or KV caches to 4 bits or fewer. That said, our results are intentionally limited to the scaled min-max quantization grid to focus our experiments on transformations and rounding; believe results could be further improved by leveraging weight and activation dis- tributions to design guantization grids that are more effective than the scaled min-man min-max grid used in this work, possibly with non-uniform grids via vector quantization.
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REPRODUCIBILITY STATEMENT
We integrate Qronos into the open source Brevitas quantization library. Our code and instructions on how to reproduce the various experiments, including hyperparameters and random seeds, can be foundhere) on GitHub. The code points to all datasets and models used for calibration and evaluation.
ACKNOWLEDGMENT
We gratefully acknowledge partial support by National Science Foundation, via the DMS-24101 20-0074177 grant. We also gratefully acknowledge partial support from AMD. From AMD, we would like to thank Nick Fraser and Giuseppe Franco for their constructive reviews and feedback that made this paper better, and Gabor Sines, Michaela Blott, Syed Naim, Yonas Bedasso, and Max Kiehn for their support.
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A PSEUDOCODE OF QRONOS
We provide the pseudocode for our efficient version of Qronos derived in Section 3.1
Algorithm Qronos (Efficient Version)
H = X H- = - LLT
P Cholesky Decomposition
for every in W (in parallel) do = NN
w()) t copy(w)
G1.21"
H1, 22
91
By Proposition E.2
H1
w 22 = Lz2,22 22,22 22,22 (G22.11 1122191--2219
(1)
>> By Lemma G.3
for = 2 N do
D By Theorem 3.1 and Lemma 3.2
a: = Q(u, (t(()
w 21+1 = 21+1 + (t)
(t)
(t-1)
A (t) = (w; 11-1)
91)
ba
end for end for return
B RESULTS ON ADDITIONAL MODELS
Our main results evaluate Llama3 foundation models and Qwen3 instruction fine-tuned models. Here, we demonstrate that Qronos maintains the quality of Llama3 instruction fine-tuned models and Qwen3 foundation models as well. We again compare against RTN, OPTQ, GPFQ, and GPTAQ.
We present weight-only PTQ results with Llama3 instruction fine-tuned models at 3 and 4 bits in Table As Section 4.1 we asymmetrically quantize weights to the scaled min-max grid with I both W3 and W4. We focus our instruction fine-tuned results on evaluating each round- ing algorithm with and without Hadamard-based incoherence processing (HIP) as the quantization transform. As Section 4.1 we find that HIP Qronos consistently provides the highest qual- ity quantized models relative to BF16 counterparts, as measured in both WikiText2 perplexity and zero-shot accuracy.
We then present weight-only PTQ results with Qwen3 foundation models in Table We asymmet- rically quantize weights to the scaled min-max grid with 0.9 for W3. We focus these results with and without Hadamard-baan incoherence processing (HIP). Again, we find HIP + Qronos consistently yields the highest quality quantized models relative to the BF16 counterparts.
C EXPERIMENT DETAILS FOR QUANTIZATION TRANSFORMS
All experiments use WikiText2 as the calibration set, aside from SpinQuant, which uses C4. To pre- process our calibration dataset, we ensure that the <bos> token always appears as the first token in an input sequence as the recent study by Barbero et al. (202) suggests removing <bos> during inference may greatly reduce performance if models were trained with <bos>> always appearing at the first token; their analysis suggests the Llama3 family of models fits this category. Thus, to quantize our models, we first load the pre-trained checkpoint, then pre-process the dataset(s), then apply the quantization pipeline visualized in Figure] [ For Section 4.1 we intentionally select SmoothQuant (Xiao et al. 2023), Hadamard based incoherence processing (HIP) (Ashkboos et al. 2024 Tseng et al. 2024). and MagR (Zhang et al. 2024) they perform fundamentally different transformations. For Section 4.2 we study QuaRot, SmoothRot, and SpinQuant. Here, we describe Ciettrrss-SSmootuuuaanne MagR, SpinQuant, and Rot
hyperparameters for the data-driven
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Table 6: Weight-only quantization of instruction fine-tuned Llama3 models. We apply Haddmardd-baee incoherence processing (HIPP) as our quantization transform (stage in Figure isolate the impact of different rounding functions (stage 2) when quantizing and bits, spectively denoted W3 and W4. We also evaluate quantization transform (i.e. "None").
W3
W4
WikiTexx2 (I)
0-shot (1)
Wiki'Text2 (J)
O-shot (f)
Stage
Stage
1B
8B
IB
1B
3B
1B
8B
BF16
12.0
9.2
6.7
59.5
66.4
74.1
12.0
9.2
6.7
59.5
66.4
74.1
RTN
2e4
4e3
3e4
32.6
33.0
32.2
21.4
12.6
9.1
51.0
62.3
67.6
OPTQ
60.0
16.1
12.2
37.4
49.9
58.2
14.3
9.9
7.3
54.5
63.6
71.8
None
GPFQ
2c2
16.6
12.9
33.8
50.8
55.3
15.4
9.9
7.3
53.3
71.5
GPTAQ
52.0
14.9
11.4
37.4
49.8
57.5
13.8
9.9
7.3
55.5
63.1
71.2
Qronos
43.8
14.3
10.6
37.5
52.1
60.6
13.8
9.8
7.2
55.5
64.8
72.2
RTN
lc3
3e2
33.4
35.0
36.9
16.6
10.8
8.0
54.6
63.6
70.8
OPTQ
19.1
12.8
9.3
48.0
58.2
59.0
13.2
9.6
7.1
56.6
64.5
72.1
HIP
GPFQ
20.4
12.8
9.6
47.6
57.1
61.1
13.2
9.8
7.2
57.0
65.3
71.9
GPTAQ
18.0
12.2
9.1
49.2
57.4
63.2
12.9
9.8
7.1
56.9
63.9
72.7
Qronos
16.6
11.6
8.8
49.9
58.4
64.1
12.8
9.6
57.6
72.1
Table 7: Weight-only quantization of Qwen3 foundation models to bits with 3 0.9. We apply Hadamard based incoherence processing (HIP) as our quantization transform (stage in Figure isolate the impact of different rounding functions (stage when quantizing to 3 bits.
WikiTeet22
(4)
O-shot (t)
Stage
Stage
1.7B
4B
8B
1.7B
4B
8B
BF16
8.6
7.3
6.5
63.9
70.1
73.6
RTN
3e5
82.0
3e3
32.9
45.3
37.1
OPTQ
37.5
10.4
8.8
35.7
57.2
61.7
Nonc
GPFQ
1c2
10.8
9.3
33.0
56.3
58.9
GPTAQ
33.8
10.1
8.5
36.1
63.7
58.5
Qronos
33.0
9.5
8.3
36.0
59.9
61.5
RTN
le3
26.3
30.1
35.1
50.8
50.3
OPTQ
10.8
8.8
7.6
54.4
64.4
67.6
HIP
GPFQ
11.4
9.1
7.9
52.7
61.6
62.2
GPTAQ
10.6
8.6
7.5
54.9
63.6
66.0
Qronos
10.1
8.4
7.4
57.2
63.5
68.0
When applying SmoothQuant, we do so before quantizing weights activations. In practice, SmoothQuan requires the selection hyperparameter control the scaling optimiza- tion criteria. We refer to the SmoothQuant hyperparameter as so as to not clash with our use of a Section 4 note that y E [o, 1]. In Table we provide the results of uniform grid search over y when quantizing Llama3.2 --tt....BBBBBB 1B- Instruct to 4 bits using round-to-neare (RTN). These results moti- vate our decision to use 7 = 0.3 all our weight-only PTQ experiments that apply SmoothQuant.
SmoothQuant.
MagR. When applying MagR, we also so before quantizing weights and activations. When coupled with HIP. we do after inserting rotations into the compute graph. In practice, MagR requires tuning the lo penalty; we refer to this hyperparameter as 0, again so as to not clash with use of in Section 4 Zhang al. (2024) tune 0 Llama2 models, settling on O 0.001 for their experiments. In Table 10, we provide new results for Llama3.2 2-11B-Instrue These results motivate decision to 0 0.01 in all our weight-only PTQ experiments that apply MagR.
SpinQuant. When applying SpinQuant, Liu et al. (025) do so after activation (and KV cache) quantization but before weight quantization using 800-sample calibration dataset: their ablation
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Table 3-bit weight-activation (W3A3) quantization of Llama3 foundation models. We apply
QuaRot as quantization transformation (stage 1) and compare different rounding functions (stage 2).
Wiki'eex22 (4)
O-shot (t)
1B
3B
8B
1B
3B
8B
RTN
2e3
9e2
le3
33.0
32.3
32.8
OPTQ
9e2
2e2
le2
32.3
33.2
35,9
GPFQ
60.0
30.1
27.9
35.6
39.2
40.3
GPTAQ
2e2
40.5
46.0
35.0
36.9
41.9
Qronos
46.8
22.0
20.4
37.0
43.4
47.4
Table 9: Impact of SmoothQuant's y on Llama3.2-IB-Instrue. We evaluate the impact of the smoothing parameter Y both WikiText2 perplexity and normalized average sero accuracy when quantizing Llama3.2- 1 to bits using round-to-nearest (RTN).
0.2
0.3
0.4
0.5
0.6
0.7
0.8
WikiText2 (L)
24.6
18.6
18.9
21.4
87.0
4e2
3e4
0-shot (1)
50.8
53.3
52.8
52.5
42.8
36.6
32.3
study demonstrates negligible degradation when using 128 samples. Thus, we employ Cayley SGD on network where only activations are quantized optimize the learnable rotations for 100 itera- tions using a calibration dataset constructed of 128 random samples from the C4 dataset.
Rot When applying SmoothRot Czakó et al. (205)), we do so before quantizing weights or activations. Similar to SmoothQuant, SmoothRot requires the selection of a hyperparameter (i.e., migration strength) to control the scaling optimization criteria. In our experiments, we use migration strength of 0.6 as empirically performed well for Ilama3 1B.
C.1
GRID SCALING ABLATION STUDY FOR 2 BITS AND FEWER
we have presented weight-only PTQ results when quantizing to 2 bits or fewer on the scaled min-max grid with / = 0.8. Here, in TT we provide additional results that demonstrate Qronos outperforms other rounding algorithms on another choice B = 1. Recall that we jointly apply Hadamard-baas
In Section 4.1
incoherence processing (HIP) and weight magnitude reduction (MagR) as quantization transforms before each rounding algorithm. Our results highlight that 3 0.8 (see Table is an overall better choice for scaling the min-max grid in this setting, which consistent with Zhang et al. (2024), and that Qronos provides the best results on both grids all bit widths and model sizes. Our results with β = also show that Qronos more robust than GPTAQ when B is not carefully selected.
D MORE ON WHY QRONOS OUTPERFORMS OPTQ
Let W be the full-precision weights of a layer, and Q their quantized counterparts, Let X be the input to the layer and let its (possibly quantized) counterpart be X; importantly, X reflects both activation quantization and the residual error propagated from previously quantized layers (possibly from previous blocks). Let Y, Y denote the respective outputs resulting from inputs X.
For any single layer, OPTQ only attempts to minimize [ - F.. which ignores the mismatch between X and X. In contrast, Qronos attempts to minimize | - XQlIe, which is the actual discrepancy between the full-precision outputs and their quantized counterparts.
A simple triangle inequality intuitively explains the distinction between OPTQ and Qronos:
||Y - E| = || - -XQI P < |[(X - X|W + || - Q)
While OPTQ only corrects the second term, Qronos corrects both terms. Thus, OPTQ only corrects quantization error in the weights given layer while Qronos corrects not only quantization error
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Table 10: Impact of MagR's 9 on LLama3.2-IB-Instruee We evaluate the impact of the penalty parameter & on both WikiText2 perplexity and normalized average zero-shot accuracy when quan- tizing Llama3.2-1 12-IB-Instrutt bits using round-to-neares. (RTN).
0.1
0.01
0.001
0.0001
WikiTex12 (U)
74.5
25.4
105.0
216.0
0-shot (t)
44.2
53.0
44.7
42.6
Table 11: Weight-only quantization of Llama3 models to 2 bits fewer with We jointly apply HIP and Magk as quantization transforms (stage in Figure 1 and compare different
rounding functions (stage 2) on the scaled min-max grid (see Section Note that these results complement Table which presents results with 3 = 0.8.
WikiText2 (4)
O-shot (1)
IB
3B
8B
IB
3B
8B
BF16
8.9
7.1
5.9
59.4
67.5
74.4
RTN
le4
le4
2e4
32.4
32.4
32.9
OPTQ
45.3
20.8
18.9
35.2
39.3
41.2
2-bit
GPFQ
47.5
22.4
17.8
33.9
38.4
39.2
GPTAQ
33.8
18.0
16.4
36.3
40.7
41.3
Qronos
24.6
14.9
12.4
38,4
43.4
45.6
RTN
2c5
3e5
6e5
32.0
32.6
32.1
OPTQ
5e3
4e2
3e2
32.5
32.4
32.2
1.58-bit
GPFQ
6e2
7e2
5e2
31.2
32.5
32.7
GPTAQ
2e3
3e2
2e2
32.2
32.5
33.2
Qronos
79.5
48.3
34.8
32.9
32.8
34.3
in both the weights and activations at given layer, but also residual quantization error coming from previous layers, possibly from previous blocks.
Furthermore, tuning the quantization grid (i.e., scaling factors and zeros points) cannot effectively
minimize our objective in Equation As before, decomposing XW XQ [(X X)W + [X(W- - Q)] isolates two error sources. Tuning quantization grids of the current layer only adjusts Q. and thus can affect only the second term, while the first term is untouched by any choice of quantization grids. Hence, it cannot close the performance gap between OPTQ and Qronos.
To illustrate this, we empirically compare quantization error accumulation by measuring the relative
l eтror, given by I|Y - || after each transformer block in Llama3.2 1B when quantizing weights bits, as Section 4.1 Here, in Figure we report the relative & error averaged over each token in our calibration dataset (i.e., 128 samples of 2048 tokens from WikiText2). Qronos yields the lowest average relative calibration error for each block, with 16% and 13% improvement over OPTQ and GPFQ, respectively, at the output of the final block.
PRELIMINARY PROPOSITIONS
Proposition E.1. The update rule given by
N Σ
= argginn
Nxw
Σ
X3X,
px,,
(t-1)
PEA
1=1
j=
N Σ
W 2141
argmin
|
Σ X, -
0z 12
(
j=1
j=t+1
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OPTO GPFQ
Average Relative Emnr
Gronos
0.6
0.4
0.2
Tra Transfomeer Black Index
Figure 3: We visualize the evolution of the average relative error over transformer blocks when quantizing the Llama3 IB foundation model bits, further discussed in Appendix D
has closed-form expressions
(Xw- Eju xt-1 12,, Lj-e41 wj (1 Xx)
qe = Q
||
and
w 2011 = (Xw KW--x
Proof. For gs, the corresponding optimization objective function is one-dimensional quadratic function of p. Since minimizing a quadratic function over a discrete set A reduces to rounding its real-valued minimizer, we compute the real-valued minimizer
(Xw (Xw-Efi &x Ej=1+1 W/ X,,"
||,.|2
Thus, we obtain the closed-form expression of Zt.
qe Q (Xw (XW-Ejj
((XW--22 Σ &x Ej=et wj
(1 X.)
||X,11
where is the round-to-nearest operator.
For 22411 the corresponding optimization problem is an unconstrained least-square problem the form of min.eR--4 ||Az 2 with X2441 and X Xzigse. Thus, the minimizer is given by A which gives the desired closed-form expression.
(t)
Proposition E.2. The update rule given by
Xi (Xw - x22
41 =
||
w 22
= _t12 (Xw - 8191
is equivalent to
-12.1--H,22 - -11,2 (0)
91 0
H1
w 22 (H>2222) (G H>211)))
where G = XT TTTxRRNXL and H = XT x NxN
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Proof. For 91, we have X = (X G1 Also, XTXx₂ (XTX)1.2 = H1.22
Thus, XX'(xw X22w22) = G1,21 H 21 Further, || (XTX))1 = -n. This
(0)
gives the equivalence for updating 91-
For w 22.
X22 is given by (X (Xz2 N (H22,22) - XT Then
222
x 22 Xw Xin) = (H22.22) -1 (Xw- - X191)
- -HH22222 (X'T X22xx) - (X X)22.11)
=H22222 (G (1221i)))211))
This gives the equivalence for updating (1)
''2-
PROOF OF THEOREM 3.
Proof. We use induction to prove the theorem. Since at t = 1 equations Equation Equation 4 and equations Equation 7 Equation are identical, the base case is trivially true. Now we proceed with (2)
the induction, assuming vi 21+1 2x+1 and de Qt-
Using definition Equation and Proposition we can obtain the closed-form expression,
( ((XW
Σソ= ((X X2 Ez=12 w/ (4) Xz. X(t1)
gc+1 Q
||
where Q is the RTN operator. Next we note that which is used to compute w (() 2141 implies that
Xw E/=1 Xx Σ w j is orthogonal to the column space of X X2441- This in turn implies that (Xw Xw-Ei 4; -ÿ wj
N (1)
Xj,Xx+1) 0. Then we can compute.
(Xw Σj=1 Nx Ejer4 wj
(1) N, Xe+1)
a++1 9+1 = 0
|1
(Xw Σi=1 Ej=14,&, -ESe++1 wj
2x + wi Xet1 Keti)
(1)
0
1|
(t) :
o
||xx+1112
o w++1 = - -
(()(()) -
(10+1))
where in the last two inequalities, we used the induction hypothesis w rule
21+1 W2t+1 and the update
(t)
Next, we prove d 2t+2 w
((+1)
(++1) 22+2
We first compute
t+1
N
2t42
((+1)
argmin 02442
1xw-
9j
Σ
x;X,112
/=1
j=l+2
NN Σ
N
= argming
Xw
Σ
Oj X,
wy
x,
+ (w/²)
- ---1))111 1) +
Σ (w)) -ww -1y) x
2
#2442
j=1
j=++1
j=t+2
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Due to the update rule (4 Xw-
(1) Ej=1 yj E Lij=+11 "jj Xj is orthogonal to the column span
of X X2441, hence to (w
9(+1)
E [)=(+2(w) xy) Then, we have
N
N
W 2242
(++1) argmin X uge+2
Σ
yjjK
Σ j=t+1
wj
Xj + (w qr qc- (0-1)x++1 +
Σ
j=1
jut+2
N
aaggming
1( 22.42
4(+1) e+1
+
Σ j=
(1)) - 2z X 1
21+22
(t++1)
where we used the Pythagorean theorem, the induction hypothesis w 21+1 4++1 14+1- This completes the induction.
(1) 2t+1 and the fact
L
G PROOF OF LEMMA 3.2
Throughout this section, we denote H Hz2.21 X2, E ((N-++))(N-++) and 20,2 (x1) E IR(N=++1)(N-++1)) We will begin with a few preliminary lemmas before we prove Lemma 3.2 While some of these lemmas may already be known, we are not aware of any rigorous proof' in the literature. Thus, we provide our proofs here for completeness.
Lemma G.1. Denote by [H (H2izo0 the first entry of Hig. and by (Hzl.2ll2. E R-- the first column of H 22 albeit with the first entry removed. Then
H 12z222221
(X (2441))))) X
H H2222d11
Proof. We denote [1 20 and b [1 [1222222. Then
is just the first column of
b
X X
X X
1.
15 so we have H2r,2 b ei. Let write H
By
21-11 X2
X X21+1X24+1
comparing the two sides of b ei we can observe rXT 2 X₂ + XI X24t1b which implies
(2
-XT
r
and finishes the proof.
The next lemma establishes how one can efficiently compute H H=1 2141,2411 from H-1
21,2"
Lemma G.2. H 2,411 can be efficiently computed from H H via
H H=1 1
Hzg.2
HZ 22221111
H
H
22.22
We note that this is simple rank-1 update followed by a submatrix slicing.
Proof. We first recall more general inverse formula for 2 x block matrix using the Schur com- plement. Consider the x 2 block matrix
M
D)
C
When A is invertible, the inverse of M is given by
M (A
(A + A CA-
BS-1
(19)
-S-ICA-1
8-1
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where 5 D CA-'BBB is the Schur complement of A in M.
When A is a scalar and M is symmetric, i.c.
0 b
M
b
D
this formula becomes
a1
S
-a
-a
M
1 S-1b
s-1
a
where S a Lb
By the Sherman-Morrison formula (Horn and Johnson 2012). we have
D-1 = S- S-1
a+b78-14 8-16
2s 2- 16 S
=S- a= + a 2b S-16
Returning to our setting where M-1 H
and D-1 H -22
21+122411 we have
2022
H H2-11244 =
22,2
[H
21.2
22.1
H 22222
=I
22222202222
H [2
|
222,1
22.22
H1 21,2
1 20,21111
[ 2222222111 1
21.211.211
22,22
Using the above lemma and Cholesky decomposition (Horn and Johnson 2012). simplify the right hand side in Lemma G.1 via the following lemma.
can further
Lemma G.3. Let H-1 = (X- and H LLT be its Cholesky decomposition where lower triangular matrix, then
L H 2222222211
e N-
|1 |1
Lu
holds for all t E [N - 1].
Proof. We first prove that given the Cholesky decomposition H-1 = LLT,, the Cholesky decompo-
sition of H= is Hz (12422)))22224) for all f. E [N], where Hz (X Xzx)
R(N-++1)x(N-+H1)
holds by assumption, and assume the result holds for t. By Lemma G.2 the updated inverse Hessian H
Let proceed by induction. The base-case when
H 1
Thus,
22.2
(Z2120)( La (222=2200n1222z222111)))((122321 (12222211)) 22,22
((1212))(22200) - 22,2
=(1212212222) =(1241224)4(44122))
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This finishes the induction and we have Cholesky decomposition H2 (121.2)) for all [N]. To finish the proof, let M RR be the Cholesky decomposition of any positive definite matrix M. By direct computation, the first column of M is R[R 121,1 Ru 21,1
and the first entry M11 = R- Then we have My1. [R]21,1 which implies that My21 Mi1 Rn
2
Mu
Rn
In our case, we have H 21 (L2.22) in the place of M RR Thus,
H 2222d222
L24.212221
L24414
H 21.201111
12022111
Tux.
With the above preliminary lemmas, now are ready to prove Lemma 3.2
Proof of Lemma 3.2
Since we initialize with w()) w, q1 ((wr) always holds. Thus the two iterations produce the same and u We proceed by induction. Assume at step t that and
(t-1) resulting from the update rules Equation and Equation 12 match those following update rules Equation 13 and Equation [4 In order to complete the induction, it suffices to show that (12)
"zzt
and 14 produce the same 26+11 which naturally results in the same Qt+1 ((w
(f)
To that end, we note that the optimization problem defined by Equation 12 has unique least-square solution as X2e+1 has full column rank. The minimizer given by
w
(t)
w (t-1) 1) 21+1 + (w(--1) (::) 4:)
20+1
#2t+1 + (w) (t-1) qx) 20+1 X24+1)
-X 2x X
By Lemma G.1 have
H |H | 122222221
(X 2141 1x -XX
20-1
1 11
Lastly, Lemma G.3 gives us
H 22222221111 L2t+11 NR-- Lu This matches A4t1 in Equation 14 and completes induction.
[ 21,2111
H PROOF OF COROLLARY 3.4
Algorithm 2 OPTQ: Quantize layer W given inverse Hessian H-1 (XT x)-1
1: for every in W parallel do
2:
q %
> Initialize quantized neuron
3: H I.L
Perform Cholesky decomposition
4: for
to N do 5: a: = =Q(w:)
Iterate over rows
6: W>t W21 L21, (w, 9t) Lut.
Update remaining weights
7: end for
8:
end for
9: return Q
For our final result of this paper, observe that updates of w 22+1 via Equation can be interpreted
by observing that the term (g: w. (t-1) 1) Xx represents the error introduced by quantizing ( The optimization problem Equation 12 seeks to mitigate this error by adjusting future weights so as minimize the resulting distortion, measured in the norm. Notably, this step does not explicitly
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attempt to correct errors introduced by earlier quantization steps I.....t f- - 1. However, by combin-
ing the proof of Theorem 3.1 in the case when X X with Lemma 3.2 arrive Corollary 3.4
which provides novel interpretation of OPTQ. It shows-perhape unexpectedly-thall Algorithm
optimally corrects the cumulative weight quantization error incurred over the first t entries of w.
Proof. The proof is based on induction on both arguments of the trajectory. Let {(w5) -()) denote the trajectory generated by update rules Equation
(1-1)
17 Equation
18
And let {(w2) (t-1) -i)
be the trajectory generated by Algorithm Our goal is to prove (w0 21
(e)
(w2 4t) for
(t-1)
1
By Lemma 3.2 the trajectory ((w (t-1
(b)) generated using Cholesky decomposition in Algo-
rithm can be equivalently regarded as generated from Equation 11
Equation
12 Thus, we will use
Equation
Equation 12 as the update rule of W2r and qt in the rest of our proof. In the base are both initialized with w and w waz
(t-1)
(0)
(0)
case, 20=11
④ = argmin,
|| pX1
10) x,11 argmin [1 - p) 2(w1) = 91- w
PEd
PEA 2
.qe) holds true. Now we proceed to Thus (w21. (1 = (w/21. )1). Assume (vo .4) = =Ww2o (w)t-1)
prove (w 2t+11 h++1) = (w 2t+1: (t+1).
Step 1: We first prove (1)
21+1 = 21+1- By construction,
argmin 1 511X U2
Σ 9j
Σ u,X,
s=1+1
For an arbitrary U2441 E N-
X
X
Σ , =
j=t-1
1-1 (Xw
N
N
Σ
"jXب -
Σ
(t-
X
+
(wi) (t((
6. +
Σ (1oft
15)
j=1
j=t
-=
(
(II
(f-1) Since in 22t1 is minimizer of Equation 18 the first term (I) X X and clearly the second term
(I E ( Xx). Thus, we have
N Σ 5-1+1
Xw Σ X
Xy
||
||(DD]]
Notice that (I) does not depend on U2 Furthermore, i 21+1 and de in (II) can be replaced by (t-1)
w. 2241 and d. respectively using our induction hypothesis. Thus,
(6) argmin 18w Σ 9x Σ
2141
j=t+1
= argmin 1
|110((
(1-1) - Xx. +
Σ
(20 (w0()-1) - X113
j=t+1
N
[1 (1-1) 9x) + Σ
(w (t-1) x11
= argmin
U2441ERN-
2
m
w 2141"
23

---

Published as a conference paper at ICLR 2026
Step 2: Now we prove q++1 = 9t+1- We just constructed
N 0j
w 21-1
argmin 7 [l 1EU1ERNN-12
Σ
Σ ug
1=1
-=++1
This implies
N Σ
Xw- Σ
X
Xj= Pxt
(Xw Σ
(20)
j=1
j=++1
j=1
By construction, we have
N
||1 Σ
Kj -9x++1- - Σ wy x Xjl
0r+1 = argmin
PEA
1=1
j=1+2
= - ( - -Ej= X3 - Σi=r+2 w/) Xj)
||(21112
Then we can use Equation 20 deduce
(X141, Xw Σ =1 X E W Xj)
(1)
| 1
(X41, Σj= qj
ΣN -jj=t1 X, + X a
|x++1 12
(X41,, Xw Ej= Qj
DN 2ij=t1 (((x,))
(Xe+1, X 1 (( [x411 12
|
(X4+1, X441
(t)
Wi41
(t)
= =iifl-
The last step W/41 Wi41 follows from what we just proved in Step 1 that (0
(t)
=22411 Thus
(t)
we know
h++1 = Q(a(() = Q(w(2)) = -9+11
This completes our induction.
24


Reviews:
[{"review_id": "jQIecqfTks", "reviewer": "Reviewer_uY12", "summary": "This paper introduces Qronos, a new post-training quantization (PTQ) algorithm for large language models that explicitly corrects errors accumulated across layers during quantization. Unlike previous methods such as OPTQ, which assume identical activations before and after quantization, Qronos accounts for the drift that occurs when earlier layers are already quantized. It alternates between two key steps: correcting current quantization errors and diffusing residual errors into future weights to prevent accumulation.\nThe authors provide a theoretical interpretation linking Qronos to OPTQ and derive an efficient implementation based on Cholesky updates. Extensive experiments are conducted to compare with other PTQ methods.", "questions": "See weakness", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. Introduces a clear mismatch-aware formulation addressing activation drift across layers.\n2. Provides an efficient implementation for the proposed algorithm. \n3. Shows consistent empirical gains over prior PTQ methods across various models and benchmarks. \n4. Can be easily used with various quantization algorithms and improve their performance. \n5. Offers useful theoretical insights linking and generalizing existing algorithms like OPTQ.", "weaknesses": "1. The proposed method mainly extends OPTQ rather than introducing a fundamentally new algorithmic framework.\n2. The improvement on weight–activation quantization appears smaller than that on weight-only quantization; additional analysis would help clarify the underlying reason.\n3. The main paper reports results primarily for 3- and 4-bit settings, where existing methods already perform well. Including 2-bit quantization experiments would better demonstrate the robustness of the proposed approach under more challenging conditions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qOOocaSum7", "reviewer": "Reviewer_4LxD", "summary": "This paper investigates post-training quantization following a similar approach to the popular OPTQ/GPTQ framework. Specifically, the paper derives an optimization method that accounts for the accumulated errors from previously quantized layers. In addition, it proposes using Cholesky decomposition to accelerate the implementation. The method is evaluated on popular LLMs and combined with well-known quantization schemes, demonstrating improved accuracy in low-precision settings.", "questions": "1). The paper does not clearly explain how the algorithm is implemented. Could the authors include implementation details and pseudocode?\n\n2). For the W4A4KV4 setting, the weight quantizer uses a linear search. How is this linear search integrated into the proposed optimization process?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1). The paper is reasonably well written.\n\n2). The intuition and analysis are solid, helping to clarify the motivation behind the proposed techniques as well as offering insights into the strengths and weaknesses of the widely used GPTQ method.\n\n3). The experimental evaluation is comprehensive, particularly in combination with state-of-the-art techniques such as rotation and MagR.\n\n4). The results generally show improved accuracy compared to previous methods.", "weaknesses": "1). The overall contribution is incremental.\n\n2). The method introduces additional computational overhead to achieve accuracy gains over GPTQ. Since GPTQ also uses an approximate inverse Hessian to balance accuracy and speed, it is difficult to conclude that this method is definitively better than GPTQ.\n\n3). The improvements are limited for 8B models, and no results are reported for very large models.\n\n4). Some claims—particularly regarding implementation speedup and memory savings—seem misleading. The optimization process takes longer than GPTQ, yet the comparison is placed in the appendix. It would be more informative to present a detailed discussion of these trade-offs in the main paper to help assess the method’s advantages and limitations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MYVBPK09up", "reviewer": "Reviewer_vydk", "summary": "This paper studies post-training quantization (PTQ) problem and presents a method named Qronos for 4bits or lower. The key idea behind Qronos is to alternate between error correction and error diffusion via optimal update rules. Promising experimental results are reported for LLAMA3 and Qwen 3 models.", "questions": "1. Why did you name the proposed method Qronos? What is the purpose of adding \"...\" in the paper title?\n2. Did authors compare Qronos with SmoothRot [1] at 4bit?\n3. Is Theorem 3.1, Lemma 3.2, and Corollary 3.3 your own results or borrowed from the literature? If later, reference should be given.\n[1] https://arxiv.org/pdf/2506.05413", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "+ I have found the idea of alternating between error correction and diffusion novel and sensible.\n+ The reported experimental results are promising and seem to be among the current SOTA.\n+ The appendix contains detailed proofs for the theoretical results reported in Sec. 3.", "weaknesses": "- The literary presentation is still lacking. For example, \"GPTAQ has been observed to be unstable in other reproductions.\" reads weird and the paper's introduction section needs to be revised significantly (too much emphasis on experimental results and little space for explaining the motivation or significance).\n- Lemma 3.2 seems like a known result. I understand you provided rigorous proof for this lemma. But I believe that the connection between LS and Cholesky decomposition has been known in the literature. \n- A minor weakness is the notable artifacts left by AI models (the notorious dash). For example, four dashes can be found on page 1 alone.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)