Predicting Resource Efficient Hamiltonian Decomposition for Continuous-Time Quantum Walk Simulations

Mostafa Atallah; Rebekah Herrman; Zain H. Saleem

quant-phAug 21, 2026

Predicting Resource Efficient Hamiltonian Decomposition for Continuous-Time Quantum Walk Simulations

Authors: Mostafa Atallah, Rebekah Herrman, Zain H. Saleem

Organizations: University of Tennessee, Knoxville · Argonne National Laboratory

Abstract

Simulating a continuous-time quantum walk (CTQW) on a graph in the circuit model of quantum computing requires decomposing its Hamiltonian into terms that can be Trotterized into hardware-native gates. We consider two such decompositions: the standard Pauli decomposition and the recently introduced matching decomposition. Prior work suggests that the matching decomposition uses fewer CX gates on sparse graphs, while the Pauli decomposition uses fewer on denser graphs. Since CX gates dominate error and runtime on current hardware, we train machine learning models to predict, for a given graph, which of the two decompositions produces the smaller CX gate count. We train and evaluate on the complete population of all 11,117 connected eight-vertex graphs from Brendan McKay's database, so the class balance and overlap are measured directly rather than estimated. We use twelve features: ten topological properties of the graph and two that count the terms the Pauli and matching decompositions produce (n_Pauli and n_match), both computable without transpiling the simulation circuit. Standard topological properties alone provide little predictive power. Instead, the dominant signal comes from n_Pauli, a property of the Hamiltonian decomposition rather than an intrinsic property of the graph; degree variance is the only other feature that carries signal. Across a range of models the Matthews correlation coefficient (MCC) falls in a narrow band, from 0.569 untuned to 0.593 after tuning, so no single architecture stands out. We adopt a single-hidden-layer neural network at MCC 0.593. Applied frozen to a held-out, class-balanced test set of larger graphs (up to 256 vertices) from structured and Erdos-Renyi families, the model transfers, with MCC rising from 0.785 at N=8 to 1 at N>=64.

Figures & tables

operation	scaling in $N$	per graph at $N{=}256$
$n_{\mathrm{Pauli}}$ (tensorized Pauli count)	$O(N^{1.2})$	$0.003$ s
ten topological features	$O(N^{1.3})$	$0.05$ s
$n_{\mathrm{match}}$ (matching count)	$O(N^{1.5})$	$0.50$ s
all $12$ features	$O(N^{1.5})$	$\mathbf{0.55}$ s
build $+$ transpile (labeling)	$O(N^{3.2})$	$\mathbf{157}$ s

TABLE I : Per-graph cost of each feature block against labeling, with empirical scaling fit over

N\in\{8,16,32,64,128,256\}

(single core, hardware in the Reproducibility section).

dataset	graphs	match-wins	Pauli-wins	win %	train	val	test
McKay $N{=}8$	10,978	418	10,560	3.8%	7,025	1,757	2,196
Erdős-Rényi (ER)	10,981	20	10,961	0.2%	n/a	n/a	n/a
structured	9,026	6,879	2,147	76.2%	n/a	n/a	n/a
balanced	13,442	6,721	6,721	50%	n/a	n/a	13,442

TABLE II : Datasets. “win %” is the percentage of graphs where matching is cheaper, with ties (

\Delta_{\mathrm{CX}}=0

) excluded from all counts. The train/val/test columns give the

64/16/20

stratified split applied to the McKay population. The balanced dataset is used only as a held-out test set and the ER and structured families only as its two sources, so these three receive no such split (marked n/a).

Fig. 1 : Permutation feature importance on the training population, measured as the mean drop in predictive performance when each feature is shuffled (five seeds).

Fig. 2 : First two principal components of the twelve standardized features on the McKay population, with each graph colored by class. The matching-wins overlap the Pauli-wins rather than separating from them, so the two classes are not separated in these two principal directions.

Fig. 3 : Pairwise Pearson correlation between the twelve features on the McKay population, with features ordered by hierarchical clustering so that correlated groups appear adjacent.

model	acc.	F1	MCC	ROC-AUC	PR-AUC
gradient boosting	$0.973\,{\scriptstyle\pm 0.001}$	$0.558\,{\scriptstyle\pm 0.034}$	$\mathbf{0.569}\,{\scriptstyle\pm 0.023}$	$0.957\,{\scriptstyle\pm 0.01}$	$0.636\,{\scriptstyle\pm 0.015}$
logistic	$0.972\,{\scriptstyle\pm 0.001}$	$0.546\,{\scriptstyle\pm 0.04}$	$0.554\,{\scriptstyle\pm 0.03}$	$0.961\,{\scriptstyle\pm 0.01}$	$0.653\,{\scriptstyle\pm 0.032}$
LightGBM	$0.969\,{\scriptstyle\pm 0.001}$	$0.519\,{\scriptstyle\pm 0.036}$	$0.515\,{\scriptstyle\pm 0.031}$	$0.952\,{\scriptstyle\pm 0.009}$	$0.574\,{\scriptstyle\pm 0.021}$
random forest	$0.97\,{\scriptstyle\pm 0.001}$	$0.5\,{\scriptstyle\pm 0.054}$	$0.513\,{\scriptstyle\pm 0.041}$	$0.939\,{\scriptstyle\pm 0.015}$	$0.599\,{\scriptstyle\pm 0.026}$
ANN	$0.969\,{\scriptstyle\pm 0.003}$	$0.515\,{\scriptstyle\pm 0.043}$	$0.512\,{\scriptstyle\pm 0.045}$	$0.949\,{\scriptstyle\pm 0.012}$	$0.563\,{\scriptstyle\pm 0.055}$
XGBoost	$0.968\,{\scriptstyle\pm 0.001}$	$0.516\,{\scriptstyle\pm 0.025}$	$0.506\,{\scriptstyle\pm 0.023}$	$0.947\,{\scriptstyle\pm 0.01}$	$0.556\,{\scriptstyle\pm 0.02}$

TABLE III : Model comparison on the McKay population, full feature set, sorted by MCC (mean

\pm

std over five seeds). Remaining metrics in Table VIII .

Fig. 4 : Training and validation convergence of the tuned ANN on the McKay split (seed

0

): log-loss (left) and MCC (right) versus epoch. Train and validation track together, indicating the single

32

-unit layer generalizes rather than memorizes.

model	acc.	F1	MCC	ROC-AUC	PR-AUC
ANN	$0.974\,{\scriptstyle\pm 0.001}$	$0.586\,{\scriptstyle\pm 0.045}$	$\mathbf{0.593}\,{\scriptstyle\pm 0.036}$	$0.961\,{\scriptstyle\pm 0.009}$	$0.659\,{\scriptstyle\pm 0.026}$
gradient boosting	$0.974\,{\scriptstyle\pm 0.001}$	$0.574\,{\scriptstyle\pm 0.017}$	$0.583\,{\scriptstyle\pm 0.013}$	$0.955\,{\scriptstyle\pm 0.011}$	$0.633\,{\scriptstyle\pm 0.026}$
random forest	$0.959\,{\scriptstyle\pm 0.003}$	$0.568\,{\scriptstyle\pm 0.029}$	$0.558\,{\scriptstyle\pm 0.032}$	$0.95\,{\scriptstyle\pm 0.01}$	$0.628\,{\scriptstyle\pm 0.029}$
XGBoost	$0.971\,{\scriptstyle\pm 0.002}$	$0.544\,{\scriptstyle\pm 0.026}$	$0.543\,{\scriptstyle\pm 0.025}$	$0.956\,{\scriptstyle\pm 0.007}$	$0.596\,{\scriptstyle\pm 0.016}$
LightGBM	$0.971\,{\scriptstyle\pm 0.001}$	$0.53\,{\scriptstyle\pm 0.033}$	$0.536\,{\scriptstyle\pm 0.026}$	$0.957\,{\scriptstyle\pm 0.009}$	$0.603\,{\scriptstyle\pm 0.026}$
decision tree	$0.937\,{\scriptstyle\pm 0.005}$	$0.436\,{\scriptstyle\pm 0.021}$	$0.431\,{\scriptstyle\pm 0.02}$	$0.802\,{\scriptstyle\pm 0.008}$	$0.421\,{\scriptstyle\pm 0.026}$

TABLE IV : Per-model tuned sweep on the McKay population, best configuration each, sorted by MCC. Remaining metrics in Table X .

model	best hyperparameters	model	best hyperparameters
ANN	one $32$ -unit hidden layer, tanh, $\alpha=10^{-4}$ , lr $10^{-3}$	decision tree	entropy split, unbounded depth, min. leaf $5$
gradient boosting	$400$ trees, depth $2$ , learning rate $0.1$ , subsample $0.8$	SVM (RBF)	$C=10$ , $\gamma=0.01$
random forest	$400$ trees, unbounded depth, min. leaf $5$	logistic	$\ell_{2}$ penalty, $C\approx 3.16$ , lbfgs solver
XGBoost	$400$ trees, depth $3$ , learning rate $0.1$ , subsample $1$	$k$ -NN	$3$ neighbors, uniform weights, Euclidean
LightGBM	$400$ trees, $15$ leaves, learning rate $0.02$ , min. child $20$	naive Bayes	Gaussian, variance smoothing $10^{-3}$

TABLE V : Best configuration selected for each model by the per-model grid search on the McKay population (maximizing MCC), sorted by MCC.

Panel of Figure 10 — Fig. 5 : Tuned ANN evaluated on the held-out McKay test graphs, averaged over five random splits. In (a), PR-AUC is the area under the precision-recall curve. A value close to zero indicates a model does no better than random guessing. In (b), each cell counts the test graphs of a given true class (rows) assigned to a given predicted class (columns), with the diagonal holding the correct predictions.

feature subset	acc.	F1	MCC	ROC-AUC	PR-AUC
topological only ( $10$ )	$0.962\,{\scriptstyle\pm 0}$	$0.019\,{\scriptstyle\pm 0.009}$	$0.062\,{\scriptstyle\pm 0.035}$	$0.726\,{\scriptstyle\pm 0.017}$	$0.166\,{\scriptstyle\pm 0.025}$
$n_{\mathrm{Pauli}}$ only	$0.967\,{\scriptstyle\pm 0.001}$	$0.402\,{\scriptstyle\pm 0.051}$	$0.423\,{\scriptstyle\pm 0.038}$	$0.938\,{\scriptstyle\pm 0.01}$	$0.452\,{\scriptstyle\pm 0.037}$
$n_{\mathrm{Pauli}}+n_{\mathrm{match}}$	$0.967\,{\scriptstyle\pm 0.002}$	$0.415\,{\scriptstyle\pm 0.075}$	$0.433\,{\scriptstyle\pm 0.057}$	$0.946\,{\scriptstyle\pm 0.011}$	$0.5\,{\scriptstyle\pm 0.042}$
$n_{\mathrm{Pauli}}+$ degree var.	$0.974\,{\scriptstyle\pm 0.001}$	$0.564\,{\scriptstyle\pm 0.031}$	$0.574\,{\scriptstyle\pm 0.022}$	$0.961\,{\scriptstyle\pm 0.01}$	$0.637\,{\scriptstyle\pm 0.033}$
full set ( $12$ )	$0.974\,{\scriptstyle\pm 0.001}$	$0.586\,{\scriptstyle\pm 0.045}$	$\mathbf{0.593}\,{\scriptstyle\pm 0.036}$	$0.961\,{\scriptstyle\pm 0.009}$	$0.659\,{\scriptstyle\pm 0.026}$
threshold on $n_{\mathrm{Pauli}}$	$0.958\,{\scriptstyle\pm 0.005}$	$0.484\,{\scriptstyle\pm 0.03}$	$0.466\,{\scriptstyle\pm 0.032}$	$\cdot$	$\cdot$

TABLE VI : Feature-subset ablation for the tuned ANN on the McKay population (mean

\pm

std over five seeds). The last row is a one-rule threshold on

n_{\mathrm{Pauli}}

, which emits no score (

\cdot

). Remaining metrics in Table IX .

$n$	graphs	acc.	F1	MCC	ROC-AUC	PR-AUC
8	198	$0.89\,{\scriptstyle\pm 0.041}$	$0.887\,{\scriptstyle\pm 0.044}$	$0.785\,{\scriptstyle\pm 0.08}$	$0.962\,{\scriptstyle\pm 0.035}$	$0.961\,{\scriptstyle\pm 0.037}$
16	658	$0.852\,{\scriptstyle\pm 0.023}$	$0.862\,{\scriptstyle\pm 0.022}$	$0.711\,{\scriptstyle\pm 0.046}$	$0.933\,{\scriptstyle\pm 0.01}$	$0.924\,{\scriptstyle\pm 0.011}$
32	3,510	$0.996\,{\scriptstyle\pm 0.002}$	$0.996\,{\scriptstyle\pm 0.002}$	$0.993\,{\scriptstyle\pm 0.005}$	$0.996\,{\scriptstyle\pm 0.003}$	$0.994\,{\scriptstyle\pm 0.005}$
64	3,492	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$
128	3,426	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$
256	2,158	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$

TABLE VII : Tuned ANN tested per vertex size on the balanced held-out dataset (frozen model, mean

\pm

std over five seeds). Remaining metrics in Table XI .

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

model	prec.	recall	spec.	NPV
gradient boosting	$0.766\,{\scriptstyle\pm 0.06}$	$0.445\,{\scriptstyle\pm 0.059}$	$0.994\,{\scriptstyle\pm 0.003}$	$0.978\,{\scriptstyle\pm 0.002}$
logistic	$0.742\,{\scriptstyle\pm 0.027}$	$0.436\,{\scriptstyle\pm 0.059}$	$0.994\,{\scriptstyle\pm 0.002}$	$0.978\,{\scriptstyle\pm 0.002}$
LightGBM	$0.645\,{\scriptstyle\pm 0.025}$	$0.436\,{\scriptstyle\pm 0.046}$	$0.99\,{\scriptstyle\pm 0.001}$	$0.978\,{\scriptstyle\pm 0.002}$
random forest	$0.715\,{\scriptstyle\pm 0.051}$	$0.393\,{\scriptstyle\pm 0.076}$	$0.993\,{\scriptstyle\pm 0.003}$	$0.976\,{\scriptstyle\pm 0.003}$
ANN	$0.645\,{\scriptstyle\pm 0.08}$	$0.433\,{\scriptstyle\pm 0.051}$	$0.99\,{\scriptstyle\pm 0.003}$	$0.978\,{\scriptstyle\pm 0.002}$
XGBoost	$0.609\,{\scriptstyle\pm 0.032}$	$0.45\,{\scriptstyle\pm 0.041}$	$0.988\,{\scriptstyle\pm 0.002}$	$0.978\,{\scriptstyle\pm 0.002}$

Appendix

TABLE VIII : Other metrics for the model comparison (Table III ).

feature subset	prec.	recall	spec.	NPV
topological only ( $10$ )	$0.467\,{\scriptstyle\pm 0.323}$	$0.01\,{\scriptstyle\pm 0.005}$	$1\,{\scriptstyle\pm 0}$	$0.962\,{\scriptstyle\pm 0}$
$n_{\mathrm{Pauli}}$ only	$0.659\,{\scriptstyle\pm 0.04}$	$0.293\,{\scriptstyle\pm 0.051}$	$0.994\,{\scriptstyle\pm 0.002}$	$0.972\,{\scriptstyle\pm 0.002}$
$n_{\mathrm{Pauli}}+n_{\mathrm{match}}$	$0.651\,{\scriptstyle\pm 0.029}$	$0.312\,{\scriptstyle\pm 0.075}$	$0.993\,{\scriptstyle\pm 0.002}$	$0.973\,{\scriptstyle\pm 0.003}$
$n_{\mathrm{Pauli}}+$ degree var.	$0.771\,{\scriptstyle\pm 0.026}$	$0.448\,{\scriptstyle\pm 0.049}$	$0.995\,{\scriptstyle\pm 0.001}$	$0.978\,{\scriptstyle\pm 0.002}$
full set ( $12$ )	$0.774\,{\scriptstyle\pm 0.036}$	$0.476\,{\scriptstyle\pm 0.066}$	$0.994\,{\scriptstyle\pm 0.002}$	$0.979\,{\scriptstyle\pm 0.003}$
threshold on $n_{\mathrm{Pauli}}$	$0.47\,{\scriptstyle\pm 0.047}$	$0.512\,{\scriptstyle\pm 0.082}$	$0.976\,{\scriptstyle\pm 0.007}$	$0.981\,{\scriptstyle\pm 0.003}$

Appendix

TABLE IX : Other metrics for the feature-subset ablation (Table VI ).

model	prec.	recall	spec.	NPV
ANN	$0.774\,{\scriptstyle\pm 0.036}$	$0.476\,{\scriptstyle\pm 0.066}$	$0.994\,{\scriptstyle\pm 0.002}$	$0.979\,{\scriptstyle\pm 0.003}$
gradient boosting	$0.772\,{\scriptstyle\pm 0.049}$	$0.46\,{\scriptstyle\pm 0.034}$	$0.994\,{\scriptstyle\pm 0.002}$	$0.979\,{\scriptstyle\pm 0.001}$
random forest	$0.48\,{\scriptstyle\pm 0.024}$	$0.698\,{\scriptstyle\pm 0.051}$	$0.97\,{\scriptstyle\pm 0.003}$	$0.988\,{\scriptstyle\pm 0.002}$
XGBoost	$0.689\,{\scriptstyle\pm 0.038}$	$0.45\,{\scriptstyle\pm 0.03}$	$0.992\,{\scriptstyle\pm 0.002}$	$0.978\,{\scriptstyle\pm 0.001}$
LightGBM	$0.711\,{\scriptstyle\pm 0.032}$	$0.426\,{\scriptstyle\pm 0.048}$	$0.993\,{\scriptstyle\pm 0.002}$	$0.978\,{\scriptstyle\pm 0.002}$
decision tree	$0.332\,{\scriptstyle\pm 0.021}$	$0.638\,{\scriptstyle\pm 0.014}$	$0.949\,{\scriptstyle\pm 0.005}$	$0.985\,{\scriptstyle\pm 0.001}$

Appendix

TABLE X : Other metrics for the per-model tuned sweep (Table IV ).

$n$	prec.	recall	spec.	NPV
8	$0.911\,{\scriptstyle\pm 0.059}$	$0.87\,{\scriptstyle\pm 0.068}$	$0.91\,{\scriptstyle\pm 0.066}$	$0.879\,{\scriptstyle\pm 0.054}$
16	$0.807\,{\scriptstyle\pm 0.025}$	$0.924\,{\scriptstyle\pm 0.032}$	$0.779\,{\scriptstyle\pm 0.034}$	$0.912\,{\scriptstyle\pm 0.032}$
32	$0.993\,{\scriptstyle\pm 0.005}$	$1\,{\scriptstyle\pm 0}$	$0.993\,{\scriptstyle\pm 0.005}$	$1\,{\scriptstyle\pm 0}$
64	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$
128	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$
256	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$	$1\,{\scriptstyle\pm 0}$

Appendix

TABLE XI : Other metrics for the per-size transfer test (Table VII ).

Quantum Circuit Simulation

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Graph Neural Networks (GNN) and Transformer-based architectures have achieved remarkable progress in graph learning, yet they still struggle to capture both global structural dependencies and model the dynamic information propagation. In this paper, we propose CTQWformer, a hybrid graph learning framework that integrates continuous-time quantum walks (CTQW) with GNN. CTQWformer employs a trainable Hamiltonian that fuses graph topology and node features, enabling physically grounded modeling of quantum walk dynamics that captures rich and intricate graph structure information. The extracted CTQW-based representations are incorporated into two complementary modules:(i) a Graph Transformer module that embeds final-time propagation probabilities as structural biases in the self-attention mechanism, and (ii) a Graph Recurrent Module that captures temporal evolution patterns with bidirectional recurrent networks. Extensive experiments on benchmark graph classification datasets demonstrate that CTQWformer outperforms graph kernel and GNN-based methods, demonstrating the potential of integrating quantum dynamics into trainable deep learning frameworks for graph representation learning. To the best of our knowledge, CTQWformer is the first hybrid CTQW-based Transformer, integrating CTQW-derived structural bias with temporal evolution modeling to advance graph learning.

Zhan Li, Wuqing Yu, Yusen Wu +1

school of Artificial Intelligence, Beijing Normal University, Beijing, China

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Graph Neural Networks (GNNs) are widely used on graph-structured data, but most suffer from two key weaknesses. First, message passing behaves as a low-pass filter under the homophily assumption, leading to poor performance on heterophilic graphs. Second, stacking layers drives node features toward constants, causing over-smoothing. Existing methods usually address these issues separately, while the few joint solutions rely largely on empirical heuristics, and many over-smoothing remedies sacrifice model expressiveness. We propose CTQW-GNN, a GNN based on Continuous-Time Quantum Walks (CTQW), to address both issues with theoretical justification. Its design exploits two properties of the CTQW propagator

e^{-\mathrm{i}Ht}

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Yuliang Zhan, Zefeng Gao, Jian Li +2

Renmin University of China Beijing, China

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May 13, 2026

Physics Guided Generative Optimization for Trotter Suzuki Decomposition

Trotter Suzuki product formulas are the standard route to Hamiltonian evolution on noisy intermediate-scale quantum (\NISQ{}) hardware, but their accuracy depends on three coupled choices: term grouping, product-formula order, and time-step allocation. Grouping and order are discrete, which makes direct gradient optimization infeasible and forces existing compilers to rely on static heuristics. We describe P-GONE, a method that combines a conditional diffusion model (D3PM + DDPM), a graph neural network (\GNN{}) encoder, and closed-loop REINFORCE fine-tuning to jointly learn grouping, order, and time-step optimization over a mixed discrete-continuous space. Under fidelity-matched conditions (

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19.4\times

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1.6\times

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T=0.90

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T=0.95

the teacher still leads -- a stratified pattern that points toward fidelity-aware fine-tuning. Under a standard depolarizing noise model, the method achieves noisy fidelity roughly

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>

time allocation

>

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N=32

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T \geq 0.95

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WenBin Yan

University of Colorado Boulder[quant-ph]

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