VQIL: Visible Quantum Interference Learning for Perturbation Response Ranking on Graphs

Seokjun Lee; Sergio A. Ortega; Daniel K. Park; Namshik Han

cs.LGOct 9, 2026

VQIL: Visible Quantum Interference Learning for Perturbation Response Ranking on Graphs

Authors: Seokjun Lee, Sergio A. Ortega, Daniel K. Park, Namshik Han

Organizations: Milner Therapeutics Institute, University of Cambridge, Cambridge, United Kingdom · Cambridge Stem Cell Institute, University of Cambridge, Cambridge, United Kingdom · Department of Applied Mathematics and Theoretical Physics, Cambridge Centre for AI in Medicine, University of Cambridge, Cambridge, United Kingdom · Department of Statistics and Data Science, Yonsei University, Seoul, Republic of Korea · Department of Applied Statistics, Yonsei University, Seoul, Republic of Korea · Department of Quantum Information, Yonsei University, Seoul, Republic of Korea · Urological Malignancies Virtual Institute, University of Cambridge, Cambridge, United Kingdom · Department of Nano Biomedical Engineering, Advanced Science Institute, Yonsei University, Seoul, Republic of Korea · Institute for Basic Science (IBS) Center for Nanomedicine, Yonsei University, Seoul, Republic of Korea · Department of Artificial Intelligence, Yonsei University, Seoul, Republic of Korea · College of Pharmacy, Yonsei University, Incheon, Republic of Korea · Institute for Convergence Research and Education in Advanced Technology, Yonsei University, Seoul, Republic of Korea

Abstract

Learning how local perturbations affect a network is important for identifying which components drive or absorb a system's response. We formulate this as seed-conditioned response ranking: given a perturbation at one seed node, ranking all other nodes by response strength. This is challenging when the model is given only the interaction topology, while the mechanism governing how perturbations propagate must be learned from data. We present VQIL (Visible Quantum Interference Learning), a trainable Szegedy quantum walk for seed-conditioned ranking. Edge weights, link phases, and per-node phase rotations are optimized end-to-end by backpropagation while the walk operator remains unitary throughout training. Node scores are obtained from a geometrically discounted coherent sum of quantum-walk amplitudes, enabling quantum interference before the final squared-norm readout. Because each learned parameter is attached to a named edge or node, the trained model provides directly inspectable hypotheses about the roles of individual graph components in the propagation mechanism. On AC power-flow perturbations over six transmission grids, with electrical parameters withheld from the model, VQIL improves Spearman rank correlation from 0.147 for the untrained walk to 0.831 and outperforms an identically trained random-walk model. On gene perturbation over five protein-protein interaction networks with Perturb-seq supervision, VQIL improves rank correlation from 0.171 to 0.620, outperforming the strongest learned classical and neural baselines. These results show that trainable quantum-walk interference provides a scalable and interpretable inductive bias for learning seed-conditioned response rankings on graphs.

Figures & tables

	Phase-angle response $\|\Delta\theta\|$		Voltage response $\|\Delta V\|$
Model	Spearman $\rho$	AUPRC@10%	Spearman $\rho$	AUPRC@10%
Naive-VQIL (untrained)	0.312 $\pm$ 0.269	0.541 $\pm$ 0.210	0.147 $\pm$ 0.154	0.310 $\pm$ 0.074
Standard-VQIL	0.880 $\pm$ 0.049	0.750 $\pm$ 0.107	0.799 $\pm$ 0.066	0.573 $\pm$ 0.132
Link-phased-VQIL	0.881 $\pm$ 0.048	0.749 $\pm$ 0.108	0.799 $\pm$ 0.066	0.572 $\pm$ 0.133
Vertex-phased-VQIL	0.911 $\pm$ 0.043	0.830 $\pm$ 0.062	0.831 $\pm$ 0.056	0.624 $\pm$ 0.144
Graph-phased-VQIL	0.917 $\pm$ 0.042	0.835 $\pm$ 0.065	0.829 $\pm$ 0.056	0.623 $\pm$ 0.144

Table 1: Power-grid perturbation-response ranking. Mean

\pm

sd over 30 cells (six power grids

\times

five seed-disjoint folds). VQIL variants use symmetric edge weights and link phases with discount

\gamma=0.9

; the classical control uses the matched restart

r=1-\gamma=0.1

. Results at

\gamma=0.7

are in Appendix Table 8 . Bold : main VQIL variant.

	Gene expression response $\|\mathrm{logFC}\|$
Model	Spearman $\rho$	AUPRC@10%	AUROC@10%	NDCG@10%
Naive-VQIL (untrained)	0.171 $\pm$ 0.050	0.155 $\pm$ 0.026	0.616 $\pm$ 0.040	0.330 $\pm$ 0.053
Standard-VQIL	0.587 $\pm$ 0.157	0.336 $\pm$ 0.080	0.824 $\pm$ 0.039	0.532 $\pm$ 0.045
Link-phased-VQIL	0.587 $\pm$ 0.157	0.336 $\pm$ 0.080	0.824 $\pm$ 0.039	0.532 $\pm$ 0.044
Vertex-phased-VQIL	0.620 $\pm$ 0.167	0.361 $\pm$ 0.086	0.849 $\pm$ 0.042	0.552 $\pm$ 0.045
Graph-phased-VQIL	0.620 $\pm$ 0.167	0.361 $\pm$ 0.085	0.849 $\pm$ 0.042	0.554 $\pm$ 0.047

Table 2: Gene-perturbation response ranking on protein–protein interaction graphs. Mean

\pm

sd over 25 cells (five Perturb-seq datasets

\times

five seed-disjoint folds). Model configurations match Table 1 (

\gamma=0.9

). Results at

\gamma=0.7

are in Appendix Table 9 . Bold : main VQIL variant.

	Power grids		PPI networks
Readout	Phase-angle $\|\Delta\theta\|$	Voltage $\|\Delta V\|$	Gene expression
Uniform, add-then-square	0.703 $\pm$ 0.165	0.773 $\pm$ 0.099	0.515 $\pm$ 0.136
Uniform, square-then-add	0.782 $\pm$ 0.191	0.692 $\pm$ 0.149	0.453 $\pm$ 0.155
Geometric, add-then-square	0.911 $\pm$ 0.043	0.831 $\pm$ 0.056	0.620 $\pm$ 0.167
Geometric, square-then-add	0.839 $\pm$ 0.126	0.733 $\pm$ 0.093	0.517 $\pm$ 0.156

Table 3: Readout ablation at the

\gamma=0.9

horizon: the

2\times 2

factorial over step weighting (

K=109

) and the order in which the squared norm is taken. All arms are Vertex-phased-VQIL trained under the protocol of Tables 1 and 2 . Spearman

\rho

, mean

\pm

sd over five seed-disjoint folds of six grids (30 cells) and five Perturb-seq datasets (25 cells). Bold : coherent resolvent readout of VQIL.

Figure 1: Trained fields on the 30-bus grid (case30; voltage response, $\gamma=0.9$ , five-fold means). (a) Ground-truth branch susceptance (edges) and bus class (markers). (b) Vertex-phased-VQIL: edge weights ( $\omega$ ), which follow susceptance, and node-phase magnitudes $|\delta|$ , smallest at regulated buses. (c) Trained-RWR learns only edge weights ( $\omega$ ). Fields show 5-fold means.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Source	Details	$\|V\|$	$\|E\|$	Avg. deg.	Density	Diam.	Queries	Train / Val / Test	$\rho_{\mathrm{hop}}$
Domain 1: power grids (pandapower); responses $\|\Delta\theta\|$ and $\|\Delta V\|$
case30	IEEE 30-bus	6 gen., 0 transf.	30	41	2.73	0.0943	6	30	18 / 6 / 6	0.344 / 0.383
case57	IEEE 57-bus	7 gen., 15 transf.	57	78	2.74	0.0489	12	52	30–32 / 10–11 / 10–11	0.213 / 0.082
case118	IEEE 118-bus	54 gen., 13 transf.	118	179	3.03	0.0259	14	117	69–71 / 23–24 / 23–24	0.571 / 0.214
case145	IEEE 145-bus	59 gen., 64 transf.	145	422	5.82	0.0404	11	144	86–87 / 28–29 / 28–29	0.339 / 0.153
case300	IEEE 300-bus	77 gen., 128 transf.	300	409	2.73	0.0091	24	281	168–169 / 56–57 / 56–57	0.485 / 0.381

Appendix

Table 4: Datasets and graphs. The model sees only the unweighted, undirected graph and the seed. Everything else is withheld. Domain 1: bus graph of in-service lines and transformers (largest connected component). A query injects

+20

MW /

+6

MVAr at the seed bus and records the AC power-flow response at every other bus (non-converged probes are dropped). gen./transf.: generator buses and transformer branches, both withheld from the model. Domain 2: largest connected component of STRING v12.0 human interactions with combined score

\geq 400

over the measured genes (no edges added). A query is one perturbed gene (Frangieh: control arm only). In both domains all non-seed nodes are candidates, the top 10% by response are positives, and splits are seed-disjoint five-fold partitions (per-fold query counts shown).

\rho_{\mathrm{hop}}

: Spearman correlation between

-

(hop distance to the seed) and the response, averaged over queries (angle / voltage for grids); a proximity-only baseline.

	GCN	GAT	Sheaf- $\mathcal{O}(2)$
Architecture (both domains)
Input features (50-dim)	seed indicator, degree, log-degree, hop distance to seed (one-hot, 0–12, $\geq$ 13), RWSE-16, LapPE-16; one RWR-landing channel held at zero
Message passing	sym.-normalised, self-loops	single-head additive attention, self-loops	$\mathcal{O}(2)$ sheaf diffusion, 2-dim stalks
Hidden width	64	64	16 channels $\times$ 2
Layer details	residual, LayerNorm, LeakyReLU(0.1)	residual, LayerNorm, LeakyReLU(0.1)	per-edge rotation angle from 2-layer MLP (32), learned step $\varepsilon$
Dropout	0.2	0.2	0.2

Appendix

Table 5: Neural baseline configurations. Entries of the form a / b give Domain 1 / Domain 2. Architecture and training protocol are shared. Depth and learning rate were selected per domain on validation Spearman

\rho

from

\{0.005,0.01,0.05\}\times\{3,6\}

layers (Domain 1: case118, fold 0, mean over both responses; Domain 2: Adamson, fold 0) and then fixed for all datasets of that domain. The training protocol (lower block) is identical for every VQIL, RWR and neural arm, except that VQIL and RWR use learning rate

0.05

	Domain 1: power grids						Domain 2: PPI graphs
Model	case30	case57	case118	case145	case300	case1354	K562	RPE1	Frangieh	Tian-i	Adamson
(a) Seconds per epoch
Standard-VQIL	0.20	0.28	0.62	0.76	1.49	3.08	40.1	34.1	8.49	3.10	3.57
Link-phased-VQIL	0.21	0.28	0.63	0.77	1.51	3.08	40.3	34.3	8.54	3.11	3.59
Vertex-phased-VQIL	0.21	0.28	0.63	0.79	1.54	3.09	41.1	34.8	8.68	3.15	3.65
Graph-phased-VQIL	0.20	0.29	0.64	0.78	1.56	3.13	41.2	35.1	8.72	3.16	3.66

Appendix

Table 6: Training cost (

\gamma=0.9

for VQIL variants,

r=0.1

for RWR). (a) Wall-clock seconds per epoch, computed as total cell time divided by epochs run (so fixed evaluation cost is included); (b) mean total seconds per cell; (c) median epochs to early stopping (at most 300, patience 10). Means and medians over the five folds (and, for Domain 1, both responses). † Untrained models: test-set inference time only.

		Domain 1: power grids						Domain 2: PPI graphs
Model	Scaling	case30	case57	case118	case145	case300	case1354	K562	RPE1	Frangieh	Tian-i	Adamson
Nodes $\|V\|$		30	57	118	145	300	1,354	5,787	5,655	5,023	2,965	4,950
Edges $\|E\|$		41	78	179	422	409	1,710	218,844	194,587	99,522	29,059	105,599
Naive-VQIL (untrained)	–	0	0	0	0	0	0	0	0	0	0	0
Standard-VQIL	$\|E\|$	41	78	179	422	409	1,710	218.8K	194.6K	99.5K	29.1K	105.6K
Link-phased-VQIL	$2\|E\|$	82	156	358	844	818	3,420	437.7K	389.2K	199.0K	58.1K	211.2K

Appendix

Table 7: Trainable parameters per model and dataset (

\mathrm{K}=10^{3}

). VQIL and RWR learn one weight per undirected edge, plus one phase per node (vertex-phased) and/or per edge (link-phased); Phased-only-VQIL freezes the weights and trains node and edge phases. Neural baselines have a size independent of the graph but set by the depth selected per domain (Table 5 : 3 layers for Domain 1; 6 for GCN and GAT in Domain 2). Counts do not depend on

\gamma

or on the response.

	Phase-angle response $\|\Delta\theta\|$		Voltage response $\|\Delta V\|$
Model	Spearman $\rho$	AUPRC@10%	Spearman $\rho$	AUPRC@10%
Naive-VQIL (untrained)	0.441 $\pm$ 0.160	0.715 $\pm$ 0.104	0.238 $\pm$ 0.099	0.427 $\pm$ 0.123
Standard-VQIL	0.755 $\pm$ 0.119	0.735 $\pm$ 0.094	0.620 $\pm$ 0.099	0.511 $\pm$ 0.128
Link-phased-VQIL	0.755 $\pm$ 0.119	0.735 $\pm$ 0.094	0.620 $\pm$ 0.099	0.511 $\pm$ 0.128
Vertex-phased-VQIL	0.849 $\pm$ 0.085	0.777 $\pm$ 0.087	0.704 $\pm$ 0.080	0.548 $\pm$ 0.137
Graph-phased-VQIL	0.855 $\pm$ 0.083	0.780 $\pm$ 0.082	0.704 $\pm$ 0.076	0.548 $\pm$ 0.133

Appendix

Table 8: Power-grid perturbation-response ranking at the shorter readout horizon,

\gamma=0.7

(matched restart

r=1-\gamma=0.3

for RWR); all other settings as in Table 1 . Mean

\pm

sd over 30 cells (six power grids

\times

five seed-disjoint folds). The neural baselines have no horizon parameter, so their rows repeat Table 1 . Bold : main VQIL variant.

	Gene expression response $\|\mathrm{logFC}\|$
Model	Spearman $\rho$	AUPRC@10%	AUROC@10%	NDCG@10%
Naive-VQIL (untrained)	0.171 $\pm$ 0.057	0.153 $\pm$ 0.029	0.615 $\pm$ 0.042	0.323 $\pm$ 0.039
Standard-VQIL	0.554 $\pm$ 0.147	0.269 $\pm$ 0.062	0.798 $\pm$ 0.041	0.454 $\pm$ 0.043
Link-phased-VQIL	0.554 $\pm$ 0.147	0.269 $\pm$ 0.062	0.798 $\pm$ 0.041	0.454 $\pm$ 0.043
Vertex-phased-VQIL	0.587 $\pm$ 0.161	0.278 $\pm$ 0.061	0.819 $\pm$ 0.042	0.456 $\pm$ 0.041
Graph-phased-VQIL	0.587 $\pm$ 0.160	0.278 $\pm$ 0.060	0.819 $\pm$ 0.041	0.456 $\pm$ 0.042

Appendix

Table 9: Gene-perturbation response ranking on protein–protein interaction graphs at the shorter readout horizon,

\gamma=0.7

(matched restart

r=0.3

for RWR); all other settings as in Table 2 . Mean

\pm

sd over 25 cells (five Perturb-seq datasets

\times

five seed-disjoint folds). The neural baselines have no horizon parameter, so their rows repeat Table 2 . Bold : main VQIL variant.

	Power grids		PPI networks
Readout	Phase-angle $\|\Delta\theta\|$	Voltage $\|\Delta V\|$	Gene expression
Uniform, add-then-square	0.456 $\pm$ 0.120	0.299 $\pm$ 0.168	0.508 $\pm$ 0.144
Uniform, square-then-add	0.495 $\pm$ 0.145	0.310 $\pm$ 0.178	0.516 $\pm$ 0.141
Geometric, add-then-square	0.849 $\pm$ 0.085	0.704 $\pm$ 0.080	0.587 $\pm$ 0.161
Geometric, square-then-add	0.861 $\pm$ 0.083	0.715 $\pm$ 0.086	0.536 $\pm$ 0.153

Appendix

Table 10: Readout ablation at the

\gamma=0.7

horizon: the

2\times 2

factorial over step weighting (

K=40

) and the order in which the squared norm is taken. Same protocol, cells and layout as Table 3 .Spearman

\rho

, mean

\pm

sd over five seed-disjoint folds of six grids (30 cells) and five Perturb-seq datasets (25 cells). Bold : coherent resolvent readout of VQIL.

Panel of Figure 2 — Figure 2: Agreement between the edge-weight fields learned by Vertex-phased-VQIL and Trained-RWR on identical edges. Axes show consensus weights $\bar{\omega}$ (means over the five folds; horizontal: Trained-RWR, vertical: Vertex-phased-VQIL), the dashed line is the identity, and the annotated $r$ is the Pearson correlation between the two consensus fields. Top: power grids (one point per edge) for the phase-angle and voltage responses at $\gamma=0.9$ (RWR restart $0.1$ ) and $\gamma=0.7$ (restart $0.3$ ). Bottom: PPI networks at both horizons, as hexagonal bins colored by the number of edges per bin (log scale).

Panel of Figure 3 — Figure 3: Distributions of the trained edge weights $\omega$ of Vertex-phased-VQIL (solid) and Trained-RWR (dashed), pooled over all edges and the five folds (log-scale density; weights are clamped to $[-2,2]$ ). Top: power grids (columns) for the phase-angle and voltage responses at $\gamma=0.9$ and $\gamma=0.7$ . Bottom: PPI networks at both horizons. At $\gamma=0.9$ , Trained-RWR places $46\%$ (grids) and $43\%$ (PPI networks) of its weights at the $\pm 2$ bounds, compared with $21\%$ and $25\%$ for VQIL.

Panel of Figure 4 — Figure 4: Distributions of the learned node-phase offsets $\delta$ of Vertex-phased-VQIL (Trained-RWR has no node parameter), pooled over all nodes and the five folds, at $\gamma=0.9$ (left) and $\gamma=0.7$ (right); $\delta=0$ is the standard walk. Top: power grids (one line per grid) for the phase-angle and voltage responses. Bottom: PPI networks (one line per dataset). The distributions are nearly symmetric about zero. At $\gamma=0.9$ , $0.3\%$ (grids) and $0.06\%$ (PPI networks) of the nodes lie at the $\pm\pi$ bound; at $\gamma=0.7$ , a small fraction of nodes accumulates at the bound.

Panel of Figure 5 — Figure 5: Trained fields on the remaining five power grids (case57, case118, case145, case300, and case1354), shown as in Figure 1 (voltage response, $\gamma=0.9$ , means over the five folds). (a) Withheld branch susceptance $-\log x$ (edge color) and bus class (marker: slack, generator, load, or transit bus). (b) Vertex-phased-VQIL: learned edge weights $\omega$ (edge color) and node-phase magnitudes $|\delta|$ (node color; the upper end of the color scale is set per grid). (c) Trained-RWR ( $r=0.1$ ): learned edge weights; RWR has no node parameter.

Panel of Figure 6 — Figure 6: Trained fields on the five PPI networks (K562, RPE1, Frangieh, Tian-i, and Adamson; $\gamma=0.9$ , means over the five folds). (a) Withheld biology: control-cell co-expression of the two endpoint genes (edge color) and recurrent response, the fraction of training perturbations in which a gene is among the top 10% of responders (node color and size). (b) Vertex-phased-VQIL: learned edge weights $\omega$ (edge color) and node-phase magnitudes $|\delta|$ (node color and size). (c) Trained-RWR ( $r=0.1$ ): learned edge weights; RWR has no node parameter.

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Graphs provide a natural language for relational data in chemistry, biology and optimisation. Graph neural networks (GNNs) have driven much of the recent progress in learning from such data through message passing, a single primitive that generalises convolution and attention. Quantum counterparts have been proposed, but with limited connection to message passing and few guarantees on performance or scalability. More broadly, the trainability of variational quantum circuits is a recognised bottleneck for their wide applicability, and pre-training has emerged as one way to address it. Yet for a quantum model to be useful, it must offer expressivity guarantees along with demonstrable scalability. Here we show how a quantum graph neural network can be built to perform message passing, to be permutation equivariant, and to sit at a chosen level of the Weisfeiler-Leman hierarchy, the standard measure of how finely a model can tell graphs apart. We show that, as for classical GNNs, the training can be done first on small graph instances, allowing for a pre-training that can mitigate usual training issues, and its output can be read out at a cost that stays low as the graph grows. We validate the framework in large-scale simulations of up to 56 qubits across three datasets, on synthetic graphs that ordinary message passing cannot separate, on molecular property prediction, and on the travelling salesperson problem. Our framework opens a path for near-term quantum algorithms with theoretical guarantees and practical scalability, bringing the principles of graph learning into quantum circuit design.

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