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
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 ∣Δθ∣
Voltage response ∣ΔV∣
Model
Spearman ρ
AUPRC@10%
Spearman ρ
AUPRC@10%
Naive-VQIL (untrained)
0.312 ± 0.269
0.541 ± 0.210
0.147 ± 0.154
0.310 ± 0.074
Standard-VQIL
0.880 ± 0.049
0.750 ± 0.107
0.799 ± 0.066
0.573 ± 0.132
Link-phased-VQIL
0.881 ± 0.048
0.749 ± 0.108
0.799 ± 0.066
0.572 ± 0.133
Vertex-phased-VQIL
0.911 ± 0.043
0.830 ± 0.062
0.831 ± 0.056
0.624 ± 0.144
Graph-phased-VQIL
0.917 ± 0.042
0.835 ± 0.065
0.829 ± 0.056
0.623 ± 0.144
Table 1: Power-grid perturbation-response ranking. Mean ± sd over 30 cells (six power grids × five seed-disjoint folds). VQIL variants use symmetric edge weights and link phases with discount γ=0.9 ; the classical control uses the matched restart r=1−γ=0.1 . Results at γ=0.7 are in Appendix Table 8 . Bold : main VQIL variant.
Gene expression response ∣logFC∣
Model
Spearman ρ
AUPRC@10%
AUROC@10%
NDCG@10%
Naive-VQIL (untrained)
0.171 ± 0.050
0.155 ± 0.026
0.616 ± 0.040
0.330 ± 0.053
Standard-VQIL
0.587 ± 0.157
0.336 ± 0.080
0.824 ± 0.039
0.532 ± 0.045
Link-phased-VQIL
0.587 ± 0.157
0.336 ± 0.080
0.824 ± 0.039
0.532 ± 0.044
Vertex-phased-VQIL
0.620 ± 0.167
0.361 ± 0.086
0.849 ± 0.042
0.552 ± 0.045
Graph-phased-VQIL
0.620 ± 0.167
0.361 ± 0.085
0.849 ± 0.042
0.554 ± 0.047
Table 2: Gene-perturbation response ranking on protein–protein interaction graphs. Mean ± sd over 25 cells (five Perturb-seq datasets × five seed-disjoint folds). Model configurations match Table 1 ( γ=0.9 ). Results at γ=0.7 are in Appendix Table 9 . Bold : main VQIL variant.
Power grids
PPI networks
Readout
Phase-angle ∣Δθ∣
Voltage ∣ΔV∣
Gene expression
Uniform, add-then-square
0.703 ± 0.165
0.773 ± 0.099
0.515 ± 0.136
Uniform, square-then-add
0.782 ± 0.191
0.692 ± 0.149
0.453 ± 0.155
Geometric, add-then-square
0.911 ± 0.043
0.831 ± 0.056
0.620 ± 0.167
Geometric, square-then-add
0.839 ± 0.126
0.733 ± 0.093
0.517 ± 0.156
Table 3: Readout ablation at the γ=0.9 horizon: the 2×2 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 ρ , mean ± 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, γ=0.9 , five-fold means). (a) Ground-truth branch susceptance (edges) and bus class (markers). (b) Vertex-phased-VQIL: edge weights ( ω ), which follow susceptance, and node-phase magnitudes ∣δ∣ , smallest at regulated buses. (c) Trained-RWR learns only edge weights ( ω ). 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
ρhop
Domain 1: power grids (pandapower); responses ∣Δθ∣ and ∣ΔV∣
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 ≥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). ρhop : 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- O(2)
Architecture (both domains)
Input features (50-dim)
seed indicator, degree, log-degree, hop distance to seed (one-hot, 0–12, ≥ 13), RWSE-16, LapPE-16; one RWR-landing channel held at zero
Message passing
sym.-normalised, self-loops
single-head additive attention, self-loops
O(2) sheaf diffusion, 2-dim stalks
Hidden width
64
64
16 channels × 2
Layer details
residual, LayerNorm, LeakyReLU(0.1)
residual, LayerNorm, LeakyReLU(0.1)
per-edge rotation angle from 2-layer MLP (32), learned step ε
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 ρ from {0.005,0.01,0.05}×{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 ( γ=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 ( K=103 ). 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 γ or on the response.
Phase-angle response ∣Δθ∣
Voltage response ∣ΔV∣
Model
Spearman ρ
AUPRC@10%
Spearman ρ
AUPRC@10%
Naive-VQIL (untrained)
0.441 ± 0.160
0.715 ± 0.104
0.238 ± 0.099
0.427 ± 0.123
Standard-VQIL
0.755 ± 0.119
0.735 ± 0.094
0.620 ± 0.099
0.511 ± 0.128
Link-phased-VQIL
0.755 ± 0.119
0.735 ± 0.094
0.620 ± 0.099
0.511 ± 0.128
Vertex-phased-VQIL
0.849 ± 0.085
0.777 ± 0.087
0.704 ± 0.080
0.548 ± 0.137
Graph-phased-VQIL
0.855 ± 0.083
0.780 ± 0.082
0.704 ± 0.076
0.548 ± 0.133
Appendix
Table 8: Power-grid perturbation-response ranking at the shorter readout horizon, γ=0.7 (matched restart r=1−γ=0.3 for RWR); all other settings as in Table 1 . Mean ± sd over 30 cells (six power grids × five seed-disjoint folds). The neural baselines have no horizon parameter, so their rows repeat Table 1 . Bold : main VQIL variant.
Gene expression response ∣logFC∣
Model
Spearman ρ
AUPRC@10%
AUROC@10%
NDCG@10%
Naive-VQIL (untrained)
0.171 ± 0.057
0.153 ± 0.029
0.615 ± 0.042
0.323 ± 0.039
Standard-VQIL
0.554 ± 0.147
0.269 ± 0.062
0.798 ± 0.041
0.454 ± 0.043
Link-phased-VQIL
0.554 ± 0.147
0.269 ± 0.062
0.798 ± 0.041
0.454 ± 0.043
Vertex-phased-VQIL
0.587 ± 0.161
0.278 ± 0.061
0.819 ± 0.042
0.456 ± 0.041
Graph-phased-VQIL
0.587 ± 0.160
0.278 ± 0.060
0.819 ± 0.041
0.456 ± 0.042
Appendix
Table 9: Gene-perturbation response ranking on protein–protein interaction graphs at the shorter readout horizon, γ=0.7 (matched restart r=0.3 for RWR); all other settings as in Table 2 . Mean ± sd over 25 cells (five Perturb-seq datasets × 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 ∣Δθ∣
Voltage ∣ΔV∣
Gene expression
Uniform, add-then-square
0.456 ± 0.120
0.299 ± 0.168
0.508 ± 0.144
Uniform, square-then-add
0.495 ± 0.145
0.310 ± 0.178
0.516 ± 0.141
Geometric, add-then-square
0.849 ± 0.085
0.704 ± 0.080
0.587 ± 0.161
Geometric, square-then-add
0.861 ± 0.083
0.715 ± 0.086
0.536 ± 0.153
Appendix
Table 10: Readout ablation at the γ=0.7 horizon: the 2×2 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 ρ , mean ± 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 2: Agreement between the edge-weight fields learned by Vertex-phased-VQIL and Trained-RWR on identical edges. Axes show consensus weights ωˉ (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 γ=0.9 (RWR restart 0.1 ) and γ=0.7 (restart 0.3 ). Bottom: PPI networks at both horizons, as hexagonal bins colored by the number of edges per bin (log scale).
Figure 3: Distributions of the trained edge weights ω 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 γ=0.9 and γ=0.7 . Bottom: PPI networks at both horizons. At γ=0.9 , Trained-RWR places 46% (grids) and 43% (PPI networks) of its weights at the ±2 bounds, compared with 21% and 25% for VQIL.
Figure 4: Distributions of the learned node-phase offsets δ of Vertex-phased-VQIL (Trained-RWR has no node parameter), pooled over all nodes and the five folds, at γ=0.9 (left) and γ=0.7 (right); δ=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 γ=0.9 , 0.3% (grids) and 0.06% (PPI networks) of the nodes lie at the ±π bound; at γ=0.7 , a small fraction of nodes accumulates at the bound.
Figure 5: Trained fields on the remaining five power grids (case57, case118, case145, case300, and case1354), shown as in Figure 1 (voltage response, γ=0.9 , means over the five folds). (a) Withheld branch susceptance −logx (edge color) and bus class (marker: slack, generator, load, or transit bus). (b) Vertex-phased-VQIL: learned edge weights ω (edge color) and node-phase magnitudes ∣δ∣ (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.
Figure 6: Trained fields on the five PPI networks (K562, RPE1, Frangieh, Tian-i, and Adamson; γ=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 ω (edge color) and node-phase magnitudes ∣δ∣ (node color and size). (c) Trained-RWR ( r=0.1 ): learned edge weights; RWR has no node parameter.
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