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VQIL: Visible Quantum Interference Learning for Perturbation Response Ranking on Graphs
Seokjun Lee, Sergio A. Ortega, Daniel K. Park, Namshik Han
TL;DR
Seed-conditioned response ranking asks how other nodes respond to a perturbation when the input graph gives topology but not the propagation mechanism. VQIL trains a visible Szegedy quantum walk on graph components, and improves ranking across power-grid and protein-interaction-network tasks. Its learned graph parameters provide inspectable hypotheses about propagation, though the method remains tied to fixed topologies.
Problem
Seed-conditioned response ranking must infer response strength at other nodes when the graph records structural connectivity but not the underlying functional dynamics.
Source: §1 · p. 1Method
VQIL trains a unitary Szegedy walk's edge weights and phases and node phases, then scores nodes from a geometrically discounted coherent sum of amplitudes.
Source: §1 · p. 2Results
VQIL improves ranking across power-grid and PPI tasks, with PPI-network Spearman ρ rising from 0.171 to 0.620 and post-training gains over a matched classical control.
Source: §5.2 · p. 8Takeaways & Limitations
Because learned parameters attach to named graph components, VQIL provides inspectable hypotheses about propagation and can reveal information about withheld system properties.
Source: §1 · p. 3VQIL is a transductive ranking tool bound to fixed topologies and does not predict signed continuous expression values or generalize to unseen graphs.
Source: §6 · p. 11
Abstract
from arXiv · showhide
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.
1 Introduction
Seed-conditioned response ranking asks how strongly nodes respond to a perturbation at one seed, but topology alone omits the mechanism that governs propagation. VQIL makes a Szegedy quantum walk trainable while keeping its learned parameters attached to named graph components.
1 Introduction
- A seed-conditioned ranking model must learn propagation dynamics from response supervision while keeping its mechanism exposed for interpretation.The graph records structural connectivity, not transmission capacity or functional dynamics.Source: §1 · p. 2
- Graph neural networks can learn complex perturbation mechanisms, but their learned logic is distributed across latent representations rather than explicit graph interactions.Supervised graph-propagation methods instead learn quantities attached to named edges or nodes, making them readable as hypotheses about transmission.Source: §1 · p. 2
- Quantum walks combine complex path amplitudes coherently, allowing constructive and destructive interference before probabilities are formed.This offers a richer propagation hypothesis class than classical walks that accumulate non-negative probability mass.Source: §1 · p. 2
- VQIL trains a Szegedy quantum walk for seed-conditioned ranking, learning edge weights and phases plus node phases while preserving unitarity.The model's learned quantities remain attached to named graph components for mechanistic examination.Source: §1 · p. 2
- The phased walk operator is differentiable, initialized from the standard walk, and costs O(m) per step for m directed edges.Unitarity is preserved throughout training.Source: §1 · p. 3
- VQIL scores nodes from a geometrically discounted coherent sum of walk amplitudes before the final squared-norm measurement.The discount gives a smooth, convergent horizon and a differentiable objective across walk parameters.Source: §1 · p. 3
- Each learned scalar is attached to a named edge or node, making the trained model an explicit graph-level hypothesis about propagation.The learned fields are reported to carry information about withheld system properties and can serve as a hypothesis-generating tool.Source: §1 · p. 3
2 Problem statement
The task is to predict how strongly every non-seed node responds to a perturbation at a seed, using a graph whose structure does not itself encode the functional dynamics. Models are trained on response rankings and evaluated on held-out seed queries.
2 Problem statement
- For an undirected, unweighted, connected simple graph, a query pairs a seed node with response magnitudes measured at every other node.Each undirected edge is represented by two directed edges.Source: §2 · p. 3
- A model maps the graph and perturbed seed to scores for all nodes, and evaluation compares their non-seed ranking with measured responses.Training minimizes a differentiable ranking loss; evaluation uses seed queries held out from training.Source: §2 · p. 3
- The graph records structural connectivity while the response labels arise from an external physical or biological mechanism, so prediction must learn that mechanism.Because the model is visible, its learned mechanism can be read from its parameters.Source: §2 · p. 3
3 Related work
Prior work includes classical propagation, fixed quantum-walk propagators, trainable quantum models operating on node features, and visible graph architectures. VQIL instead learns native walk parameters directly on graph edges and nodes and uses a differentiable coherent resolvent for scoring.
3 Related work
- Classical random-walk methods diffuse probability mass, and supervised extensions learn the diffusion from training queries.Fixed quantum walks have also been used as topological propagators for centrality and disease-network analysis.Source: §3 · p. 3
- Existing trainable quantum-walk neural networks learn coin operators acting on continuous node features, whereas VQIL optimizes native edge and node walk parameters.VQIL extracts scores through a differentiable coherent resolvent.Source: §3 · p. 3
- Sheaf neural networks learn per-edge restriction maps, with their O(2) planar rotations serving as a real-valued feature-space analog of VQIL's complex U(1) link phases.The comparison isolates propagation capacity by evaluating both model families without external node attributes.Source: §3 · p. 4
- VQIL enforces a bijection between learnable parameters and physical graph nodes and edges, so optimized parameters can be audited as direct hypotheses.This contrasts with message-passing GNNs, whose perturbation mechanisms are embedded in feature-space transformations.Source: §3 · pp. 3–4
4 Methods
Methods parameterize a unitary Szegedy walk with learnable edge weights and phases, then rank nodes using a geometrically discounted coherent amplitude readout and supervised optimization.
4.1 Learned parameters and unitary propagation
- The Szegedy walk evolves on directed-edge states; symmetric learned edge weights and edge/node phases parameterize it while preserving unitarity at O(m) cost per step.Edge weights are normalized over incident edges, so transition probabilities can remain directional despite symmetric log-weights.Source: §4.1 · p. 4
4.2 Coherent-resolvent readout
- A geometric discount sums walk amplitudes across lengths before one squared-norm readout, yielding smooth resolvent scores that are rankings, not probabilities or physical measurements.With γ = 0.9, K = 109 keeps the discarded tail below 10^-5; no quantum implementation is claimed or required.Source: §4.2 · p. 5
4.3 VQIL training
- Training minimizes soft-Spearman on log-transformed standardized non-seed scores, differentiates through the walk, and uses AdamW with validation-based early stopping across parameter-subset variants.The protocol trains for up to 300 epochs, with patience 10 and seed-disjoint train, validation, and test splits.Source: §4.3 · pp. 5–6
4.4 Baselines
- Baselines include matched learned-weight random walk with restart differing only in propagation, plus GCN, GAT, and O(2) neural sheaf models supplied with seed-relative topological features.The classical control shares VQIL’s weight parameterization, loss, optimizer, splits, and early stopping.Source: §4.4 · pp. 6–7
5 Experiments
The experiments test seed-conditioned response ranking on power grids and PPI networks using topology-only inputs and seed-disjoint evaluation. VQIL improves over untrained and matched classical walks, while ablations and learned parameter fields examine its readout and mechanistic interpretation.
5.1 Set-up
- Evaluation uses topology-only graphs and seed-disjoint five-fold splits across six AC grids and five Perturb-seq PPI screens, with Spearman ρ as the primary metric.The grids have 30–1,354 nodes; PPI graphs have 2,965–5,787 nodes, and each screen contributes its own graph.Source: §5.1 · p. 7
5.2 Results
- After training, Vertex-phased-VQIL exceeds matched classical RWR by 0.134 and 0.149 in power-grid ρ and by 0.050 on PPI networks.Before training, the classical walk ranks better on unweighted graphs across all three blocks.Source: §5.2 · p. 8
- On PPI networks, Vertex-phased-VQIL beats GCN, GAT, and Sheaf-O(2) by 0.038, 0.091, and 0.255 in ρ, respectively.It leads GCN, the strongest neural baseline, on every dataset and all four metrics; on power-grid angle response it matches GCN and Sheaf-O(2), trails GAT by 0.020, and trails all three on voltage response.Source: §5.2 · p. 8
- Power-grid ρ reaches 0.917/0.831 (angle/voltage) in the best phase-bearing variants, while PPI ρ rises from 0.171 to 0.620.Weight-only training reaches 0.880/0.799 on power grids; the reported power-grid values are means across six grids and PPI values span five networks.Source: Table 1 · p. 8, §5.2 · p. 8, Table 2 · p. 9
- Learned edge weights and node-phase magnitudes provide graph-attached hypotheses about propagation, with power-grid phases lowest at regulated buses and PPI phases higher at broadly responsive genes.The overall sign of node phases is unidentifiable and relative signs are empirically unstable, so the authors interpret |δ| only; on power grids, |δ| separates generator buses with AUC 0.74 versus a degree-matched null of 0.56, while PPI phase magnitudes capture stress-responsive genes with AUC 0.71–0.87.Source: §5.2 · pp. 9–11, Figure 1 · p. 10
- At γ = 0.9, the coherent geometric readout leads alternatives in every task, including PPI ρ gains of +0.103 over square-then-add and +0.105 over a uniform window.On power grids, its corresponding gains are +0.072/+0.098 for readout order and +0.208/+0.057 for weighting (angle/voltage); its geometric discount also yields a smoother objective, while coherent summation further reduces curvature around trained solutions.Source: §5.2 · p. 9, Table 3 · p. 10
6 Discussion and Conclusion
The discussion presents VQIL’s learned graph parameters as mechanistic hypotheses and quantum interference as a way to route signals around hubs. It also identifies limits in topology generalization and response prediction.
6 Discussion and Conclusion
- Learned local node phases are presented as a way to bypass high-degree bottlenecks and preserve responsive gene modules or regulated electrical buses.The discussion identifies amplitude superposition in the coherent resolvent readout as the mechanism for targeted signal routing.Source: §6 · p. 11
- VQIL maps edge weights to transmission corridors and node-phase magnitudes to sinks, making its parameters identifiable hypotheses about graph components.The authors describe this visibility as useful for discovery-driven fields such as computational biology.Source: §6 · p. 11
- VQIL is currently transductive on fixed topologies and cannot predict signed continuous expression values or generalize to unseen graphs.The authors propose feature-based encoders and anti-symmetric link phases as future extensions.Source: §6 · p. 11
Funding
The work acknowledges support from Korean government agencies and research programs, alongside a jointly funded PhD studentship.
Funding
- The project lists support from Korean ARPA-H through KHIDI and the Brain Pool Plus Fellowship Program.The stated funding identifiers are RS-2025-25456722 and RS-2025-25427881, respectively.Source: Funding · p. 11
- Additional support came from IITP-ITRC, the National Research Foundation of Korea, and MOTIE’s Industrial Innovation Infrastructure Development Project.The passage lists grant identifiers for each program.Source: Funding · p. 11
- S.L.’s PhD studentship was jointly funded by the Cambridge Stem Cell Institute and the Milner Therapeutics Institute.Source: Funding · p. 11
A.1 Datasets and Models
The appendix documents the graph datasets, neural baseline configurations, training-cost measures, and parameter counts used in the experiments.
A.1 Datasets and Models
- The model receives only an unweighted, undirected graph and seed in both domains, while all other information is withheld.The domains are power-grid buses with AC power-flow responses and STRING human interactions with Perturb-seq gene perturbations; folds are seed-disjoint.Source: Table 4 · p. 16
- Neural baseline depth and learning rate are selected on validation Spearman ρ per domain, then fixed across that domain’s datasets.The appendix reports the configurations and shared training protocol in Table 5.Source: Table 5 · p. 16, §A.1 · p. 16
- Training cost is summarized by wall-clock seconds per epoch, mean seconds per cell, and median epochs to early stopping.The reported means and medians span five folds and, for Domain 1, both response types.Source: §A.1 · p. 16, Table 6 · p. 17
- VQIL and RWR parameter counts scale with graph edges and learned node or edge phases, whereas neural baseline size is graph-independent.The counts are reported per model and dataset; they do not depend on γ or response type.Source: §A.1 · p. 16, Table 7 · p. 18
A.2 Results for γ = 0.7
The appendix reports power-grid and gene-perturbation results at γ = 0.7 and describes a readout ablation at that horizon. The supplied table descriptions specify evaluation scope but not result values.
A.2 Results for γ = 0.7
- The appendix includes the shorter-horizon results alongside the main paper’s γ = 0.9 setting.The appendix identifies γ = 0.7 as an alternative horizon and reports its results in Tables 8 and 9.Source: §A.2 · p. 19
- At γ = 0.7, Tables 8 and 9 report power-grid and PPI ranking results, respectively, as mean ± sd over five seed-disjoint folds.Table 8 covers 30 cells across six grids; Table 9 covers 25 cells across five Perturb-seq datasets.Source: Table 8 · p. 19, Table 9 · p. 19
- The γ = 0.7 readout ablation is a 2 × 2 factorial over step weighting and the order of squared-norm measurement.Table 10 covers five-fold results for six grids and five Perturb-seq datasets.Source: Table 10 · p. 20
A.3 Learned Parameter Fields
The appendix compares VQIL and trained random-walk edge-weight fields and distributions, shows VQIL node-phase offsets, and visualizes learned fields alongside withheld physics and biology.
A.3 Learned Parameter Fields
- The edge-field comparison plots five-fold consensus weights for Trained-RWR against Vertex-phased-VQIL, with the identity line and Pearson correlation r shown.Power-grid points and PPI hexagonal bins are shown at γ = 0.9 and γ = 0.7.Source: Figure 2 · p. 22
- At γ = 0.9, Trained-RWR places 46% of grid weights and 43% of PPI weights at the ±2 bounds, versus 21% and 25% for VQIL.The distributions pool weights across edges and five folds; weights are clamped to [−2, 2].Source: Figure 3 · p. 23
- VQIL node-phase offset distributions are nearly symmetric about zero; at γ = 0.9, 0.3% of grid nodes and 0.06% of PPI nodes reach the ±π bound.At γ = 0.7, a small fraction of nodes accumulates at the bound; δ = 0 is the standard walk.Source: Figure 4 · p. 24
- For the five remaining power grids, visualizations place learned VQIL edge weights and node-phase magnitudes beside withheld branch susceptance and bus class, and compare trained RWR edge weights.The fields shown are five-fold means for voltage response at γ = 0.9; RWR has no node parameter.Source: Figure 5 · p. 25
- For five PPI networks, visualizations place learned VQIL edge weights and node-phase magnitudes beside withheld co-expression and recurrent-response information, and compare trained RWR edge weights.The fields shown are five-fold means at γ = 0.9; RWR has no node parameter.Source: Figure 6 · p. 26