IPQSSL: Hybrid Quantum Poisson Learning

• IPQSSL is a hybrid quantum–classical method that combines improved Poisson label propagation with QR-based quantum embeddings for graph-based semi-supervised learning.
• The method refines classical propagation by subtracting stationary components and adding self-influence, enhancing convergence and robustness on imbalanced datasets.
• It employs variational quantum circuits alongside classical updates, achieving notable accuracy improvements on benchmark datasets compared to traditional approaches.

Improved Poisson Quantum Semi-Supervised Learning (IPQSSL) is a hybrid quantum–classical graph-based semi-supervised learning method that combines an improved Poisson label-propagation scheme with QR-based unitary embeddings and variational quantum circuits. Introduced together with ILQSSL, it is designed for settings in which labeled data are scarce, and it produces a soft label matrix by iterating a stabilized Poisson-style propagation on a similarity graph before assigning labels by classwise maximization (Gholipour et al., 4 Aug 2025). Its lineage runs from graph Poisson learning at very low label rates (Calder et al., 2020), through improved graph-based semi-supervised schemes that subtract stationary components and add self-influence (Bozorgnia, 2024), and it is also situated against analyses arguing that plain Poisson learning remains tied to Laplacian regularization and therefore does not automatically overcome large-graph global information loss (Nguyen, 2022).

1. Foundational lineage in graph Poisson learning

Graph Poisson learning was introduced as an alternative to standard Laplacian semi-supervised learning in the very low label rate regime. In the original formulation, labeled values are replaced by sources and sinks in a graph Poisson equation. For a weighted undirected graph with Laplacian $L=D-W$, centered source term $f$, and degree-weighted zero-mean constraint, one solves

$L u = f, \qquad \sum_{i=1}^n d_i\,u(x_i)=0,$

with $f(x_i)=y_i-\bar y$ on labeled nodes and $f(x_i)=0$ on unlabeled nodes. The resulting method was motivated by the observation that Laplacian label propagation becomes nearly constant across the graph when the hitting time to the labeled set exceeds the mixing time, whereas Poisson learning remains more stable and informative in that regime (Calder et al., 2020).

This background matters directly for IPQSSL because the quantum method does not define a new semi-supervised objective from scratch; rather, it imports an improved Poisson propagation core into a hybrid quantum workflow. The 2024 improved classical schemes paper modifies the Poisson iteration to remove the stationary component that prevents convergence and to add stabilizing self-influence, specifically to improve robustness and accuracy on large and imbalanced datasets (Bozorgnia, 2024).

At the same time, the broader Poisson literature is not uniform in its conclusions. A separate analysis of Poisson learning on large graphs argues that plain Poisson learning is “simply Laplace regularization with thresholding,” since its solution can be written as $u=L^\dagger t+c\,1_n$, and concludes that it cannot avoid global information loss because the core operator remains $L^\dagger$ (Nguyen, 2022). This establishes an important conceptual boundary: claims about IPQSSL must be interpreted in relation to the improved fixed-point scheme it adopts, not as a blanket vindication of every Poisson-based SSL method.

2. Graph formulation and fixed-point equation

IPQSSL assumes that data are represented by an undirected weighted graph with similarity matrix $W\in\mathbb{R}^{n\times n}$, degree matrix $D=\operatorname{diag}(d_i)$ where $d_i=\sum_j W_{ij}$, and random-walk transition matrix

$f$0

Supervision is encoded by a label matrix $f$1 for $f$2 classes, with one-hot rows on labeled nodes and zero rows on unlabeled nodes. The model outputs a soft label matrix $f$3 (Gholipour et al., 4 Aug 2025).

The improved Poisson core used by IPQSSL is the affine iteration

$f$4

initialized by

$f$5

Here $f$6 is a rank-one degree-derived matrix, $f$7 are tunable hyperparameters, and $f$8 encodes class-distribution information. The stopping rule is

$f$9

after which the fixed point $L u = f, \qquad \sum_{i=1}^n d_i\,u(x_i)=0,$0 is used for prediction via

$L u = f, \qquad \sum_{i=1}^n d_i\,u(x_i)=0,$1

When the operator is invertible, the converged solution is written in closed form as

$L u = f, \qquad \sum_{i=1}^n d_i\,u(x_i)=0,$2

This formulation makes IPQSSL, at its mathematical core, a fixed-point graph propagation method whose quantum layer is wrapped around a classical update rather than replacing it (Gholipour et al., 4 Aug 2025).

The same paper describes the improved iteration as a modification of Poisson-style diffusion rather than a direct Dirichlet boundary-value solve on the unlabeled subgraph. In that sense, IPQSSL belongs to the family of iterative label-propagation methods expressed through $L u = f, \qquad \sum_{i=1}^n d_i\,u(x_i)=0,$3, augmented by low-rank correction, self-transition, and class-distribution injection.

3. Improved Poisson correction and imbalance robustness

The immediate classical antecedent of IPQSSL is the Improved Poisson Learning (IPL) scheme. In that formulation, the baseline Jacobi-like iteration

$L u = f, \qquad \sum_{i=1}^n d_i\,u(x_i)=0,$4

does not converge because $L u = f, \qquad \sum_{i=1}^n d_i\,u(x_i)=0,$5 is row-stochastic and retains an eigenvalue at $L u = f, \qquad \sum_{i=1}^n d_i\,u(x_i)=0,$6. The proposed correction subtracts a stationary rank-one component and, in the final version, adds an identity term and forcing scale:

$L u = f, \qquad \sum_{i=1}^n d_i\,u(x_i)=0,$7

The cited rationale is threefold: subtracting $L u = f, \qquad \sum_{i=1}^n d_i\,u(x_i)=0,$8 removes the global drift toward the stationary distribution, adding $L u = f, \qquad \sum_{i=1}^n d_i\,u(x_i)=0,$9 preserves more of a node’s current state and counters oversmoothing, and $f(x_i)=y_i-\bar y$0 tunes the influence of labels relative to graph structure (Bozorgnia, 2024).

In the improved classical presentation, the stationary correction is defined using the random-walk stationary distribution $f(x_i)=y_i-\bar y$1 and $f(x_i)=y_i-\bar y$2, so that $f(x_i)=y_i-\bar y$3 is a rank-one projector and $f(x_i)=y_i-\bar y$4. In the IPQSSL paper, by contrast, $f(x_i)=y_i-\bar y$5 is described as a rank-one matrix with rows equal to the degree vector $f(x_i)=y_i-\bar y$6 “as in [28]” (Gholipour et al., 4 Aug 2025). This suggests that the quantum paper inherits the Bozorgnia-style stationary subtraction at the level of intent, but the exact normalization of $f(x_i)=y_i-\bar y$7 should be checked in the original algorithm statement when reproducing results.

A central practical theme is robustness under class imbalance. The improved classical scheme uses centered forcing based on $f(x_i)=y_i-\bar y$8, and it also describes a post-iteration class-prior reweighting,

$f(x_i)=y_i-\bar y$9

where $f(x_i)=0$0 is the empirical class-fraction vector. The IPQSSL paper likewise states that the $f(x_i)=0$1 term injects class-distribution information to respect class proportions (Bozorgnia, 2024). This places imbalance handling inside the propagation mechanism rather than treating it purely as post hoc threshold calibration.

4. Quantum embedding and hybrid circuit architecture

IPQSSL integrates quantum computation in two stated locations: QR-based embedding of graph structure into unitary operators and a variational “quantum evolution” block that accompanies the improved Poisson iteration. The global embedding begins with a QR decomposition of the similarity matrix,

$f(x_i)=0$2

where the orthonormal factor $f(x_i)=0$3 is used as a unitary block in a quantum circuit. The paper also writes an amplitude encoding

$f(x_i)=0$4

but presents the QR-derived unitary as the hardware-friendly route. A node-level alternative takes a local matrix $f(x_i)=0$5, decomposes it as $f(x_i)=0$6, and prepares

$f(x_i)=0$7

to encode node-local structure or features (Gholipour et al., 4 Aug 2025).

The hybrid workflow proceeds by graph preprocessing, initialization of $f(x_i)=0$8, quantum embedding through global or local QR, and repeated classical updates of $f(x_i)=0$9 while optionally encoding $u=L^\dagger t+c\,1_n$0 as a quantum state and applying a variational unitary $u=L^\dagger t+c\,1_n$1. The stated role of this variational block is to emulate the linear map $u=L^\dagger t+c\,1_n$2 and/or to provide discriminative quantum features during propagation. Circuit architecture is described as a layered parameterized quantum circuit with alternating single-qubit rotations and entangling layers; the study varies depths in $u=L^\dagger t+c\,1_n$3 and qubit counts in $u=L^\dagger t+c\,1_n$4 (Gholipour et al., 4 Aug 2025).

Prediction remains classical. After convergence to $u=L^\dagger t+c\,1_n$5, node labels are assigned by $u=L^\dagger t+c\,1_n$6 over class scores. The paper also refers to “Quantum Evaluation,” in which observables of the encoded states are analyzed, but it does not specify a dedicated variational cost function for the quantum block. A plausible implication is that IPQSSL is best interpreted as a hybrid augmentation of an already convergent classical propagation scheme, rather than as a purely variational quantum classifier.

5. Experimental evidence and hardware diagnostics

The reported evaluation covers Iris, Wine, Heart Disease, and German Credit Card. The single-model metrics given for IPQSSL are summarized below (Gholipour et al., 4 Aug 2025).

Dataset	IPQSSL performance	Best classical accuracy
Iris	Acc 0.97, F1 0.96, Recall 0.96, Precision 0.96	0.9111
Wine	Acc 0.94, F1 0.94, Recall 0.94, Precision 0.95	0.7222
Heart Disease	Acc 0.83, F1 0.77, Recall 0.70, Precision 0.85	0.5333
German Credit Card	Acc 0.75, F1 0.71, Recall 0.74, Precision 0.72	0.7100

The same source separately summarizes “IPQSSL vs best classical accuracy” as 0.97 vs 0.91 on Iris, 0.94 vs 0.72 on Wine, 0.83 vs 0.53 on Heart Disease, and 0.77 vs 0.71 on German Credit. The German Credit figure therefore appears as 0.75 in one table and 0.77 in another summary, a discrepancy that should be checked against the original tables (Gholipour et al., 4 Aug 2025).

Architectural sweeps show dataset-dependent stability. Iris accuracy is reported to remain at 0.97 across all tested depths and qubit counts. Wine remains at 0.94 for many settings but drops to 0.88 above 10 qubits, which the paper attributes to noise sensitivity. German Credit stays at 0.67 in the layer/qubit sweeps, while Heart Disease stays at 0.88 in those sweeps; the paper notes that these sweeps isolate architectural effects and therefore differ from the headline table (Gholipour et al., 4 Aug 2025).

The hardware analysis focuses on entanglement entropy

$u=L^\dagger t+c\,1_n$7

and Randomized Benchmarking with

$u=L^\dagger t+c\,1_n$8

where $u=L^\dagger t+c\,1_n$9. The reported interpretation is that moderate entanglement correlates with better generalization, while increased depth and width reduce RB fidelity and can undermine performance on current hardware. The practical recommendation is therefore shallow-to-moderate depth and moderate qubit counts, with approximately $L^\dagger$0–$L^\dagger$1 and $L^\dagger$2–$L^\dagger$3 described as effective on structured datasets such as Iris and Wine (Gholipour et al., 4 Aug 2025).

6. Relation to the broader literature, misconceptions, and limitations

A common misconception is that Poisson-based SSL denotes a single fixed algorithm. The cited literature distinguishes at least four relevant regimes. First, the original graph Poisson method solves a centered Poisson system or equivalent random-walk iteration and was designed specifically for very low label rates (Calder et al., 2020). Second, the 2024 improved schemes modify Poisson propagation by subtracting a stationary component, adding self-influence, and incorporating class-distribution corrections (Bozorgnia, 2024). Third, IPQSSL embeds that improved propagation into a QR-based hybrid quantum circuit (Gholipour et al., 4 Aug 2025). Fourth, Poisson learning has also been adapted to domain-specific architectures, as in semi-supervised few-shot medical image segmentation, where unlabeled samples contribute through graph-based propagation but do not participate in end-to-end gradient updates (Shen et al., 2021).

Another misconception is that any Poisson variant automatically resolves the asymptotic pathologies of Laplacian SSL on large graphs. The 2022 analysis explicitly argues that plain Poisson learning is still governed by $L^\dagger$4 and is therefore unable to overcome global information loss: kernel entries vanish for fixed nodes as $L^\dagger$5, commute-time distances degenerate to local degree terms, and the learned function becomes peaky near labels and nearly constant elsewhere (Nguyen, 2022). For IPQSSL, the available sources establish improved convergence, better robustness under imbalance, and empirical gains over classical baselines, but they do not establish that the hybrid quantum construction removes those large-graph asymptotics. This suggests that IPQSSL’s primary novelty lies in stabilized propagation plus QR-based quantum embedding, not in a proof that the fundamental geometry of large-graph Poisson learning has been altered.

The principal limitations reported for IPQSSL are correspondingly concrete. The paper does not provide exact asymptotic complexity bounds for the hybrid routine, does not specify a dedicated quantum training objective tied to propagation residuals, and shows that wider or deeper circuits can degrade RB fidelity and fail to improve accuracy on noisier tasks such as German Credit (Gholipour et al., 4 Aug 2025). In that sense, IPQSSL occupies a specific place in the Poisson-learning literature: it is an improved and quantum-augmented propagation framework for low-label graph SSL, with strong reported empirical performance on small benchmark datasets, but its theoretical status relative to large-graph degeneracy remains an open question in the cited sources.