Improved Laplacian Quantum Semi-Supervised Learning

• ILQSSL is a hybrid quantum-classical method that integrates graph Laplacian regularization with variational quantum circuits for semi-supervised learning.
• It embeds graph structure into quantum states using both block-encoded and QR-based techniques for effective label propagation under limited supervision.
• The framework optimizes circuit depth and qubit count to balance entanglement and noise, achieving competitive performance on benchmark datasets.

Searching arXiv for the specified ILQSSL papers and closely related source material. Improved Laplacian Quantum Semi-Supervised Learning (ILQSSL) denotes a hybrid quantum approach to graph-based semi-supervised learning in which Laplacian-style label propagation is integrated with variational quantum circuits under scarce supervision. In the reported formulations, ILQSSL extends Laplacian-based Quantum Semi-Supervised Learning by embedding graph structure into quantum states, coupling graph regularization to trainable quantum models, and analyzing the interplay among qubit count, circuit depth, entanglement, and noise on standard benchmark datasets (Gholipour et al., 2024, Gholipour et al., 4 Aug 2025). The available descriptions present two closely related implementations under the same name: one centered on a unified graph-regularized variational loss with block-encoded Laplacians, and another centered on an improved propagation operator with QR-based quantum embedding. This suggests that ILQSSL is best understood as a methodological line rather than a single immutable circuit specification.

1. Conceptual lineage and problem setting

ILQSSL emerges from the observation that Laplacian learning is a well-established technique in classical graph-based semi-supervised learning, whereas its quantum analogue remained comparatively underexplored in the Laplacian-based Quantum Graph Neural Network for Semi-Supervised Learning framework of Gholipour and collaborators (Gholipour et al., 2024). The target regime is the standard semi-supervised setting in which a dataset $X=\{x_i\}_{i=1}^n\subset\mathbb R^d$ contains only a small labeled subset, while the remaining unlabeled points are expected to contribute through graph structure and smoothness constraints (Gholipour et al., 4 Aug 2025).

The 2024 description of ILQSSL is formulated as an improvement over the “Laplacian-based Quantum Graph Neural Network for Semi-Supervised Learning” by Gholipour et al. It states that the key idea is to marry a graph-regularized loss, specifically the Laplacian smoothing term, with a hybrid quantum-classical variational circuit whose architecture and cost have been enhanced to reduce shot noise, mitigate barren plateaus, and absorb unlabeled-data propagation directly into the variational loss (Gholipour et al., 2024). The 2025 description frames ILQSSL together with Improved Poisson Quantum Semi-Supervised Learning as enhanced quantum models that incorporate advanced label propagation strategies within variational quantum circuits and utilize QR decomposition to embed graph structure directly into quantum states (Gholipour et al., 4 Aug 2025).

A consistent feature across these descriptions is that ILQSSL treats the graph not merely as a preprocessing device but as an operator that shapes the quantum learning dynamics. The unlabeled nodes are therefore not external to the optimization objective; they enter through either an explicit Laplacian regularizer over predicted outputs or through iterative propagation updates governed by an improved graph operator. A plausible implication is that ILQSSL should be read as a quantum graph-SSL framework whose design space includes both direct variational regularization and alternating propagation-evaluation schemes.

2. Graph operators and quantum embedding schemes

In the 2024 formulation, the graph is an undirected graph $G=(V,E)$ with adjacency matrix $A\in\mathbb R^{n\times n}$ and degree matrix $D=\mathrm{diag}(d_1,\dots,d_n)$, where $d_i=\sum_j A_{ij}$. The method uses the symmetric normalized Laplacian

$\mathcal{L} \;=\; I \;-\; D^{-1/2} A D^{-1/2},$

whose eigenvalues lie in $[0,2]$. In amplitude encoding, the entries of the Laplacian are stored as amplitudes of an $n$-qubit basis state, and a block-encoding $U_L$ is constructed such that

$$U_L \;=\; \begin{pmatrix} \mathcal{L}/\alpha & \cdot \ \cdot & \cdot \end{pmatrix}, \qquad \alpha\ge \|\mathcal{L}\|_\infty.$$

This block-encoding can be realized via known Hamiltonian-simulation techniques (Gholipour et al., 2024).

In the 2025 formulation, preprocessing begins from a weighted similarity matrix $G=(V,E)$0, obtained either from a kernel such as RBF or from $G=(V,E)$1-nearest neighbors, with entries

$G=(V,E)$2

From this, one forms the degree matrix $G=(V,E)$3, the unnormalized graph Laplacian $G=(V,E)$4, and the random-walk transition matrix $G=(V,E)$5. ILQSSL then introduces an improved propagation operator

$G=(V,E)$6

where $G=(V,E)$7 is a rank-one “residual” matrix and the diagonal shift is intended to gain robustness in very low-label regimes. Label propagation is written as

$G=(V,E)$8

The same formulation also states that the similarity matrix is QR-factorized as $G=(V,E)$9, the factor $A\in\mathbb R^{n\times n}$0 is discarded, and $A\in\mathbb R^{n\times n}$1 is promoted, after padding or truncation, to a unitary $A\in\mathbb R^{n\times n}$2, yielding the graph state

$A\in\mathbb R^{n\times n}$3

It further notes that $A\in\mathbb R^{n\times n}$4 itself can be amplitude-encoded as

$A\in\mathbb R^{n\times n}$5

In practice, however, propagation is carried out by alternating classical updates of $A\in\mathbb R^{n\times n}$6 with quantum evaluations of observables on $A\in\mathbb R^{n\times n}$7 and re-encoded label vectors (Gholipour et al., 4 Aug 2025).

Aspect	2024 formulation	2025 formulation
Primary graph operator	Symmetric normalized Laplacian $A\in\mathbb R^{n\times n}$8	Improved operator $A\in\mathbb R^{n\times n}$9
Quantum embedding	Block-encoding $D=\mathrm{diag}(d_1,\dots,d_n)$0 of $D=\mathrm{diag}(d_1,\dots,d_n)$1	QR-based unitary $D=\mathrm{diag}(d_1,\dots,d_n)$2 from $D=\mathrm{diag}(d_1,\dots,d_n)$3
Propagation mechanism	Laplacian regularization folded into the variational loss	Iterative update $D=\mathrm{diag}(d_1,\dots,d_n)$4

This comparison indicates that “improved Laplacian” refers to different technical mechanisms in the two descriptions: in one case to a multi-scale block-encoded Laplacian regularizer, and in the other to a modified propagation operator augmented by $D=\mathrm{diag}(d_1,\dots,d_n)$5 and a diagonal shift.

3. Variational circuit design and measurement model

The 2024 ILQSSL circuit lives on $D=\mathrm{diag}(d_1,\dots,d_n)$6 qubits, typically $D=\mathrm{diag}(d_1,\dots,d_n)$7 or chosen heuristically, and uses $D=\mathrm{diag}(d_1,\dots,d_n)$8 entangling layers. Each layer consists of angle-embedding gates $D=\mathrm{diag}(d_1,\dots,d_n)$9 on wire $d_i=\sum_j A_{ij}$0, a “strongly entangling” block of two-qubit gates $d_i=\sum_j A_{ij}$1 followed by $d_i=\sum_j A_{ij}$2, and single-qubit rotations $d_i=\sum_j A_{ij}$3 on each wire. The layer action is written as

$d_i=\sum_j A_{ij}$4

After $d_i=\sum_j A_{ij}$5 such blocks, the block-encoded Laplacian $d_i=\sum_j A_{ij}$6 is applied once, or multiple times for multi-scale smoothing, and the final measurement is $d_i=\sum_j A_{ij}$7 on the first qubit to produce a probability $d_i=\sum_j A_{ij}$8 (Gholipour et al., 2024).

The 2025 ILQSSL ansatz instead fixes $d_i=\sum_j A_{ij}$9, with experiments typically using 4–12 qubits and circuit depth $\mathcal{L} \;=\; I \;-\; D^{-1/2} A D^{-1/2},$0 to $\mathcal{L} \;=\; I \;-\; D^{-1/2} A D^{-1/2},$1 layers. Each layer applies single-qubit rotations $\mathcal{L} \;=\; I \;-\; D^{-1/2} A D^{-1/2},$2 to every qubit, followed by a fixed nearest-neighbor entangling pattern of CNOT or CZ gates around the ring. The full variational unitary is

$\mathcal{L} \;=\; I \;-\; D^{-1/2} A D^{-1/2},$3

Quantum label propagation is then described procedurally: prepare $\mathcal{L} \;=\; I \;-\; D^{-1/2} A D^{-1/2},$4, evolve under $\mathcal{L} \;=\; I \;-\; D^{-1/2} A D^{-1/2},$5, measure commuting observables $\mathcal{L} \;=\; I \;-\; D^{-1/2} A D^{-1/2},$6 whose expectations encode the current label estimate $\mathcal{L} \;=\; I \;-\; D^{-1/2} A D^{-1/2},$7, perform the classical update of $\mathcal{L} \;=\; I \;-\; D^{-1/2} A D^{-1/2},$8, re-encode $\mathcal{L} \;=\; I \;-\; D^{-1/2} A D^{-1/2},$9 into the quantum register via a small-angle rotation layer, and iterate until convergence in Frobenius norm (Gholipour et al., 4 Aug 2025).

Both descriptions emphasize hardware-efficient nearest-neighbor entanglement and low- to moderate-width circuits. The 2024 account adds an explicit adaptive-entangling mechanism in which the entangling-angle parameters $[0,2]$0 are adjusted during training through an auxiliary penalty on two-qubit gate counts, discouraging overly deep entanglement that leads to noise accumulation (Gholipour et al., 2024). The 2025 account does not state the same auxiliary penalty, but it converges on the same practical conclusion that circuit complexity must be constrained under NISQ conditions (Gholipour et al., 4 Aug 2025).

4. Objective functions, propagation, and optimization

A central distinction of the 2024 formulation is its explicit unification of supervised classification and graph smoothness in a single cost. Writing $[0,2]$1 and $[0,2]$2, the loss is

$[0,2]$3

where $[0,2]$4 trades off smoothness against classification accuracy. The description explicitly states that, unlike prior QSSL, this version does not separate quantum classification and classical Laplacian smoothing into two disjoint steps; instead, the Laplacian is folded into the same variational cost so that the optimizer simultaneously enforces label propagation and quantum feature discrimination. The same formulation also introduces multi-scale Laplacian block-encoding by replacing a single $[0,2]$5 with

$[0,2]$6

to capture both local and global graph structure (Gholipour et al., 2024).

The 2025 description adopts a different training target. The variational parameters $[0,2]$7 are optimized to minimize the discrepancy between the converged propagated label matrix $[0,2]$8 and ground truth on the labeled subset, with an $[0,2]$9 regularizer: $n$0 Gradients are estimated via the parameter-shift rule and sent to a classical optimizer such as Adam or COBYLA. The practical implementation alternates quantum evaluations with classical propagation updates until the stopping criterion

$n$1

is met (Gholipour et al., 4 Aug 2025).

These two objective constructions differ in where the graph regularization enters the optimization. In the 2024 setup it appears directly as $n$2 inside the loss; in the 2025 setup it appears through the converged propagation dynamics that define $n$3. This suggests two distinct but compatible readings of ILQSSL: one as direct graph-regularized variational learning, and one as variationally assisted iterative label propagation.

5. Workflow and practical deployment

The 2024 procedural recipe for a new dataset begins with preprocessing: fill missing $n$4-entries by column median and standardize features to zero mean and unit variance. A graph is then built using

$n$5

followed by construction of $n$6 and the normalized Laplacian $n$7. Circuit initialization chooses $n$8 and $n$9, with random rotation parameters $U_L$0. Training samples mini-batches of labeled nodes $U_L$1, runs the quantum circuit to estimate $U_L$2 for all nodes in the batch and their entangled neighbors, computes the cross-entropy term on $U_L$3 and $U_L$4 on the batch via classical multiplication, and updates $U_L$5 by Adam using the parameter-shift rule. Evaluation on held-out test nodes reports Accuracy, Precision, Recall, and F1, while the entanglement entropy $U_L$6 is monitored to detect barren-plateau onset (Gholipour et al., 2024).

The 2025 workflow starts from the partially labeled dataset $U_L$7, builds a similarity graph via an RBF kernel or $U_L$8-nearest neighbors, and forms $U_L$9, $$U_L \;=\; \begin{pmatrix} \mathcal{L}/\alpha & \cdot \ \cdot & \cdot \end{pmatrix}, \qquad \alpha\ge \|\mathcal{L}\|_\infty.$$0, $$U_L \;=\; \begin{pmatrix} \mathcal{L}/\alpha & \cdot \ \cdot & \cdot \end{pmatrix}, \qquad \alpha\ge \|\mathcal{L}\|_\infty.$$1, and $$U_L \;=\; \begin{pmatrix} \mathcal{L}/\alpha & \cdot \ \cdot & \cdot \end{pmatrix}, \qquad \alpha\ge \|\mathcal{L}\|_\infty.$$2. QR decomposition of $$U_L \;=\; \begin{pmatrix} \mathcal{L}/\alpha & \cdot \ \cdot & \cdot \end{pmatrix}, \qquad \alpha\ge \|\mathcal{L}\|_\infty.$$3 produces the unitary graph embedding $$U_L \;=\; \begin{pmatrix} \mathcal{L}/\alpha & \cdot \ \cdot & \cdot \end{pmatrix}, \qquad \alpha\ge \|\mathcal{L}\|_\infty.$$4. The quantum-classical loop then alternates three elements: quantum state preparation and evaluation, classical application of the improved propagation operator $$U_L \;=\; \begin{pmatrix} \mathcal{L}/\alpha & \cdot \ \cdot & \cdot \end{pmatrix}, \qquad \alpha\ge \|\mathcal{L}\|_\infty.$$5, and re-encoding of the updated labels back into the quantum register. The description adds several implementation remarks for NISQ devices: circuit depth $$U_L \;=\; \begin{pmatrix} \mathcal{L}/\alpha & \cdot \ \cdot & \cdot \end{pmatrix}, \qquad \alpha\ge \|\mathcal{L}\|_\infty.$$6–$$U_L \;=\; \begin{pmatrix} \mathcal{L}/\alpha & \cdot \ \cdot & \cdot \end{pmatrix}, \qquad \alpha\ge \|\mathcal{L}\|_\infty.$$7 layers is identified as a “sweet spot,” 4–8 qubits are said to suffice for graphs up to $$U_L \;=\; \begin{pmatrix} \mathcal{L}/\alpha & \cdot \ \cdot & \cdot \end{pmatrix}, \qquad \alpha\ge \|\mathcal{L}\|_\infty.$$8 nodes via amplitude or QR embedding in block-diagonal form, hardware-efficient nearest-neighbor CZ/CNOT is preferred to minimize SWAP overhead on linear or ibm-like backends, and 2–5 classical update steps for $$U_L \;=\; \begin{pmatrix} \mathcal{L}/\alpha & \cdot \ \cdot & \cdot \end{pmatrix}, \qquad \alpha\ge \|\mathcal{L}\|_\infty.$$9 per quantum evaluation reduce end-to-end circuit count (Gholipour et al., 4 Aug 2025).

Taken together, these workflows depict ILQSSL as a strongly hybrid procedure rather than a fully coherent end-to-end quantum routine. Graph construction, matrix factorization, propagation updates, and some regularization terms remain classical, while the variational circuit supplies trainable nonlinear feature discrimination and measurement-based label estimation.

6. Empirical results, scaling behavior, and diagnostics

The 2024 study reports results on Iris, Wine, Breast Cancer Wisconsin, and Heart Disease. It compares ILQSSL against classical Laplacian SSL, the original QSSL of Gholipour et al., and label-propagation baselines, with the following accuracies: Iris $G=(V,E)$00, Wine $G=(V,E)$01, BreastCancer $G=(V,E)$02, and HeartDisease $G=(V,E)$03, where each triplet is classical Laplacian, original QSSL, and ILQSSL, respectively. It further states that on Iris with 6 qubits and $G=(V,E)$04, ILQSSL reaches $G=(V,E)$05 accuracy, and that the entangling-entropy optimum is $G=(V,E)$06 bits. The same source notes that ILQSSL “consistently closes the gap” to the best classical semi-supervised methods, giving Label Propagation as $G=(V,E)$07 on Iris and $G=(V,E)$08 on Wine (Gholipour et al., 2024).

The 2025 study evaluates ILQSSL on Iris, Wine, Heart Disease, and German Credit Card with label fractions $G=(V,E)$09. At $G=(V,E)$10 labeled, the reported test metrics for ILQSSL are: Iris accuracy $G=(V,E)$11, F1 $G=(V,E)$12, recall $G=(V,E)$13, precision $G=(V,E)$14; Wine accuracy $G=(V,E)$15, F1 $G=(V,E)$16, recall $G=(V,E)$17, precision $G=(V,E)$18; Heart Disease accuracy $G=(V,E)$19, F1 $G=(V,E)$20, recall $G=(V,E)$21, precision $G=(V,E)$22; German Credit Card accuracy $G=(V,E)$23, F1 $G=(V,E)$24, recall $G=(V,E)$25, precision $G=(V,E)$26. The corresponding original LQSSL accuracies are $G=(V,E)$27, $G=(V,E)$28, $G=(V,E)$29, and $G=(V,E)$30. Under $G=(V,E)$31 and $G=(V,E)$32 labels, the absolute accuracy drops by 2–5 points, but ILQSSL is said to outperform the original by 8–30 points on structured sets such as Iris and Wine and by 5–10 points on noisier sets such as Heart and German (Gholipour et al., 4 Aug 2025).

Both sources emphasize that larger quantum resources do not monotonically improve learning quality. The 2024 hyperparameter sweep over $G=(V,E)$33 identifies a sweet spot at $G=(V,E)$34–$G=(V,E)$35, while beyond 10 qubits gradients vanish, described as barren plateaus. For entangling layers $G=(V,E)$36, the best Iris test accuracy $G=(V,E)$37 occurs at $G=(V,E)$38; additional layers produce larger entanglement but over-fitting. The same study monitors the von Neumann entropy

$G=(V,E)$39

with $G=(V,E)$40 the reduced density on the first $G=(V,E)$41 qubits, and reports a characteristic “rise and fall” of test accuracy versus $G=(V,E)$42, with moderate $G=(V,E)$43 optimal (Gholipour et al., 2024).

The 2025 study adds Randomized Benchmarking diagnostics. It defines half-register entanglement entropy through

$G=(V,E)$44

and RB fidelity from the fit

$G=(V,E)$45

On Iris, increasing depth from $G=(V,E)$46 to $G=(V,E)$47 layers raises $G=(V,E)$48 from approximately $G=(V,E)$49 to $G=(V,E)$50 but lowers $G=(V,E)$51 from approximately $G=(V,E)$52 to $G=(V,E)$53, while increasing qubits from $G=(V,E)$54 to $G=(V,E)$55 raises $G=(V,E)$56 from $G=(V,E)$57 to $G=(V,E)$58 and slightly lowers $G=(V,E)$59 from $G=(V,E)$60 to $G=(V,E)$61; accuracy remains flat at $G=(V,E)$62. The same trend is reported for Wine, German Credit, and Heart Disease: deeper circuits become marginally more entangled, RB fidelity decreases, and accuracy does not improve; wider circuits substantially increase entanglement, RB drops slightly, and on less-structured datasets accuracy can decrease (Gholipour et al., 4 Aug 2025).

7. Interpretation, limits, and relation to adjacent methods

A recurring result across the available formulations is the expressivity–noise trade-off. The 2024 description states that adding more qubits to a quantum system does not always improve performance, that the effectiveness of additional qubits depends on the quantum algorithm and how well it matches the dataset, and that moderate levels of entanglement offer the best balance between model complexity and generalization capabilities (Gholipour et al., 2024). The 2025 description sharpens the same point with RB analysis, concluding that increased circuit complexity may introduce noise that undermines performance on current quantum hardware (Gholipour et al., 4 Aug 2025).

This directly addresses a common misconception that better semi-supervised performance in quantum graph models should follow automatically from deeper circuits, more entanglers, or more qubits. The reported results do not support that view. Instead, both sources converge on moderate circuit scale as the preferred operating regime. The 2025 implementation guidance explicitly recommends 10–30 layers, 4–8 qubits, a small $G=(V,E)$63 penalty $G=(V,E)$64, and error-mitigation methods such as zero-noise extrapolation or readout calibration to reduce SPAM terms in RB fits (Gholipour et al., 4 Aug 2025). The 2024 account similarly stresses precise hyperparameter tuning tailored to each dataset and identifies barren-plateau onset and over-fitting as practical failure modes (Gholipour et al., 2024).

Another important interpretive point is that ILQSSL is described in relation to both classical and quantum baselines. In the 2024 benchmarks it improves over classical Laplacian SSL and the original QSSL but does not uniformly surpass the best classical label propagation baseline on Iris. In the 2025 study it is described as consistently outperforming both classical semi-supervised methods and the original Laplacian QSSL, particularly under limited supervision. Because the two sources use different datasets, operators, and evaluation setups, these statements should be read as benchmark-specific rather than as a universal ranking claim.

The relation to Improved Poisson Quantum Semi-Supervised Learning is also noteworthy. The 2025 work introduces ILQSSL and IPQSSL together as enhanced quantum models for low-label settings (Gholipour et al., 4 Aug 2025). Although the present topic is ILQSSL, this pairing suggests a broader research direction in which classical graph-based propagation operators, Laplacian and Poisson among them, are being reformulated as hybrid quantum-classical variational procedures. A plausible implication is that future work may compare operator choice, embedding method, and propagation dynamics within a common QSSL framework rather than treating ILQSSL as an isolated model family.