We report a scalable hybrid quantum-classical machine learning framework to build Bayesian networks (BN) that captures the conditional dependence and causal relationships of random variables. The generation of a BN consists of finding a directed acyclic graph (DAG) and the associated joint probability distribution of the nodes consistent with a given dataset. This is a combinatorial problem of structural learning of the underlying graph, starting from a single node and building one arc at a time, that fits a given ensemble using maximum likelihood estimators (MLE). It is cast as an optimization problem that consists of a scoring step performed on a classical computer, penalties for acyclicity and number of parents allowed constraints, and a search step implemented using a quantum annealer. We have assumed uniform priors in deriving the Bayesian network that can be relaxed by formulating the problem as an estimation Dirichlet parameters. We demonstrate the utility of the framework by applying to the problem of elucidating the gene regulatory network for the MAPK/Raf pathway in human T-cells using proteomics data where the concentration of proteins, nodes of the BN, are interpreted as probabilities.
We report a scalable hybrid quantum-classical machine learning framework to build Bayesian networks (BN) that captures the conditional dependence and causal relationships of random variables. The generation of a BN consists of finding a directed acyclic graph (DAG) and the associated joint probability distribution of the nodes consistent with a given dataset. This is a combinatorial problem of structural learning of the underlying graph, starting from a single node and building one arc at a time, that fits a given ensemble using maximum likelihood estimators (MLE). It is cast as an optimization problem that consists of a scoring step performed on a classical computer, penalties for acyclicity and number of parents allowed constraints, and a search step implemented using a quantum annealer. We have assumed uniform priors in deriving the Bayesian network that can be relaxed by formulating the problem as an estimation Dirichlet parameters. We demonstrate the utility of the framework by applying to the problem of elucidating the gene regulatory network for the MAPK/Raf pathway in human T-cells using proteomics data where the concentration of proteins, nodes of the BN, are interpreted as probabilities.
Gene regulatory networks (GRNs) encode the combinatorial dependencies that drive cellular programs but are difficult to learn from CRISPR perturbation data using classical models with pairwise features. We present a parameterized quantum circuit (PQC) implemented in Qiskit that represents each gene as a qubit, encodes expression with single-qubitRy(πx)rotations, and models directed regulation with controlled-Ry (θ−{i→j})gates. Circuit parameters are trained by a hybrid classical-quantum loop (SPSA/COBYLA on Aer), and the learnedθvalues yield a weighted, directed GRN. Experiments on synthetic and curated immune-pathway datasets (4−8genes) show consistent gains over strong classical baselines. The PQC achieves 95.2% accuracy, 94.8% precision, 95.6 % recall,F1=95.2%, andAUC=0.98, improving performance by 5-6 percentage points while requiring fewer samples to reach a given accuracy (≈90%using only 50 % of the training set). Graph analysis of the inferred network ranks TP53, NFKB1, IL6, MAPK1, and VEGFA as top regulators, aligning with biological priors. We also evaluate sparse interaction topologies that reduce two-qubit depth while retaining accuracy, underscoring near-term hardware viability using Ry and CNOT-based decompositions. Our results provide an interpretable and data-efficient quantum blueprint for CRISPR-guided network inference and motivate larger-scale tests on real quantum devices. Code and circuits are available for reproducibility.
This paper investigates the application of quantum-inspired optimization techniques to the problem of inference and learning within large-scale Bayesian Networks (BNs). Traditional Bayesian Network learning and inference methods often struggle with scalability, particularly as the number of nodes and dependencies increases. We propose leveraging concepts from quantum algorithms, specifically drawing inspiration from quantum annealing and quantum Monte Carlo methods, to efficiently explore the complex search space associated with BN structure optimization and parameter estimation. The core idea is to mimic the exploration capabilities of quantum systems to accelerate the process. The proposed methodology aims to improve the efficiency of BN learning and inference, potentially unlocking new capabilities for probabilistic reasoning in complex systems. This work presents a theoretical framework and a conceptual outline for such an approach, laying the groundwork for future research and experimentation. The effectiveness of this approach is predicated on the ability to translate the inherent parallelism of quantum algorithms into an efficient optimization strategy for Bayesian Networks.
We introduce a novel quantum Hamiltonian-based gene expression model (QHGM), a generative framework for modeling pseudotime-ordered single-cell gene expression data. In QHGM, gene interactions are encoded via a parameterized Hamiltonian, and the outcomes of quantum measurements provide a discrete representation of the gene expression profile. To learn the Hamiltonian parameters and infer gene regulatory networks (GRNs), we develop a scalable variational quantum algorithm for network inference (VQ-Net) based on empirical risk minimization. We derive finite-sample recovery guarantees for accurate parameter estimation, demonstrating polynomial scaling with the number of genes. Experiments on synthetic data demonstrate accurate GRN recovery, with VQ-NET achieving over 25% improvement in edge recovery and over 50% improvement in parameter-sign recovery compared to state-of-the-art classical methods. We further apply the framework to glioblastoma scRNA-seq data, where it identifies biologically plausible regulatory interactions associated with cancer progression, highlighting the potential of quantum-like modeling beyond classical probabilistic frameworks.
Abstract We introduce a novel quantum Hamiltonian-based gene expression model (QHGM), a generative frame-work for modeling pseudotime-ordered single-cell gene expression data. In QHGM, gene interactions are encoded via a parameterized Hamiltonian, and the outcomes of quantum measurements provide a discrete representation of the gene expression profile. To learn the Hamiltonian parameters and infer gene regulatory networks (GRNs), we develop a scalable variational quantum algorithm for network inference (VQ-Net) based on empirical risk minimization. We derive finite-sample recovery guarantees for accurate parameter estimation, demonstrating polynomial scaling with the number of genes. Experiments on synthetic data demonstrate accurate GRN recovery, with VQ-NET achieving over 25% improvement in edge recovery and over 50% improvement in parameter-sign recovery compared to state-of-the-art classical methods. We further apply the framework to glioblastoma scRNA-seq data, where it identifies biologically plausible regulatory interactions associated with cancer progression, highlighting the potential of quantum-like modeling beyond classical probabilistic frameworks.
• Published on bioRxiv, proposing the 'Quantum Hamiltonian-based gene-expression model (QHGM)' for Gene Regulatory Network (GRN) inference. • Treats genes as 'qubits', interpreting gene interactions over pseudotime by encoding them into quantum entanglement and Hamiltonian matrices. • Applied to glioblastoma (GBM) single-cell RNA sequencing data, successfully elucidating complex cancer cell developmental networks by overcoming the limitations of classical machine learning models. [Quantum Biology Society] Gene Regulatory Networks (GRNs), which determine the cell fate and phenotype of living organisms, are highly dynamic systems where numerous genes and proteins intricately entangle and interact. Despite the massive influx of data from recent advancements in single-cell RNA sequencing (scRNA-seq) technology, existing classical machine learning models have faced limitations in perfectly capturing the delicate context of biological regulation and non-classical probabilistic features. A recent study has been published that dramatically solves this challenge in systems biology through the 'Quantum Hamiltonian Learning (QHL)' framework, which infers the parameters of a many-body quantum system. The paper, titled "Quantum Hamiltonian Learning using Time-Resolved Measurement Data and its Application to Gene Regulatory Network Inference," published on the preprint repository bioRxiv, showcases an unconventional approach of modeling gene networks as macroscopic quantum systems. A joint research team, including Mohammad Aamir Sohail and Professor Arvind Rao from the University of Michigan, USA, led this innovative study. ■ Genes as Qubits, Regulatory Networks as Hamiltonians The 'Quantum Hamiltonian-based gene-expression model (QHGM)' proposed by the research team treats individual genes as qubits. The unexpressed state of a gene is defined as the basis state |0>, and the expressed state as |1>. Regulatory interactions, such as activation or repression between genes, are encoded as weights within a parameterized Hamiltonian matrix constructed using tensor products. Gene qubits, which initially start in independent separable states, evolve according to the Schrödinger equation along pseudotime, an indicator of cell development that acts as physical time. During this evolution process, complex correlations akin to quantum entanglement are formed between genes according to the interaction rules inputted into the Hamiltonian. ■ Combination of IC-POVM Measurements and VQ-Net AI To extract gene expression data from these entangled quantum states, the researchers introduced single-qubit IC-POVM, an informationally complete quantum measurement technique. These measurement results are linked to the discretized gene expression profiles observed in actual scRNA-seq data. Subsequently, the team developed VQ-Net, a variational quantum network inference algorithm based on a mini-batch empirical risk minimization framework. Based on the scRNA-seq data collected over pseudotime, this algorithm minimizes the negative log-likelihood loss and learns by inversely tracing the weight parameters of the Hamiltonian. ■ Elucidating the Glioblastoma (GBM) Network… Outperforming Classical Models This quantum-biology hybrid model did not stop at mere theory but was deployed in actual cancer research. By analyzing scRNA-seq data from patients with IDH-wild-type glioblastoma (GBM), a malignant brain tumor, the researchers inferred the gene regulatory program governing the differentiation of OPC-like cells. The analysis revealed that the QHGM model accurately pinpointed how core genes such as ASCL1, CDK4, and STMN1 interact with cell-cycle proliferation and extracellular matrix (ECM) components, identifying dense feedback loops and directionality consistent with biological literature. Notably, this method scales polynomially with system size, and recorded overwhelmingly higher accuracy and F1 scores in network edge recovery and weight sign identification than representative classical inference algorithms such as ARACNE, GENIE3, and GeneNet. This study is a groundbreaking achievement demonstrating how powerfully "quantum-like" modeling, which interprets complex biological systems using the mathematical tools of quantum information theory, can be utilized in analyzing actual clinical and multi-omics data. The researchers emphasized that this framework will provide crucial insights for understanding the heterogeneity and plasticity of cancer cells and discovering new therapeutic targets in the future.
The Adaptive Bayesian Networks (ABN) represent a powerful framework for modeling complex systems by leveraging probabilistic inference and Bayesian reasoning. This paper introduces a novel quantumized algorithm for ABN, designed to enhance learning and prediction accuracy, particularly for systems exhibiting intricate probability distributions. We explore the utilization of quantum phenomena – superposition and entanglement – to accelerate the network's learning process and improve the precision of risk assessments. The core mechanism focuses on applying quantum-enhanced optimization techniques to the ABN structure, leading to a more efficient and accurate representation of the underlying probability landscape. This research addresses the limitations of classical ABN algorithms, offering a promising pathway toward enhanced predictive capabilities in diverse domains.