[Paper Review] Quantum Hamiltonian Learning using Time-Resolved Measurement Data and its Application to Gene Regulatory Network Inference

Mohammad Aamir Sohail, Ranga R. Sudharshan+2 more|arXiv (Cornell University)|Feb 23, 2026

Gene Regulatory Network Analysis0 citations

TL;DR

This paper develops a quantum Hamiltonian learning framework using time-resolved IC-POVM data to infer gene regulatory networks, introducing the QHGM model and a scalable variational learning algorithm (VQ-Net) validated on synthetic data and glioblastoma scRNA-seq.

ABSTRACT

We present a new Hamiltonian-learning framework based on time-resolved measurement data from a fixed local IC-POVM and its application to inferring gene regulatory networks. We introduce the quantum Hamiltonian-based gene-expression model (QHGM), in which gene interactions are encoded as a parameterized Hamiltonian that governs gene expression evolution over pseudotime. We derive finite-sample recovery guarantees and establish upper bounds on the number of time and measurement samples required for accurate parameter estimation with high probability, scaling polynomially with system size. To recover the QHGM parameters, we develop a scalable variational learning algorithm based on empirical risk minimization. Our method recovers network structure efficiently on synthetic benchmarks and reveals novel, biologically plausible regulatory connections in Glioblastoma single-cell RNA sequencing data, highlighting its potential in cancer research. This framework opens new directions for applying quantum-like modeling to biological systems beyond the limits of classical inference.

Motivation & Objective

• Formulate a Hamiltonian-learning problem from time-resolved IC-POVM data starting from a fixed initial state.
• Instantiate a quantum-inspired model for gene expression (QHGM) that encodes GRN interactions as a parameterized Hamiltonian.
• Develop a scalable variational learning algorithm (VQ-Net) to estimate Hamiltonian parameters from scRNA-seq data.
• Provide finite-sample guarantees and analyze sample complexity.
• Demonstrate the approach on synthetic data and apply to Glioblastoma scRNA-seq to uncover plausible regulatory connections.

Proposed method

• Define H(w)=sum_j w_j H_j with w in a bounded set and evolve rho_t(w)=U_t(w) rho_0 U_t(w)† where U_t(w)=exp(-i t H(w)).
• Perform IC-POVM measurements on each qudit to obtain outcomes with probability phi(m|t,w)=tr(Lambda_m rho_t(w)).
• Collect N_t time samples with N_c outcomes per time; minimize empirical risk L_hat(w) = (1/N_t)(1/N_c) sum_i sum_k ell(phi(m_(i,k)|t_i,w)) using empirical risk minimization.
• Prove that under assumptions, N_t = O~(c^3/ε^2 log(1/δ)) and N_c = O~(c^3/ε^2 log(N_t/δ)) yield (1-2ε)μ_0-strong convexity of L_hat and error bound ||w_hat−w*|| = O~( (1/((1-2ε)μ_0)) sqrt(c/(N_c) log(N_t/δ)) ).
• research_questionsRelated to the following:
• How many time samples and measurements per time are needed to accurately recover Hamiltonian parameters in QHL?
• Can a quantum Hamiltonian framework be effectively used to model and infer GRNs from scRNA-seq data?
• What are the identifiability and convergence properties of empirical risk minimization in this setting?
• How does the method perform on synthetic data and real GBM scRNA-seq data in recovering known regulatory interactions?
• key_findings
• VQ-Net recovers Hamiltonian weights efficiently on synthetic data, with improved accuracy as N_t and N_c increase.
• Finite-sample bounds show polynomial scaling of required samples with system size when c scales polynomially in n.
• Empirical risk minimization with time-resolved IC-POVM data yields strong convexity and provable error bounds for w_hat.
• VQ-Net applied to GBMap GBM scRNA-seq data identifies regulatory interactions consistent with known biology, including ASCL1 and targets like BCAN, CDK4, and CKB.
• On synthetic data, increasing N_t beyond 15 and N_c beyond 1000 improves weight recovery; insufficient N_t leads to non-identifiability.
• The GBM analysis reveals a GRN with discernible activation and repression patterns and shows the method can reveal biologically plausible regulatory connections.

Figure 1 : Overview of the quantum Hamiltonian-based gene-expression model (QHGM). A gene regulatory network (GRN) is mapped to a parameterized Hamiltonian $\mathrm{H}(\mathbf{w})$ , where the presence of gene $\mathsf{g}_{i}$ induces the action of a Pauli- $\mathrm{Y}$ operator on gene $\mathsf{g}_

Experimental results

Research questions

• RQ1 What is the sample complexity (N_t, N_c) required for reliable Hamiltonian parameter estimation under time-resolved IC-POVM measurements?
• RQ2 Can the QHL framework be instantiated as a generative model for gene expression (QHGM) and learned from scRNA-seq data?
• RQ3 What are the uniform convergence properties of the empirical loss L_hat to the expected loss L in this setting?
• RQ4 Does VQ-Net recover known regulatory interactions in real GBM scRNA-seq data and reveal plausible new connections?

Key findings

• Theoretical results show N_t and N_c scale polynomially with the number of Hamiltonian parameters and provide an error bound for w_hat.
• Empirical results on synthetic data demonstrate weight recovery improves with larger N_t and N_c and that low N_t can cause identifiability issues.
• VQ-Net successfully learns initial-state parameters alongside weights, with initial-state estimates being more robust to sample variations.
• Application to GBMap GBM scRNA-seq data recovers known regulatory interactions and highlights regulatory patterns consistent with differentiation trajectories in OPC-like cells.
• The GBM analysis demonstrates the framework can uncover biologically plausible GRNs and suggests potential regulatory structures in cancer research.

Figure 2 : VQ-Net. (A) Raw scRNA-seq data are preprocessed, normalized, and assigned pseudotime values, providing a temporal ordering of cells along a developmental trajectory. (B) The normalized pseudotime-ordered scRNA-seq data are converted into four discrete values, denoted as $\mathcal{G}$ , wh