Quantum generative learning based on instantaneous quantum polynomial-time (IQP) circuits can benefit from efficient classical training strategies. A recent latent adaptation framework for IQP-based generative modeling transfers shared circuit parameters across instances of the same task with different hyperparameters while adapting only a low-dimensional latent variable for each new instance. However, existing approaches initialize this latent variable randomly, which can limit optimization efficiency and performance. In this work, we introduce a physics-informed latent initialization scheme for IQP generative learning to improve upon existing random initialization schemes. Motivated by the platonic representation hypothesis, we use latent representations extracted from a classical physics-informed neural network (PINN) surrogate to initialize the latent variables of the quantum model for the solution of the Burgers' equation. The initialized IQP model is then adapted on a higher-resolution solution domain. We find that this structured initialization consistently outperforms random latent initialization, yielding improved adaptation behavior and stronger generative accuracy across multiple viscosity settings. These results show that classical surrogate representations can provide useful inductive bias for quantum generative models and offer a practical route to improved initialization in IQP-based learning.
Quantum generative learning based on instantaneous quantum polynomial-time (IQP) circuits can benefit from efficient classical training strategies. A recent latent adaptation framework for IQP-based generative modeling transfers shared circuit parameters across instances of the same task with different hyperparameters while adapting only a low-dimensional latent variable for each new instance. However, existing approaches initialize this latent variable randomly, which can limit optimization efficiency and performance. In this work, we introduce a physics-informed latent initialization scheme for IQP generative learning to improve upon existing random initialization schemes. Motivated by the platonic representation hypothesis, we use latent representations extracted from a classical physics-informed neural network (PINN) surrogate to initialize the latent variables of the quantum model for the solution of the Burgers' equation. The initialized IQP model is then adapted on a higher-resolution solution domain. We find that this structured initialization consistently outperforms random latent initialization, yielding improved adaptation behavior and stronger generative accuracy across multiple viscosity settings. These results show that classical surrogate representations can provide useful inductive bias for quantum generative models and offer a practical route to improved initialization in IQP-based learning.
Quantum circuit Born machines based on instantaneous quantum polynomial-time (IQP) circuits are natural candidates for quantum generative modeling, both because of their probabilistic structure and because IQP sampling is provably classically hard in certain regimes. Recent proposals focus on training IQP-QCBMs using Maximum Mean Discrepancy (MMD) losses built from low-body Pauli-Z correlators, but the effect of initialization on the resulting optimization landscape remains poorly understood. In this work, we address this by first proving that the MMD loss landscape suffers from barren plateaus for random full-angle-range initializations of IQP circuits. We then establish lower bounds on the loss variance for identity and an unbiased data-agnostic initialization. We then additionally consider a data-dependent initialization that is better aligned with the target distribution and, under suitable assumptions, yields provable gradients and generally converges quicker to a good minimum (as indicated by our training of circuits with 150 qubits on genomic data). Finally, as a by-product, the developed variance lower bound framework is applicable to a general class of non-linear losses, offering a broader toolset for analyzing warm-starts in quantum machine learning.
Generative modeling is one of the most promising applications of quantum machine learning, yet training and deploying Quantum Generative Models (QGMs) on near-term hardware remains effectively intractable due to prohibitive gradient estimation and implementation costs. We propose a resource-efficient approach based on shallow Instantaneous Quantum Polynomial-time (IQP) circuits that circumvents these bottlenecks by leveraging efficient classical training while retaining the guarantee of sampling hardness. To validate this approach, we formalize graph generation as a hierarchy of physical correlations, allowing us to map abstract data features, such as edge density and bipartiteness, directly to the quantum observables required to learn them. We validate our protocol through demonstrations both on real hardware (from 28 to 153 qubits) and simulations (28 qubits). Results show that while global structural features exhibit significant degradation beyond 91 qubits, our models achieve high-precision reproduction of local correlations, even up to 153 qubits. These findings establish shallow IQP circuits as a robust, scalable candidate for generative tasks on current Noisy Intermediate-Scale Quantum (NISQ) devices.
Physics-informed neural networks (PINNs) have emerged as a versatile approach for solving nonlinear partial differential equations (PDEs), yet achieving high accuracy efficiently using these techniques remains challenging for high-dimensional or multiscale systems. Here, we present a hybrid quantum-classical framework that enhances Quantum PINNs (QPINNs) through adaptive collocation point sampling and loss-aware attention mechanisms. By dynamically prioritizing points in regions with large PDE residuals or steep solution gradients, our method mitigates the spectral bias inherent in conventional PINNs. Current Quantum Physics-Informed Neural Networks are commonly assumed to be limited by the expressive power of quantum circuits. In our work, we observed that, across diverse differential equations, optimization - not only expressivity - can be an important bottleneck. Furthermore, a trainable loss-weighting scheme balances contributions from physics residuals, boundary conditions, and data fidelity during training. Integrating these strategies with quantum computing techniques (including variational quantum circuits and quantum gradient estimation) can yield at least a 60% improvement in solution accuracy under specific regimes for benchmark fluid flows and reaction-diffusion systems. Finally, we argue that merely increasing model expressivity is insufficient for resolving complex PDEs via QPINNs, as they remain constrained by the structural optimization limitations of classical PINNs. This framework provides a scalable pathway for quantum-enhanced scientific machine learning, bridging physics-based modeling with emerging quantum computational capabilities.
Partial differential equations (PDEs) play a crucial role in financial mathematics, particularly in portfolio optimization, and solving them using classical numerical or neural network methods has always posed significant challenges. Here, we investigate the potential role of quantum circuits for solving PDEs. We design a parameterized quantum circuit (PQC) for implementing a polynomial based on tensor rank decomposition, reducing the quantum resource complexity from exponential to polynomial when the corresponding tensor rank is moderate. Building on this circuit, we develop a Quantum Physics-Informed Neural Network (QPINN) and a Quantum-inspired PINN, both of which guarantee the existence of an approximation of the PDE solution, and this approximation can be represented as a polynomial that incorporates tensor rank decomposition. Numerical experiments are conducted on the Hamilton--Jacobi--Bellman (HJB) PDE arising from the Merton portfolio optimization problem, which determines the optimal investment fraction between a risky and a risk-free asset. The results show that our quantum models achieve lower losses and approximation errors than a classical fully connected PINN while using substantially fewer trainable parameters. Our quantum models further outperform a classical PINN constructed to share a similar inductive bias, providing experimental evidence of quantum-induced improvement in the tested settings and highlighting a resource-efficient pathway toward classical and near-term quantum solvers for PDEs with exploitable solution structure.
We study a practical question in generative quantum machine learning: given a classical dataset, can we determine, before training, whether it is well suited to a quantum generative model? We focus on a class of quantum circuits known as instantaneous quantum polynomial-time (IQP) circuits, whose output distributions are widely believed to be difficult to sample from using classical methods. These circuits are used to build our quantum generative models. We introduce a Correlation-Complexity Map, a simple diagnostic built from two quantities computed from data samples. The first measures how closely the dataset's spectral correlation patterns resemble those naturally produced by IQP circuits, while the second quantifies how much of the dataset's structural correlation cannot be captured by simple pairwise models. In other words, we can estimate beforehand how well a dataset can be approximated by our model family and also how complex its correlations are, indicating possible failures of classical models. Applying this framework, we identify turbulence data as a promising target for quantum generative modeling. Guided by this analysis, we use a latent-parameter adaptation scheme that reuses a compact IQP circuit over a temporal sequence by learning and interpolating a low-dimensional latent trajectory, and observe competitive performance against classical baselines in a low-data, low-parameter regime. These results suggest that dataset-level diagnostics can help prioritize problems where quantum generative models are most likely to be useful, with improvements in data and parameter efficiency.
Quantum physics-informed neural networks (QPINNs) have recently emerged as a promising framework for the solution of partial differential equations (PDEs), with several studies reporting improved convergence and accuracy relative to classical physics-informed neural networks (PINNs) at reduced training cost. Motivated by these advances, we introduce geometric quantum physics-informed neural networks (GQPINNs), a symmetry-aware extension of QPINNs in which the geometric structure of the underlying PDE is incorporated directly into the quantum-circuit ansatz. Building on the framework of geometric quantum machine learning, we construct parametrized circuits that encode finite-group and compact Lie-group symmetries as inductive biases through problem-specific equivariant generator sets . Using a twirling-based construction, we derive symmetry-preserving gates that ensure that the model predictions respect the symmetries of the governing equation whenever the boundary and initial data are symmetry compatible. We benchmark GQPINNs against standard QPINNs and symmetry-adapted classical PINN baselines under matched training protocols across a representative set of linear and nonlinear PDEs. Across these benchmarks, GQPINNs achieve improved solution accuracy, as quantified by lower mean absolute error, while requiring substantially fewer trainable parameters. These results identify symmetry-aware quantum-circuit design as an effective route toward improved efficiency and generalization in quantum PDE solvers and provide a systematic framework for incorporating geometric inductive biases into quantum-enhanced scientific machine learning.