Quantum Circuit Born Machines

A single observation can admit several plausible futures, making uncertainty an essential part of world modelling. However, quantum generative models remain underexplored for this task. We introduce Quantum-JEPA, a latent-action JEPA framework that learns distributions of future representations using a conditional quantum Born prior. Circuit measurements generate binary latent actions, which a shared predictor maps to alternative future representations. We train the quantum prior through the quality and diversity of these predictions using a sampled energy objective. Our model learns distinct continuations on constructed multimodal tasks and captures latent-action correlations that support coordinated multi-agent futures. On D3IL, a benchmark of diverse human demonstrations, our trained models generate multiple obstacle-crossing routes from a common starting point without observed motion. On the controlled tasks, our classical and quantum prior implementations improve prediction validity and distributional fidelity over the evaluated variational baselines. Our quantum priors combine competitive predictive quality with a parameter-efficient sampling structure: they avoid an exponentially sized categorical output layer while supporting correlated latent actions. We further validate finite-shot parameter-shift training, including training with readout noise. Unlike most prior work, which focuses on simulation, we also verify the feasibility of inference with trained priors on a real QPU. These results establish a trainable quantum component for JEPA future prediction and a basis for exploring quantum-enhanced world models at larger scales.

We study variational quantum distribution learning through a hierarchy of Walsh--Fourier approximations on the Boolean cube. At each level, a selected set of target Fourier coefficients defines a spectral truncation, which is projected onto the probability simplex and used as the target of a quantum circuit Born machine. Parameters learned at one level initialize the next through a warm-start map. We prove an end-to-end expected learning guarantee where the approximation term is determined by the omitted Fourier mass, while a normalized unbiased estimator yields an explicit statistical bound for empirical truncations. We then instantiate the abstract discrepancy conditions for total variation distance and relate the resulting distributional error to quantum-state fidelity. The total-variation specialization incurs the explicit factor $2^{n-1}$ under our normalized $\ell_2$ convention and is therefore informative only for sufficiently concentrated Fourier tails. The framework does not establish global trainability or eliminate barren plateaus; rather, it identifies the conditions under which low-to-high spectral training admits a approximation--estimation--optimization analysis.

Spectral filtering has been proposed as a route to regularization in quantum generative models: the quantum Fourier transform exposes the amplitude spectrum of a quantum circuit Born machine, and a diagonal filter suppresses the high frequencies associated with finite-sample noise, an operation whose classical counterpart seemingly requires manipulating an exponentially long amplitude vector. We examine whether this coherent operation produces anything that classical post-processing of samples from the unfiltered model cannot match. Measuring the filter against convolution with a symmetric probability kernel at matched sampling cost, which accounts for the post-selection overhead of attenuation, we derive necessary and sufficient conditions for the gap between the two to vanish. Magnitude (attenuating) filters obey a dichotomy: at a fixed affordability threshold, the filtered output is either a constant-size Fourier object with an efficient classical sampler, or the passband must widen until no fixed frequency is attenuated and the filter no longer smooths. In neither case does the filter create a quantum-classical separation. Whatever separation survives is inherited from the spectral phase of the input state. Numerical experiments on trained circuit Born machines confirm the classification and show that the deciding phases are invisible to the Born-rule training loss and set by the initialization. Within the diagonal family, pure phase filters remain the only spectral operations exempt from these constraints.

Data scarcity and class imbalance are persistent challenges in machine learning that degrade model generalization and introduce predictive bias. We present a hybrid quantum-classical framework for synthetic data generation using a Quantum Circuit Born Machine (QCBM) to address these limitations. The proposed approach exploits quantum mechanical properties -- superposition and entanglement -- within a parameterized variational quantum circuit to model complex probability distributions that are difficult for classical generative methods to capture. Experiments are conducted on two tabular benchmark datasets: the Iris dataset and the Telco Customer Churn dataset. Preprocessing includes normalization and PCA-based dimensionality reduction to enable efficient basis encoding for quantum circuits. The QCBM is trained by minimizing Kullback-Leibler (KL) divergence between real and generated data distributions using a gradient-based parameter-shift optimization rule. Augmenting training data with QCBM-generated synthetic samples at 40-50% of the minority class improves F1-score by approximately 5-15% and minority-class recall by 10-25%. Cross-domain evaluations (Train on Synthetic, Test on Real; and Train on Real, Test on Synthetic) reveal a performance gap of only 3-10%, indicating strong distributional fidelity. Comparative analysis against classical oversampling methods -- SMOTE, Borderline-SMOTE, KMeansSMOTE, and SVM-SMOTE -- shows that QCBM achieves competitive classification performance and produces lower Maximum Mean Discrepancy (MMD) on the Telco dataset, suggesting superior structural similarity in certain imbalanced settings. These findings establish QCBM as a viable complementary tool for data augmentation, particularly for low-dimensional structured tabular data with class imbalance.

Quantum Circuit Born Machines (QCBMs) offer a natural approach to generative machine learning by leveraging the Born rule. Recent work has provided a method to classically train QCBMs with Instantaneous Quantum Polynomial (IQP) circuits via the Maximum Mean Discrepancy (MMD) loss. Despite the assumed intractability of sampling from IQP circuits classically, their expectation values can be computed classically, enabling training of these IQP QCBMs. However, quantum machine learning (QML) models have various other challenges, including trainability issues caused by exponential concentration or barren plateaus. While these issues have been explored for parameters sampled from a uniform distribution, little work has been done to rigorously treat the use of arbitrary Gaussian initialization schemes. This work leverages Stein's lemma and Lipschitz concentration bounds for Gaussian random variables to provide an analytical lower bound of the variance of the gradient and a probabilistic concentration bound of the deviation of the gradient from its mean. It discusses strategies to either avoid or encourage exponential concentration, as well as the conditions under which barren plateaus are more likely to occur.

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.

Accurate financial volatility forecasting is crucial but challenged by the non-linear, highly correlated nature of market data. Recently, quantum computing has emerged as a promising paradigm for solving complex high-dimensional sampling problems. To harness this, we propose a novel hybrid framework combining the temporal representation power of classical neural networks with the distribution-learning capabilities of quantum models. Specifically, we integrate a Long Short-Term Memory (LSTM) network with a Quantum Circuit Born Machine (QCBM). The LSTM extracts dynamic features, while the QCBM acts as a learnable generative prior modeling complex market distributions to guide forecasting. Evaluated on 5-minute high-frequency data from the SSE Composite and CSI 300 indices, our model significantly outperforms a classical LSTM baseline across MSE, RMSE, and QLIKE metrics. Furthermore, by introducing a stochastic Drop-Prior" mechanism during training, the LSTM implicitly distills structured information from the quantum prior. This establishes a pragmatic paradigm of quantum-assisted training with classical-efficient inference", whereby the model retains its quantum-enhanced accuracy even when the quantum module is entirely disabled during deployment. This demonstrates a practical pathway for leveraging quantum computing to enhance classical models without real-time quantum inference latency.

Quantum generative modeling, where the Born rule naturally defines probability distributions through measurement of parameterized quantum states, is a promising near-term application of quantum computing. We propose a Quantum Scrambling Born Machine in which a fixed entangling unitary -- acting as a scrambling reservoir -- provides multi-qubit entanglement, while only single-qubit rotations are optimized. We consider three entangling unitaries -- a Haar random unitary and two physically realizable approximations, a finite-depth brickwork random circuit and analog time evolution under nearest-neighbor spin-chain Hamiltonians -- and show that, for the benchmark distributions and system sizes considered, once the entangler produces near-Haar-typical entanglement the model learns the target distribution with weak sensitivity to the scrambler's microscopic origin. Finally, promoting the Hamiltonian couplings to trainable parameters casts the generative task as a variational Hamiltonian problem, with performance competitive with representative classical generative models at matched parameter count.