Quantum computers can efficiently sample from probability distributions that are believed to be classically intractable, providing a foundation for quantum generative modeling. However, practical training of such models remains challenging, as gradient evaluation via the parameter-shift rule scales linearly with the number of parameters and requires repeated expectation-value estimation under finite-shot noise. We introduce a training framework that extends the principle of contrastive divergence to quantum models. By deriving the circuit structure and providing a general recipe for constructing it, we obtain quantum circuits that generate the samples required for parameter updates, yielding constant scaling with respect to the cost of a forward pass, analogous to backpropagation in classical neural networks. Numerical results demonstrate that it attains comparable accuracy to likelihood-based optimization while requiring substantially fewer samples. The framework thereby establishes a scalable route to training expressive quantum generative models directly on quantum hardware.
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10 pages, 3 figures; supplementary material 6 pages, 3 figures
Quantum computers can efficiently sample from probability distributions that are believed to be classically intractable, providing a foundation for quantum generative modeling. However, practical training of such models remains challenging, as gradient evaluation via the parameter-shift rule scales linearly with the number of parameters and requires repeated expectation-value estimation under finite-shot noise. We introduce a training framework that extends the principle of contrastive divergence to quantum models. By deriving the circuit structure and providing a general recipe for constructing it, we obtain quantum circuits that generate the samples required for parameter updates, yielding constant scaling with respect to the cost of a forward pass, analogous to backpropagation in classical neural networks. Numerical results demonstrate that it attains comparable accuracy to likelihood-based optimization while requiring substantially fewer samples. The framework thereby establishes a scalable route to training expressive quantum generative models directly on quantum hardware.
The discovery of the backpropagation algorithm ranks among one of the most important moments in the history of machine learning, and has made possible the training of large-scale neural networks through its ability to compute gradients at roughly the same computational cost as model evaluation. Despite its importance, a similar backpropagation-like scaling for gradient evaluation of parameterised quantum circuits has remained elusive. Currently, the most popular method requires sampling from a number of circuits that scales with the number of circuit parameters, making training of large-scale quantum circuits prohibitively expensive in practice. Here we address this problem by introducing a class of structured circuits that are not known to be classically simulable and admit gradient estimation with significantly fewer circuits. In the simplest case – for which the parameters feed into commuting quantum gates – these circuits allow for fast estimation of the gradient, higher order partial derivatives and the Fisher information matrix. Moreover, specific families of parameterised circuits exist for which the scaling of gradient estimation is in line with classical backpropagation, and can thus be trained at scale. In a toy classification problem on 16 qubits, such circuits show competitive performance with other methods, while reducing the training cost by about two orders of magnitude.
Quantum generative modeling has emerged as a promising application of quantum computers, aiming to model complex probability distributions beyond the reach of classical methods. In practice, however, training such models often requires costly gradient estimation performed directly on the quantum hardware. Crucially, for certain structured quantum circuits, expectation values of local observables can be efficiently evaluated on a classical computer, enabling classical training without calls to the quantum hardware in the optimization loop. In these models, sampling from the resulting circuits can still be classically hard, so inference must be performed on a quantum device, yielding a potential computational advantage. In this work, we introduce a photonic quantum generative model built on parametrized Gaussian Boson Sampling circuits. The training is based on the efficient classical evaluation of expectation values enabled by the Gaussian structure of the state, allowing scalable optimization of the model parameters through the maximum mean discrepancy loss function. We demonstrate the effectiveness of the approach through numerical experiments on photonic systems with up to 805 modes and over a million trainable parameters, highlighting its scalability and suitability for near-term photonic quantum devices.
Scaling parametrized quantum circuits to a larger number of qubits is a central challenge in Quantum Machine Learning: gradient estimation via the parameter-shift rule quickly becomes prohibitively expensive, while gradient-free optimizers offer limited convergence guarantees. We apply the Pauli propagation framework to a broad class of Quantum Machine Learning problems in which the loss is expressed as a function of expectation values of the circuit's observables, while keeping the relevant quantity of interest, such as the quantum state or its samples, classically intractable to compute. By propagating the observables backward through the circuit, we obtain an explicit parametric expression for the loss that can be minimized entirely on classical hardware. We also present complementary strategies to make propagation tractable at large scale, including weight-based and degree-based truncations and pruning heuristics. We demonstrate the approach on two benchmarks run on 64-qubit circuits: Variational Quantum Eigensolver on a two-dimensional Ising model, and training of a discrete-type generative model for binary images. In both cases, the classically pre-trained parameters are deployed on the IBM quantum processor ibm_berlin. For generative learning, we additionally show that the circuits considered here require substantially fewer parametrized gates than IQP-style ansätze, reducing transpilation overhead and making deployment on current quantum hardware feasible.
Models of quantum systems faithfully map system parameters to observations, but the inverse problem of parameter inference from measurement data presents a fundamental challenge: computationally intractable likelihoods due to an exponentially large Hilbert space. Here, we introduce simulation-based quantum system inference, a unified, likelihood-free framework that learns parameter posteriors directly from classical simulation data. The central idea is to pair polynomial-cost classical simulators, such as Pauli propagation and tensor networks, with normalizing flows or other neural density estimators for accurate, reusable inference. A single model, trained once, maps any new measurement record to its posterior in one forward pass---turning per-experiment inference into a fixed, up-front cost. We numerically demonstrate the framework's versatility across Pauli noise learning, quantum error mitigation, quantum state tomography, and Hamiltonian learning, with examples involving 81-qubit shallow circuits and 735-parameter inference. In each case, the approach yields accurate estimates of identifiable parameters, while posterior uncertainty provides additional diagnostics of non-identifiability and indicates where further characterization is needed. Our framework reduces data-acquisition requirements in quantum experiments and accelerates parameter inference, providing a practical route to characterizing and improving large-scale quantum systems.
Quantum generative modeling (QGM) relies on preparing quantum states and generating samples from these states as hidden---or known---probability distributions. As distributions from some classes of quantum states (circuits) are inherently hard to sample classically, QGM represents an excellent test bed for quantum supremacy experiments. Furthermore, generative tasks are increasingly relevant for industrial machine learning applications, and thus QGM is a strong candidate for demonstrating a practical quantum advantage. However, this requires that quantum circuits are trained to represent industrially relevant distributions, and the corresponding training stage has an extensive training cost for current quantum hardware in practice. In this work, we propose protocols for classical training of QGMs based on circuits of the specific type that admit an efficient gradient computation, while remaining hard to sample. In particular, we consider instantaneous quantum polynomial (IQP) circuits and their extensions. Showing their classical simulability in terms of the time complexity, sparsity, and anticoncentration properties, we develop a classically tractable way of simulating their output probability distributions, allowing classical training to a target probability distribution. The corresponding quantum sampling from IQPs can be performed efficiently, unlike when using classical sampling. We numerically demonstrate the end-to-end training of IQP circuits using probability distributions for up to 30 qubits on a regular desktop computer. When applied to industrially relevant distributions this combination of classical training with quantum sampling represents an avenue for reaching advantage in the noisy intermediate-scale quantum (NISQ) era.
Quantum generative models use the intrinsic probabilistic nature of quantum mechanics to learn and reproduce complex probability distributions. In this paper, we present an implementation of a 3-qubit quantum circuit Born machine trained to model a 3-bit Gaussian distribution using a Kullback-Leibler (KL) divergence loss and parameter-shift gradient optimization. The variational quantum circuit consists of layers of parameterized rotations and entangling gates, and is optimized such that the Born rule output distribution closely matches the target distribution. We detail the mathematical formulation of the model distribution, the KL divergence cost function, and the parameter-shift rule for gradient evaluation. Training results on a statevector simulator show that the KL divergence is minimized to near zero, and the final generated distribution aligns quantitatively with the target probabilities. We analyze the convergence behavior and discuss the implications for scalability and quantum advantage. Our results demonstrate the feasibility of small-scale quantum generative learning and provide insight into the training dynamics of quantum circuit models.