Quantum generative modelling casts sampling as a generative task: a parametrised quantum circuit is trained such that sampling reproduces a target probability distribution. Instantaneous Quantum Polynomial-time (IQP) circuits combine structural simplicity with complexity-theoretic evidence for quantum advantage. Yet their practical value depends not only on expressivity, but on how efficiently they consume genuinely quantum resources. We study this question through the lens of magic, or non-stabiliserness, as a resource for quantum generative modelling. We show that established fidelity- and geodesic-based notions of computational progress in a projective Hilbert space are ill-suited to generative models, since operational performance is determined by output probability distributions rather than quantum states themselves. We evaluate magic-consumption directly on the probability simplex, using changes in Jensen-Shannon divergence to quantify progress. Applying this framework to trained random γ-sparse IQP circuits shows signatures of efficient magic use, with the dominant contribution arising from two-qubit gates. As IQP circuits produce remarkably low intermediate magic relative to phase-randomised states with the same sampling distributions, this renders IQP-based quantum generative models as promising candidates for resource-efficient demonstrations of quantum advantage on early fault-tolerant architectures.
Quantum generative modelling casts sampling as a generative task: a parametrised quantum circuit is trained such that sampling reproduces a target probability distribution. Instantaneous Quantum Polynomial-time (IQP) circuits combine structural simplicity with complexity-theoretic evidence for quantum advantage. Yet their practical value depends not only on expressivity, but on how efficiently they consume genuinely quantum resources. We study this question through the lens of magic, or non-stabiliserness, as a resource for quantum generative modelling. We show that established fidelity- and geodesic-based notions of computational progress in a projective Hilbert space are ill-suited to generative models, since operational performance is determined by output probability distributions rather than quantum states themselves. We evaluate magic-consumption directly on the probability simplex, using changes in Jensen-Shannon divergence to quantify progress. Applying this framework to trained random γ-sparse IQP circuits shows signatures of efficient magic use, with the dominant contribution arising from two-qubit gates. As IQP circuits produce remarkably low intermediate magic relative to phase-randomised states with the same sampling distributions, this renders IQP-based quantum generative models as promising candidates for resource-efficient demonstrations of quantum advantage on early fault-tolerant architectures.
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.
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.
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 generative models based on instantaneous quantum polynomial (IQP) circuits show great promise in learning complex distributions while maintaining classical trainability. However, current implementations suffer from two key limitations: lack of controllability over generated outputs and severe generation bias towards certain expected patterns. We present a Controllable Quantum Generative Framework, ConQuER, which addresses both challenges through a modular circuit architecture. ConQuER embeds a lightweight controller circuit that can be directly combined with pre-trained IQP circuits to precisely control the output distribution without full retraining. Leveraging the advantages of IQP, our scheme enables precise control over properties such as the Hamming Weight distribution with minimal parameter and gate overhead. In addition, inspired by the controller design, we extend this modular approach through data-driven optimization to embed implicit control paths in the underlying IQP architecture, significantly reducing generation bias on structured datasets. ConQuER retains efficient classical training properties and high scalability. We experimentally validate ConQuER on multiple quantum state datasets, demonstrating its superior control accuracy and balanced generation performance, only with very low overhead cost over original IQP circuits. Our framework bridges the gap between the advantages of quantum computing and the practical needs of controllable generation modeling.
Parameterized Instantaneous Quantum Polynomial (IQP) circuits have proven useful in quantum generative learning models, particularly for binary distributions. However, when applied to non-binary datasets, they exhibit notable limitations: mapping integer values into qubit-compatible binary representations often destroys the original metric structure of the data. In this paper we aim to extend them to a qudits formulation operating on an integer mapping of the data. The IQP quantum circuit is adapted to encode each integer valued pixel into a bit-string of fixed length and quantum gates are transformed to follow the qudit formalism. As a generative machine learning approach, a suitable loss function for the circuit training and the calculation of the covariance matrix among features are developed and validated on the energy deposits from single-particle electron showers in the electromagnetic calorimeter of the CLIC detector. The method proposed in this work can be also extended to other applications that utilize quantum generative machine learning for non-binary data.
Quantum generative models offer an inherently efficient sampling procedure, yet their training on quantum hardware is often a bottleneck. Parametrized Instantaneous Quantum Polynomial-time (IQP) circuits present a promising approach combining classical training with single-shot measurements for sample generation. However, its practical viability on Noisy Intermediate-Scale Quantum (NISQ) hardware remains an open question. In this work, we provide the first large-scale experimental validation of this method for learning graph distributions with both local (edge probabilities) and global (‘2-colorability’) constraints. While our graph encoding requires a number of qubits that scales quadratically with node count, our experiments, ranging from 28 to 153 qubits, demonstrate the approach's capabilities and limitations. For instance, a 153-qubit model successfully learned sparse ErdősRényi graphs with an average edge probability error below two percentage points. In contrast, learning the global property of bipartiteness proved more challenging and sensitive to noise, achieving 25.78 % accuracy for 18-node sparse instances. These results validate parametrized IQP circuits as a practical tool for generative modeling on near-term devices, while clearly delineating the critical challenges that must be addressed for future progress.