A generative quantum machine learning approach
July 9, 2026
Hamburg / Innsbruck, 09.07.2026 – A group of researchers at ParityQC and the University of Innsbruck have extended parameterized Instantaneous Quantum Polynomial (IQP) circuits to qudits, enabling a generative quantum machine learning approach that operates directly on integer data.
The challenge: binary circuits meet non-binary data
Generative machine learning methods aim to approximate the underlying probability distribution of a dataset, enabling the synthesis of new samples that faithfully reflect the statistical structure of the original data. Quantum systems are particularly well suited to this task, and parameterized IQP circuits have proven useful in quantum generative learning models, particularly for binary distributions. However, when applied to non-binary datasets, they exhibit a notable limitation: mapping integer values into qubit-compatible binary representations often destroys the original metric structure of the data.
A qudit approach to integer data
In the latest publication “Qudit extension of parameterized IQP circuits: A generative quantum machine learning approach to integer data”, a group of researchers at ParityQC and the University of Innsbruck – Robert J. Banks, Arianna Crippa, Matthias Traube, Josua Unger, Christian Ertler, and Wolfgang Lechner – extend IQP circuits to a qudit formulation operating on an integer mapping of the data. The work was carried out as part of the HQML (Hamburg Full Stack Quantum Machine Learning) project. Key features of the proposed approach include:
- Operating directly on integer data: The IQP quantum circuit is adapted to encode each integer-valued feature of the distribution into a bit-string of fixed length, and the quantum gates are transformed to follow the qudit formalism. This avoids the need to binarize the target data distribution and preserves the metric structure that binary encodings tend to destroy.
- Tailored training tools: 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. In particular, the Maximum Mean Discrepancy (MMD) loss function is modified to address the cyclic distance between qudit states.
- Validation on real high-energy-physics data: The method is validated by training the circuit on energy deposits from single-particle electron showers in the electromagnetic calorimeter of the CLIC detector, a fundamental task in high energy physics.
Beyond high energy physics
The method proposed in this work can also be utilized in other applications that utilize quantum generative machine learning for non-binary data.
Read the preprint
The preprint “Qudit extension of parameterized IQP circuits: A generative quantum machine learning approach to integer data”, authored by Robert J. Banks, Arianna Crippa, Matthias Traube, Josua Unger, Christian Ertler, and Wolfgang Lechner, is now available here.
FAQ
Parameterized IQP circuits have proven useful in quantum generative learning models, particularly for binary distributions. When applied to non-binary datasets, however, they show a notable limitation: mapping integer values into qubit-compatible binary representations often destroys the original metric structure of the data. This work extends IQP circuits to a qudits such that it operates directly on an integer mapping of the data, avoiding the need to binarize the target distribution.
Instantaneous Quantum Polynomial (IQP) circuits are a class of quantum circuits used in quantum generative learning. In the approach that this work builds on, the circuit parameters are learned using classical optimization methods rather than by sampling. Training is instead based on efficiently computing expectation values or moments, which means it can be carried out entirely on classical hardware. Sampling from the trained circuit is believed to be classically hard in general and therefore requires quantum hardware.
The IQP quantum circuit is adapted to encode each integer-valued feature into a bit-string of fixed length, and the quantum gates are transformed to follow the qudit formalism. The authors develop the mathematical tools required to calculate the expectation values of diagonal operators in the qudit formalism, and they modify the Maximum Mean Discrepancy (MMD) loss function to address the cyclic distance between qudit states.
The method was validated by training the circuit on the energy deposits from single-particle electron showers in the electromagnetic calorimeter (ECAL) of the CLIC detector. Simulating electromagnetic particle showers is a fundamental task in high energy physics. The target distribution is based on a publicly available dataset of calorimeter images.
Modeling the energy-deposition distribution in a calorimeter is challenging because it requires capturing correlations in the energy-deposition structure, and classical simulations based on Monte Carlo methods are computationally intensive. Quantum computing may offer a potential alternative framework for addressing these computational demands, and quantum generative models could also serve as simulators for tasks such as data reconstruction and classification in high energy physics.
Yes. While the method was applied to high-energy physics data, the authors note that it can also be utilized to other applications that utilize quantum generative machine learning for non-binary data.
