Abstract
Quantum computers promise to be a revolutionary solution to fulfill the increasing need for high-performance computing, and quantum computing has been identified as a key intervention area in the “Programma Nazionale per la Ricerca 2021-27”. Determining the problems on which quantum computers can provide major advantages with respect to classical computers is the main theoretical problem of quantum computing and constitutes the challenge that LeQun addresses. The most promising family of quantum algorithms that can be implemented on the forthcoming generation of quantum computers are variational quantum algorithms, also called quantum neural networks. Despite the promises, there is no problem of practical interest yet where quantum neural networks have a provable advantage over the best classical algorithms. A thorough theoretical study of the trainability, expressibility and generalization properties of quantum neural networks and of their potential advantages with respect to classical computers constitutes the main theoretical challenge of quantum machine learning. LeQun will tackle this extremely ambitious challenge through the following objectives: O1: - To analytically characterize the probability distribution of the functions generated by trained quantum neural networks and determine their trainability and generalization performances. - To study the training stability against imperfect outcomes of quantum measurements and the query complexity of the entire training process, namely the number of measurements that must be performed to achieve the desired accuracy. O2: -To identify the architectures of quantum neural networks that have the potential to provide major advantages with respect to classical computers and to perform a proof-of-principle validation of the advantages of the identified architectures using real quantum devices and simulators. LeQun will tackle the challenge with an interdisciplinary approach that connects quantum machine learning with probability theory and quantum many-body physics. The strategy of LeQun is based on the recent breakthrough results in classical machine learning stating that in the mean-field limit of infinite width of the hidden layers, trained deep neural networks are equivalent to Gaussian processes. These results explained the unreasonably good performances of deep neural networks, and LeQun will generalize them to the quantum setting. LeQun will constitute a successful example of interdisciplinary research and will foster a cross-fertilization among quantum computing, probability theory and many-body quantum physics. The results of LeQun will be highly relevant for all the researchers working on quantum computing and machine learning both in industry and academia, thus contributing to creating value for the whole society, and its results will constitute a fundamental contribution to the main challenge of quantum computing. Results Achieved The LeQun project has significantly advanced our understanding of quantum neural networks (QNNs), providing both new mathematical foundations and practical insights into their capabilities and limitations. The project addressed two major scientific goals: understanding how quantum neural networks learn and identifying the conditions under which they may offer a genuine quantum advantage over classical artificial intelligence. The first objective focused on the theoretical analysis of QNNs during training. A central achievement was proving that very large quantum neural networks behave in a remarkably simple and predictable way. As the number of qubits increases, the output of these networks converges to a mathematical object known as a Gaussian process, whose statistical properties can be calculated exactly. This result applies not only before training but also after learning has taken place, providing the first rigorous description of the behaviour of a broad family of trained quantum neural networks. The analysis also shows that, in this large-scale limit, the networks are capable of perfectly fitting the training data. Beyond establishing this asymptotic behaviour, the project quantified how quickly realistic, finite-size QNNs approach the Gaussian-process regime. Explicit mathematical bounds were derived, showing how the difference between finite and ideal infinite networks decreases as the system grows. Importantly, these guarantees remain valid throughout the entire training process. Such quantitative results provide valuable guidance for understanding the performance of practical quantum learning systems. Another important contribution concerned the role of noise, an unavoidable feature of current quantum hardware. Training quantum neural networks requires estimating gradients through repeated quantum measurements, which introduces statistical fluctuations. The project demonstrated that only a polynomial number of measurements is sufficient to preserve both the Gaussian-process behaviour and the excellent learning performance of the networks. This shows that training can remain computationally efficient despite measurement noise. Overall, the first objective was fully achieved. LeQun established a solid mathematical foundation for understanding why large quantum neural networks can be trained successfully, how they evolve during learning, and how their behaviour scales with system size. The second objective investigated which quantum learning architectures might deliver computational advantages over classical methods. Here, the project produced both highly encouraging and cautionary results. On the constructive side, the researchers introduced a new mixture-of-quantum-experts architecture capable of classifying handwritten images from the widely used MNIST benchmark. Using only ten qubits through amplitude encoding, the model achieved up to 97.5% classification accuracy, demonstrating that sophisticated image recognition can be performed with remarkably limited quantum resources. At the same time, one of the project’s most important theoretical discoveries revealed a fundamental limitation. The researchers developed an efficient classical algorithm capable of reproducing the behaviour of wide trained quantum neural networks by exploiting their Gaussian-process description. For a broad class of architectures, this means that sufficiently large trained QNNs can be efficiently simulated on conventional computers. Consequently, these networks cannot provide a genuine computational quantum advantage in this regime. This negative result is scientifically important because it clearly identifies the boundaries within which quantum machine learning should be expected to outperform classical approaches. Rather than closing the door on quantum advantage, these findings helped identify more promising directions. The project studied physics-informed quantum neural networks designed to solve linear partial differential equations. Unlike the previously analysed architectures, these networks cannot be efficiently simulated using the proposed classical algorithm, making them promising candidates for future demonstrations of quantum advantage. While the current work established their mathematical behaviour, determining whether they truly outperform classical methods remains an important challenge for future research. LeQun also explored machine learning directly on quantum data, developing algorithms capable of recognising different quantum phases of matter from a limited number of quantum measurements. The results indicate that meaningful learning can be achieved without reconstructing the full quantum state, a task that is typically prohibitively expensive. Numerical evidence suggests that the computational cost grows only polynomially under suitable conditions, although establishing rigorous guarantees for very large quantum systems remains an active area of investigation. Taken together, the results of LeQun provide one of the clearest theoretical pictures to date of the opportunities and limitations of quantum machine learning. The project demonstrated powerful new quantum learning architectures, established rigorous guarantees for their trainability and robustness, identified fundamental situations where quantum neural networks cannot outperform classical algorithms, and highlighted promising new directions where genuine quantum advantage may still emerge. These achievements provide an essential scientific foundation for the future development of quantum artificial intelligence and help guide the search for quantum learning methods capable of delivering practical benefits beyond classical computation.
Project details
Unibo Team Leader: Giacomo De Palma
Unibo involved Department/s:
Dipartimento di Matematica
Coordinator:
ALMA MATER STUDIORUM - Università di Bologna(Italy)
Total Eu Contribution: Euro (EUR) 187.486,00
Total Unibo Contribution: Euro (EUR) 85.770,00
Project Duration in months: 24
Start Date:
28/09/2023
End Date:
28/02/2026
