Project Description

Abstract

QUAI4EO 2 was a joint research project between CERN, ESA Φ-lab, and the University of Leiden, running from June 2022 to August 2025 and carried out by PhD candidate Alice Barthe, with the thesis defense scheduled for the end of March. While the first QUAI4EO project targeted static imagery, this project addressed dynamic systems and time-dependent data, as well as the theoretical foundations for knowing when quantum models help.

The central challenge was to map the state of a dynamic system efficiently into a Hilbert space in which time evolution acts as a linear operator. From that starting point, three questions followed: whether properties of the dynamical system can be derived from such a representation; whether models can learn from time-dependent data generated by the system; and for which dynamical systems and which tasks a genuine advantage over classical models exists.

The main result of the collaboration is a move beyond heuristic observation toward a theory-grounded characterization of algorithm behavior, delivered across four complementary contributions. Regarding trainability and expressivity, the analysis of quantum re-uploading units (QRUs) shows that data re-uploading induces Gaussian smoothing in frequency space, suppressing high-frequency components and thereby limiting expressivity. As a result, the re-uploading layers must be adjusted deliberately; absorption witnesses provide a principled way to assess and mitigate barren plateau risk. In generative modeling, two universal architectures for expectation-value samplers were established, along with their resource requirements and fundamental dimensionality limits. In learning theory, a provable quantum advantage was demonstrated through Fourier coefficient extraction and PAC-learning separations in synthetic concept classes as well as in physically meaningful ones, such as Hamiltonian dynamics. Finally, the advantage analysis was extended to continuous-variable systems through CVQC ODE solvers and bosonic circuit complexity classifications.

Together, these results map out theoretical foundations for future quantum machine learning and quantum simulation work on dynamical, time-dependent Earth observation data.

Detailed description

Context and motivation

Much of the Earth observation data that matters is not a single image but a time series: the state of a system evolving under only partially known dynamics. Applying quantum machine learning to that setting raises a representational question before any model can be trained — how to embed the state of a dynamic system in a Hilbert space such that its time evolution becomes a linear operator, which is the form quantum mechanics handles natively.

QUAI4EO 2 was hosted by CERN in collaboration with ESA Φ-lab and the University of Leiden, and carried out as a doctoral research project by Alice Barthe between June 2022 and August 2025; the thesis defense is scheduled for the end of March. Unlike the first project, whose emphasis was empirical demonstration on hardware, this project was deliberately theoretical in orientation: its aim was to replace heuristic observations about quantum model behavior with provable statements.

Challenges addressed

• Efficiently map the state of a dynamic system to a Hilbert space in which the time evolution is a linear operator.
• Determine whether properties of the dynamical system can be derived from that representation.
• Identify for which dynamical systems, and for which tasks, there would be an advantage over classical models.

QRU trainability and expressivity

The first contribution studies the effect of data embedding on the trade-off between expressive power and trainability in variational models. The analysis shows that data re-uploading — repeatedly injecting the input into the circuit — can induce a Gaussian smoothing in frequency space, which suppresses high-frequency components and therefore limits the expressivity actually available, contrary to the intuition that more re-uploading always means more expressive power. The practical consequence is that the number and placement of re-uploading layers must be adjusted rather than increased blindly.

Alongside this, absorption witnesses were introduced as a principled way to assess and mitigate barren plateau risk in quantum re-uploading units, giving a diagnostic for trainability rather than an empirical rule of thumb.

Quantum generative modeling

The second contribution establishes universal architectures for expectation-value samplers: two architectures were identified, together with their resource requirements and their fundamental dimensional limitations. This clarifies how quantum generative models scale and where the hard limits on scaling lie — again replacing case-by-case observation with an architectural statement.

Provable quantum advantage

The third contribution delivers a provable quantum advantage via Fourier coefficient extraction and PAC-learning separations. The separations were established both in synthetic concept classes and in physically meaningful ones, notably Hamiltonian dynamics, which connects the learning-theoretic result back to the physical systems the project set out to model.

Continuous-variable quantum simulation

The fourth contribution extends the quantum advantage analysis to continuous-variable systems, using continuous-variable quantum computing ODE solvers and classifications of bosonic circuit complexity. This covers the simulation of both classical and quantum systems and reveals connections between continuous-variable dynamics and quantum computational complexity hierarchies.

Outcome and significance

The thesis unifies these insights across parameterized quantum circuit machine learning, quantum generative modeling, and continuous-variable/bosonic quantum systems. The overall outcome of the collaboration is a shift from heuristic observation to a theory-grounded characterization of algorithm behavior: statements about when variational and continuous-variable quantum models can be trained, how they scale, and where an advantage over classical models can be proved. These results provide the theoretical foundations on which future quantum machine learning and quantum simulation work for dynamical, time-dependent Earth observation data can be built.

Project Outcomes

• PhD thesis of Alice Barthe (University of Leiden), completed August 2025, defense at the end of March 2026.
• Analysis and numerical code supporting the QRU expressivity/trainability results (Barthe, Alice, and Adrián Pérez-Salinas. "Gradients and frequency profiles of quantum re-uploading models." Quantum8 (2024): 1523.)
• Constructions for universal expectation-value sampler architectures and the continuous-variable ODE solver analysis (Barthe, Alice, et al. "Parameterized quantum circuits as universal generative models for continuous multivariate distributions." npj Quantum Information 11.1 (2025): 121), (Barthe, Alice, et al. "Continuous variables quantum algorithm for solving ordinary differential equations." 2023 IEEE International Conference on Quantum Computing and Engineering (QCE). Vol. 2. IEEE, 2023.)

Outcome of the collaboration

The collaboration moved the treatment of dynamic systems and time-dependent data in quantum machine learning from heuristic observation to theory-grounded characterization. Its four results — the Gaussian-smoothing limit on expressivity from data re-uploading, universal architectures for expectation-value samplers, provable advantage via Fourier extraction and PAC-learning separations, and the extension of advantage analysis to continuous-variable and bosonic systems — together define when and why quantum models can be trained, scaled and expected to outperform classical counterparts.