Quantum computing context
Current classical simulations of physical phenomena face many limitations, particularly when dealing with complex systems such as turbulent fluids, plasmas or strongly correlated quantum materials. These simulations often struggle with high computational costs, limited scalability across multiple dimensions and time scales, and difficulties in capturing inherently non-linear or chaotic behaviors. Even with access to powerful high-performance computing (HPC) resources, accurately resolving fine-scale dynamics or long-time evolutions remain a significant challenge.
The development of novel numerical methods is therefore essential for both academic research and industrial applications. Quantum computers emerge as novel numerical tools for studying such systems and solving the partial differential equations that model them. Thanks to the superposition and entanglement properties of qubits, digital computations can be performed in novel ways, offering promising theoretical speedups for certain problems.
In this context, CERFACS contributes to the development of quantum algorithm for scientific computations. CERFACS' quantum activities can be classified in four strategic axes :
- The development of efficient quantum solvers for differential equations on fault-tolerant quantum computers (FTQCs): ranging from quantum numerical analysis to establish and guarantee their efficiency, to the development of efficient quantum circuits implementing the underlying quantum routines.
- Testing on quantum hardware: Since current quantum hardware is noisy and continuously evolving, our approach is to evaluate quantum devices using simple pedagogical instances of the target problems.
- Application of the developed quantum solvers to industrial use cases: including aeronautics, combustion, weather forecasting, plasma physics, energy, and defense applications.
- Estimation of computational resources and assessment of quantum advantage: with the objective of determining to what extent quantum computing can provide practical computational benefits over classical approaches.
Quantum algorithms for differential equations
Solving differential equations (DE) on a digital FTQC computer is particularly challenging since quantum gates are unitary and linear whereas most DE have an associated non-unitary and non-linear evolution. In order to solve them quantumly, one therefore needs to map the non-linear-non-unitary problem to a unitary iterative scheme.
To do so many approaches have been developed such as linearization techniques, non-linear transformation and mappings into larger space where the dynamics is linear. Then, from a linear partial differential equation, one can either discretize space and time to obtain a linear system of equation that can be inverted with a Quantum Linear Solver Algorithm (QLSA), or either discretize only the space to get a system of ordinary differential equations (ODE) that can be mapped to a Schrodinger-type ODE. Then, many Hamiltonian simulation methods have been developed to solve Schrödinger-type ODEs that enable to solve it with a unitary iterative scheme. The different approaches are summarized in the following scheme.
For instance, solving an initial value problem requires three steps: an initialization that loads the initial condition into a qubit state, an evolution or an inversion algorithm performed by a quantum circuit that produce a qubit state close to the one encoding the solution, and a measurement protocol to extract relevant quantities of interest. The following scheme summarizes these three steps.
CERFACS' seminars on quantum computing:
24/06/2026 : Pierre Sagault, Quantum algorithms for fluid dynamics.
01/06/2026 : Martin Pujol, Quantum numerical scheme for spectrally truncated Euler flows.
05/05/2026 : Jérémie Messud, On the robustness of quantum phase estimation.
30/04/2026 : Abtin AMERI, Quantum lower bound for simulating fluid dynamics.
05/03/2026 : Matthieu Saubanère, Quantum and classical Krylov methods
