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The currently available quantum computers fall into the NISQ (Noisy Intermediate Scale Quantum) regime. These enable variational algorithms with a relatively small number of free parameters. We are now entering the FTQC (Fault Tolerant Quantum Computer)  regime where gate fidelities are high enough that error-correction schemes are effective. The UK Quantum Missions include the target for a FTQC device that can perform a million operations by 2028, and a trillion operations by 2035.

This talk will present the outcomes from assessments of  two quantum linear equation solvers for FTQCs– the Harrow–Hassidim–Lloyd (HHL) and the Quantum Singular Value Transform (QSVT) algorithms. These have used sample matrices from a Computational Fluid Dynamics (CFD) testcase. The quantum solvers have also been embedded with an outer non-linear solver to judge their impact on convergence. The analysis uses circuit emulation and is used to judge the FTQC requirements to deliver quantum utility.

Our visual perception of the world heavily relies on sophisticated and delicate biological mechanisms, and any disruption to these mechanisms negatively impacts our lives. Age-related macular degeneration (AMD) affects the central field of vision and has become increasingly common in our society, thereby generating a surge of academic and clinical interest. I will present some recent developments in the mathematical modeling of the retinal pigment epithelium (RPE) in the retina in the context of AMD; the RPE cell layer supports photoreceptor survival by providing nutrients and participating in the visual cycle and “cellular maintenance". Our objectives include modeling the aging and degeneration of the RPE with a mechanistic approach, as well as predicting the progression of atrophic lesions in the epithelial tissue. This is a joint work with the research team of Prof. M. Paques at Hôpital National des Quinze-Vingts.

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We are going to study the Ginzburg-Landau energy for two specific geometries, related to the very experiments on fermionic condensates: annuli and strips 

The specific geometry of a strip provides connections between solitons and vortices, called solitonic vortices, which are vortices with a solitonic behaviour in the infinite direction of the strip. Therefore, they are very different from classical vortices which have an algebraic decay at infinity. We show that there exist stationary solutions to the Gross-Pitaevskii equation with k vortices on a transverse line, which bifurcate from the soliton solution as the width of the strip is increased. This is motivated by recent experiments on the instability of solitons by imposing a phase shift in an elongated condensate for bosonic or fermionic atoms.

For annuli, we prescribe a very large degree on the outer boundary and find that either there is a transition from a giant vortex to vortices also in the bulk but tending to the outer boundary.

This is joint work with Ph. Gravejat and E.Sandier for solitonice vortices and Remy Rodiac for annuli.
 

In stochastic optimal control and conditional generative modelling, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward. Most existing methods require this reward to be differentiable, using gradients to steer the diffusion towards favourable outcomes. However, in many practical settings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero otherwise. We introduce a novel framework, based on Malliavin calculus and path-space integration by parts, that enables the development of methods robust to such singular rewards. This allows our approach to handle a broad range of applications, including classification, diffusion bridges, and conditioning without the need for artificial observational noise. We demonstrate that our approach offers stable and reliable training, outperforming existing techniques. 

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We propose and analyse two structure preserving finite volume schemes to approximate the solutions to a cross-diffusion system with self-consistent electric interactions introduced by Burger, Schlake & Wolfram (2012). This system has been derived thanks to probabilistic arguments and admits a thermodynamically motivated Lyapunov functional that is preserved by suitable two-point flux finite volume approximations. This allows to carry out the mathematical analysis of two schemes to be compared.

This is joint work with Maxime Herda and Annamaria Massimini.

We investigate pattern formation for a 2D PDE-ODE bulk-cell model, where one or more bulk diffusing species are coupled to nonlinear intracellular
reactions that are confined within a disjoint collection of small compartments. The bulk species are coupled to the spatially segregated
intracellular reactions through Robin conditions across the cell boundaries. For this compartmental-reaction diffusion system, we show that
symmetry-breaking bifurcations leading to stable asymmetric steady-state patterns, as regulated by a membrane binding rate ratio, occur even when
two bulk species have equal bulk diffusivities. This result is in distinct contrast to the usual, and often biologically unrealistic, large
differential diffusivity ratio requirement for Turing pattern formation from a spatially uniform state. Secondly, for the case of one-bulk
diffusing species in R^2, we derive a new memory-dependent ODE integro-differential system that characterizes how intracellular
oscillations in the collection of cells are coupled through the PDE bulk-diffusion field. By using a fast numerical approach relying on the
``sum-of-exponentials'' method to derive a time-marching scheme for this nonlocal system, diffusion induced synchrony is examined for various
spatial arrangements of cells using the Kuramoto order parameter. This theoretical modeling framework, relevant when spatially localized nonlinear
oscillators are coupled through a PDE diffusion field, is distinct from the traditional Kuramoto paradigm for studying oscillator synchronization on
networks or graphs. (Joint work with Merlin Pelz, UBC and UMinnesota).

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