Computational Mathematics and Applications Seminar

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.

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.