Sponsor: ONERA
Context
We are interested in solving problems involving hyperbolic systems of partial differential equations (PDEs)
\(\partial_t{\bf u}+\nabla\cdot{\bf f}({\bf u}) = 0, \quad (1)\)
where \(\bf u\) is the vector of conserved variables and \({\bf f}({\bf u})\) are the fluxes. Solutions to (1) may develop discontinuities in finite time and the derivatives have to be understood in the sense of distributions, which leads to difficulties in capturing the physically relevant solution with conventional discretization methods.
Simulating such problems on a quantum computer (QC) is even more challenging, in particular in the case of nonlinear equations. A common approach to solve PDEs on a QC consists in encoding the solution in the amplitude of a quantum state, and mapping the equations to a problem for which efficient quantum algorithms exist. Quantities of interest are then computed through a final measurement step.
Objectives
We here propose to use the relaxation approximation from [S. Jin and Z. Xin, Commun. Pure Appl. Math, 48 (1995)] that circumvent difficulties in the treatment of the nonlinearities by approximating (1) with a linearized enlarged system with stiff relaxation source terms:
\(\partial_t{\bf u}+\nabla\cdot{\bf v} = 0, \quad (2a)\)
\(\partial_t{\bf v}+{\bf A}\cdot\nabla{\bf u} = -\tfrac{1}{\epsilon}\big({\bf v}-{\bf f}({\bf u})\big), \quad (2b)\)
where \(0<\epsilon\ll1\) represents the relaxation rate and \({\bf A}\) is a coefficient matrix that can be chosen so that (2) is a dissipative approximation consistent with (1) in the limit \(\epsilon=0^+\). This model introduces additional variables \({\bf v}\) that approximate the nonlinearities \({\bf f}({\bf u})\). The nonlinearities are taken into account in lower-order source terms, while the transport operator in the left-hand side is linear.
The objective of this project is to evaluate the potential of this approach for solving nonlinear equations with a QC. A quantum algorithm will be designed and implemented in a QC emulator, then assessed on problems involving several PDEs. The results will be analyzed in terms of complexity, accuracy, impact of noise, theoretical performance, etc.