Can Current Quantum Algorithms Speed Up Linear Vlasov Equation Simulations?

Plasma physics is notoriously difficult to simulate. It is natural to seek alternative computational platforms that may offer speedups of such simulations. Quantum computers are an attractive option, as they have the potential to solve certain problems exponentially or polynomially faster than classical computers (Grover 1996, Shor 1999). Previous work in this area has focused on the collisionless, linearized Vlasov equation, and has claimed an exponential speedup with respect to system size (Engel, Smith, and Parker 2019). This is done by truncating the velocity space and recasting the linearized Vlasov equation as a Schrodinger-type equation, for which Hamiltonian simulation algorithms can be used (Low and Chuang 2019). We show that by expanding in velocity space using Hermite polynomials, we can solve the same problem using an exponentially smaller system, thereby yielding a classical algorithm with the same performance as the proposed quantum algorithm. We also discuss that it is unlikely that a quantum version of our classical algorithm can yield an exponential speed up, but it is likely that we can obtain polynomial speedup in some parameter regimes. It is also straightforward to add collisions to our system without changing the performance of the algorithm.