A Quantum Algorithm for the Linearized Vlasov Equation with Collisions*

Plasma physics is notoriously difficult to simulate. It is natural to seek alternative computational platforms that may speed up such simulations. Quantum computers are an attractive option, as they can solve certain problems exponentially or polynomially faster than classical computers (Grover 1996, Shor 1999). We develop a quantum algorithm for the linearized Vlasov equation with collisions (which offers a first-principles description of plasmas in the linear limit) and apply it to the canonical Landau damping problem. We show that by using a Hermite representation of velocity space and incorporating Hamiltonian simulation and quantum ODE solver algorithms (Low and Chuang 2019, Krovi 2022), we obtain a quadratic speedup in system size compared to the most efficient classical algorithms. However, previous work on this problem (without collisions) has demonstrated an exponential speedup in system size (Engel, Smith, and Parker 2019). We resolve this discrepancy by demonstrating that the Hermite representation yields an exponentially smaller system size. Thus, a classical algorithm implementing the Hermite representation has the same performance as the quantum algorithm of the previous work.