Liouville Equation on Quantum Computers*

Solving the nonlinear PDEs useful for plasma physics is one of the current challenges to fault-tolerant linear quantum computing. For Hamiltonian systems, a general solution to this problem is to replace the original non-linear PDEs by their equivalent linear Liouville equations (Koopman-Von Neumann transform).

Here, a new method to solve the Liouville equation on quantum computers will be presented. The quantum algorithm evolves an initial wave-function in discrete time and space and the probability density of this wave-function obeys the Liouville equation. The quantum circuit associated to the algorithm requires mainly the Quantum Fourier Transform and diagonal unitary operators, and these are implemented efficiently with Walsh series

The resource requirement is also discussed. Finally, the method is demonstrated on two examples form classical mechanics: (i) the $1D$ harmonic oscillator (ii) the $2D$ chaotic Hénon-Heiles potential.