Efficient Explicit Quantum Algorithms for Advection-Diffusion and the Koopman-von Neumann Approach to Nonlinear Dynamics

Quantum computers promise to solve many scientific computing problems with exponential advantage. However, because the basic operations are unitary, quantum computers lose their advantage for simulating iterative nonlinear processes. Thus, we developed the the Koopman-von Neumann approach for quantum simulation of nonlinear dynamics which is equivalent to simulating a Fokker-Planck equation for the probability distribution function. We also recently developed explicit near-optimal implementations of quantum algorithms for simulating important partial differential equations (PDEs) in mathematical physics including the advection, diffusion, Liouville, and generalized Koopman-von Neumann (KvN) equations. Our algorithm leverages a combination of state-of-the-art techniques including linear combination of Hamiltonian (LCH) simulations for simulating dissipation, quantum signal processing (QSP) for Hamiltonian simulation, quantum singular value transformation (QSVT) for initializing the wavefunction, and amplitude amplification for boosting the success probability and improving the convergence of the expectation value of observables. We will present an overview of our approach to solving these challenges as well as the key algorithmic ingredients that allowed us to design an efficient solution.