Quantum time evolution for solving the advection-diffusion equation.*

Time evolution algorithms are an effective and gate-efficient tool for implementing quantum simulations on contemporary NISQ hardware. Simulating Hamiltonians in quantum chemistry and condensed matter physics is one example of their efficiency. With Trotterization or variational (real or imaginary) time evolution algorithms, the dynamics of a quantum system can be properly implemented using quantum gates and physical qubits. Here, an approach is presented to utilize these tools for solving the linear convection-diffusion equation, which is a type of non-unitary evolution. The results show the variational solution obtained with these methods follows the accurate classical direct numerical simulation (DNS). The two-local ansatz has also been implemented on the IBM Torino quantum computer, showing that the circuit depth is suitable for present-day hardware. In general, variational time evolution algorithms can be a valid option for solving PDEs, expanding the applicability of quantum hardware to this class of problems.