Quantum Dynamics Simulation of the Advection-Diffusion Equation

The advection-diffusion equation is simulated on a superconducting quantum computer via several quantum algorithms. Three formulations are considered: (1) Trotterization, (2) variational quantum time evolution (VarQTE), and (3) adaptive variational quantum dynamics simulation (AVQDS). These schemes were originally developed for the Hamiltonian simulation of many-body quantum systems. The finite-difference discretized operator of the transport equation is formulated as a Hamiltonian and solved without the need for ancillary qubits. Computations are conducted on a quantum simulator (IBM Qiskit Aer) and an actual quantum hardware (IBM Fez). The former emulates the latter without the noise. The predicted results are compared with direct numerical simulation (DNS) data with infidelities of the order $10^{-5}$. In the quantum simulator, Trotterization is observed to have the lowest infidelity and is suitable for fault-tolerant computation. The AVQDS algorithm requires the lowest gate count and the lowest circuit depth. The VarQTE algorithm is the next best in terms of gate counts, but the number of its optimization variables is directly proportional to the number of qubits. Due to current hardware limitations, Trotterization cannot be implemented, as it has an overwhelming large number of operations. Meanwhile, AVQDS and VarQTE can be executed, but suffer from large errors due to significant hardware noise. These algorithms present a new paradigm for computational transport phenomena on quantum computers.