Quantum Dynamics Simulation of the Advection-Diffusion Equation
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In this work, the potential of quantum computing for solving linear partial differential equations is investigated. For this purpose, the advection-diffusion equation is simulated via several quantum algorithms. Three formulations are considered: (1) Trotterization, (2) variational quantum time evolution (VarQTE), and (3) adaptive variational quantum dynamics simulation (AVQDS). The finite-difference discretized operator of the transport equation is formulated as a Hamiltonian and solved without the need for ancillary qubits. The algorithms are first run on noiseless quantum simulators, and their results are compared with data from direct numerical simulation (DNS), having infidelities of the order $10^{-5}$. Trotterization is observed to have the lowest infidelity and is suitable for fault-tolerant computation. The AVQDS algorithm requires the lowest gate count and 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 quantum hardware limitations, Trotterization cannot be implemented in actual devices, as it has an overwhelming large number of operations. Meanwhile, AVQDS and VarQTE are executed at the hardware level. These algorithms present a new paradigm for computational transport phenomena on quantum computers.
