Quantum algorithms for solving a drift-diffusion equation: analysing circuit depths

Abstract: We compare the circuit depths for five different gate sets to implement a quantum algorithm solving a drift-diffusion equation in two spatial dimensions. Our algorithm uses diagonalisation by the quantum Fourier transform. The gate sets are: An unconstrained gate set, the TK1 gate set from Quantinuum, the native gate sets of IBM Heron and IonQ, and Fujitsu's space-time efficient analog rotation (STAR) gate set. Our analysis covers a set of illustrative scenarios using up to 22 qubits. We find that while scaling with spatial resolution aligns with theoretical predictions in one dimension, scaling with spatial dimension is less efficient than theorised due to overhead from block encoding. Finally, using the STAR gate set, we find that even minimal problem instances exceed the operational limits of current quantum hardware.