Quantum computing for solving a drift-diffusion equation
Ellen Devereux · Warwick Research Archive Portal (University of Warwick) · 2025
Quantum computing promises a theoretical computational advantage for solving certain problems, thus it holds significant academic and commercial interest. This thesis begins by reviewing established quantum algorithms and their applications. Then I expand this set to demonstrate quantum computational advantage for solving the drift-diffusion equation, a computationally intensive problem with applications in finance, modelling, and manufacturing. First, I introduce a complexity analysis for four classical algorithms designed to solve the multi-dimensional drift-diffusion equation. I then present four quantum algorithms, leveraging a quantum linear system solver, a quantum Hamiltonian simulation, a quantum random walk, and the quantum Fourier transform. By comparing the complexities of these methods to their classical counterparts, I find that diagonalisation via the quantum Fourier transform offers a quantum computational advantage for solving linear partial differential equations at a fixed final time. I employ a multidimensional amplitude estimation process to extract the full probability distribution from the quantum computer. Finally, I compare the circuit depths for five gate sets to implement a quantum algorithm solving the drift-diffusion equation in two spatial dimensions. This uses the diagonalisation by the quantum Fourier transform algorithm. The gate sets are: an unconstrained gate set, 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. This analysis covers a set of illustrative scenarios using up to 22 qubits. I 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, I find that even minimal problem instances exceed the operational limits of current quantum hardware. We find that for a test scenario using 22 logical qubits, requires 1 × 106 logical operations, assuming full connectivity.