Quantum Algorithms Accelerate Solutions to Drift-Diffusion Equations

Three algorithms – a linear system solver, random walk, and Fourier transform – were developed to solve multi-dimensional drift-diffusion equations. Analysis reveals the Fourier transform method provides a computational advantage for fixed-time solutions of linear partial differential equations, enabling extraction of the full probability distribution.

The simulation of carrier transport, crucial in semiconductor device modelling and materials science, often relies on computationally intensive methods to solve the drift-diffusion equation – a partial differential equation describing the movement of charge carriers. Researchers are now investigating whether quantum computation can offer a performance benefit for these simulations. In a new study, Ellen Devereux and Animesh Datta, both from the Department of Physics at the University of Warwick, alongside colleagues, detail three quantum algorithms designed to solve multi-dimensional forms of this equation. Their work, entitled ‘Quantum algorithms for solving a drift-diffusion equation’, compares the computational complexity of these quantum approaches – utilising linear system solvers, random walks, and Fourier transforms – with established classical methods, identifying potential advantages for specific solution scenarios. They demonstrate that employing a Fourier transform-based diagonalisation process, coupled with multidimensional amplitude estimation, may offer a computational benefit when solving linear partial differential equations at a fixed final time.

Numerical Stability in Multi-Dimensional Drift-Diffusion Equations

Establishing robust numerical schemes for solving drift-diffusion equations in multiple dimensions presents a significant computational challenge. Recent research focuses on deriving lower bounds on the norm of an operator, L, which governs the stability of these schemes. These bounds are critical for ensuring accurate solutions and managing computational resources.

Drift-diffusion equations model the transport of particles under the influence of both drift and diffusion, appearing in diverse fields including semiconductor physics, materials science and fluid dynamics. Numerical solutions require discretisation in both space and time, introducing potential instabilities if not carefully managed. The operator L represents the discretised form of the drift-diffusion operator and its spectral properties – specifically, its lower bound – dictate the stability of the numerical method.

The research demonstrates that these lower bounds scale directly with the dimensionality (d) of the problem. Higher dimensional systems require more stringent conditions for stability. Furthermore, the time step (τ) employed in the discretisation significantly influences the bounds; smaller time steps generally improve stability but at the cost of increased computational demand. This trade-off is a central consideration in practical applications.

Three distinct algorithms were investigated for solving the equations. A direct linear system solver provides a baseline approach. A random walk method, leveraging established principles of stochastic processes, offers an alternative. Finally, a Fourier transform method, exploiting the spectral properties of the operator, emerges as particularly efficient for a fixed final simulation time.

The methodology connects the operator L to the probability of return in a discrete random walk. This connection allows the application of known results from random walk theory to derive the lower bounds on L. The approach extends from one dimension to d dimensions under the assumption of independent random walks in each direction.

To fully characterise the solution, a multidimensional amplitude estimation process was implemented. This technique extracts the complete probability distribution of the solution, providing a more detailed understanding of system behaviour than simply obtaining a single value at a specific point. This detailed analysis enhances the accuracy of the solution and provides valuable insights into the underlying physical processes.

The research provides a theoretical framework for assessing the stability of numerical schemes for multi-dimensional drift-diffusion equations, offering guidance on selecting appropriate time steps and algorithms to achieve accurate and efficient simulations.