Research internship: Quantum computing and AI for partial differential equations 2027

The Algorithms and Applications for PDEs team focusses on the paradigm of quantum centric supercomputing. Here, the idea is to mix and match the state of the art in all three areas -- quantum computing, AI and conventional numerics -- to yield breakthrough performance increases for forward and inverse problems for PDEs. Our global team actively works on quantum solvers, neural operators and data assimilation methods with a particular focus on workflows and algorithms that bridge computational paradigms. Partial differential equations appear throughout scientific and industrial problems. The study of forward and inverse problems has been a cornerstone of scientific computing from its earliest days. While this has led to a rich landscape of powerful numerical methods and tools, there are still many challenges -- in particular where scaling limits are encountered or underlying dynamics not fully known. The AI revolutions of the 2010s and 2020s started addressing these challenges. In some domains, data-driven solvers equal or surpass the performance of conventional approaches, frequently while running on an arguably smaller computational footprint. Quantum methods with their exponential state spaces and scaling advantages hold the tantalizing promise of much more rapid advances. Yet unlocking this promise requires addressing questions such as state preparation and read-out to name just two. Bridging these gaps is at the heart of our team's research. By combining expertise in scientific computing, machine learning, and quantum algorithms, we develop workflows that leverage the strengths of each paradigm and investigate their potential for real-world PDE applications. The successful candidate will actively contribute to the team's research agenda across the entire vertical of algorithm and model development, experimentation and validation. Depending on educational background, existing skills and strengths as well as research opportunities, this can mean implementing quantum solvers, training AI models, running validation across computational paradigms or developing cross-platform workflows. Beyond the work on a primary project, the candidate will contribute to and be part of the general discourse within the group and lab. This means participating in journal clubs and preparing reports; yet also interacting with interns and researchers from other groups across IBM Research in general and the Yorktown lab in particular. All of: Working understanding of partial differential equations -- from a theoretical or application-oriented perspective. Elementary statistics. Proven experience prototyping in python (numpy). At least one of: Research experience in applied mathematics. In particular regarding differential equations and operator-theoretic methods or numerical linear algebra for large-scale systems. Research experience in AI models for PDEs. Research experience in quantum computing. One or more of the following Experience with data assimilation methods such as variational data assimilation (3D-Var, 4D-Var), Kalman filtering or AI data assimilation algorithms (score-based data assimilation). Experience with neural operators and/or AI models for inverse problems. Demonstrated experience of Hamiltonian simulation, LCU, QSP/SVT, qubitization, block-encodings; amplitude amplification/estimation, phase estimation, or quantum linear systems methods. Experience with discretization methods (FEM/FDM/spectral), PDE solvers (CG/GMRES/multigrid), and conditioning/preconditioning analysis. Background in model reduction (POD/DMD, balanced truncation) or Koopman/Carleman embeddings. Strong software engineerins skills. Publication record.