Intern 2027: Quantum Algorithms for Differential Equations (Math)

At IBM Research, we are the innovation engine of IBM. Exploring what’s next in computing and shaping the technologies the world will rely on tomorrow. From advancing AI and hybrid cloud to pioneering practical quantum computing, we anticipate challenges and unlock new opportunities for clients, partners, and society. Working in Research means joining a team that accelerates discovery at the intersection of high-performance computing, AI, quantum, and cloud. You’ll collaborate with leading scientists, engineers, and visionaries to push boundaries and turn ideas into reality. With a culture built on curiosity, creativity, and collaboration, IBM Research offers the opportunity to grow your career while contributing to breakthroughs that transform industries and change the world. IBM Research is seeking a 3 month intern for 2027 with a strong Applied Mathematics background to design and analyze quantum algorithms for solving classical differential equations. The role emphasizes:

• Embedding nonlinear dynamics into linear—potentially infinite-dimensional—systems (e.g., via Koopman/Carleman/Kolmogorov lifts, semigroup approaches), and
• Encoding linear dynamics as efficient quantum circuits that implement time evolution (e.g., block encodings, LCU, QSP). You will translate mathematical formulations (PDEs/ODEs, linear operators, discretizations, stability/error bounds) into algorithmic primitives suitable for near- and long-term quantum architectures, and benchmark against best-in-class classical methods. Key responsibilities include:
• Time-evolution & simulation: Develop and improve end-to-end quantum time-evolution algorithms for s‑sparse Hamiltoniansand linear dissipative dynamics (e.g., dissipative perturbations of Hamiltonian systems).
• Operator embeddings: Design mathematically principled embeddings of differential operators (elliptic/parabolic/hyperbolic; SPDEs where applicable) into forms amenable to block-encoding, LCU, and QSP.
• Numerics-aware design: Connect discretization choices (finite difference/finite element/spectral) to quantum complexity—analyzing conditioning, stability, and error propagation, and devising preconditioning strategies.
• Resource estimation: Perform rigorous resource estimation (query complexity, qubits, T-depth/T-count, logical error budgets), identify bottlenecks, and pursue complexity improvements and potential quantum-classical separations.
• Benchmarking: Compare quantum pipelines to classical baselines (e.g., CG/GMRES/Krylov, multigrid, PDE-constrained optimization), using fair accuracy/complexity criteria.
• Community engagement: Collaborate across quantum and applied math communities (dynamical systems, numerical linear algebra, control), contribute to working groups, and co-author publications. Why this is exciting for Applied Mathematicians
• Direct line from theory to implementation: Operator theory, semigroups, and discretization analysis carry over cleanly to block encodings and QSP/LCU-based evolutions.
• Numerical linear algebra at the core: Conditioning, sparsity, preconditioning, and model reduction (e.g., POD/DMD, balanced truncation) can materially change quantum resource counts.
• Impact across domains: Methods extend to PDEs arising in control, inverse problems, materials, and fluid/transport systems, with opportunities to show clear algorithmic separations. • Enrolled in a PhD during the full duration of the internship • Research experience in Applied Mathematics or closely related fields (e.g., numerical analysis, dynamical systems, control, optimization, probability/information theory). • Demonstrated depth in differential equations and operator-theoretic methods (e.g., semigroups, stability, dissipativity, functional analysis) or numerical linear algebra for large-scale systems (sparse operators, Krylov methods, preconditioning). • Experience prototyping in Python (NumPy/SciPy/Matplotlib) and interest in applying ideas within Qiskit or similar frameworks. If you already have experience with quantum algorithms for differential equations (e.g., QSP, LCU, sparse Hamiltonian simulation, block encodings), that’s ideal. Otherwise, a strong applied math foundation and a clear motivation to pivot into quantum algorithms is highly valued. • Publications in applied math, numerical analysis, or quantum algorithms (journals or top-tier conferences). • 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. • Familiarity with Qiskit or quantum simulation stacks (not mandatory if you have strong math fundamentals and are eager to learn). • Experience with Git/GitHub and collaborative research. • Experience running quantum demonstrations/experiments or numerical benchmarks on large-scale classical systems. • Experience writing proposals and participating in grants.