Intern 2027: Quantum Algorithms for Differential Equations

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 for a 3 month intern in 2027 with a strong background in Quantum Information and Quantum Algorithms to design and analyze algorithms for solving classical differential equations on quantum computers. The role focuses on:

• Embedding nonlinear dynamics into linear—potentially infinite-dimensional—systems (e.g., Koopman/Carleman lifts, semigroup formalisms), and
• Encoding linear dynamics as efficient quantum circuits for time evolution (e.g., block-encodings, qubitization, LCU, QSP/SVT). You will translate problem formulations (ODEs/PDEs, operator semigroups, discretizations) into query-model statements, define oracle and state-preparation assumptions, and design fault-tolerant-compatible circuits with end-to-end resource estimates. You’ll benchmark quantum pipelines against classical baselines under matched accuracy and data-loading assumptions. Key responsibilities include:
• Hamiltonian & dissipative simulation: Develop and refine end-to-end time-evolution algorithms for s‑sparse Hamiltonians and linear dissipative dynamics (e.g., dissipative perturbations of Hamiltonian flows).
• Block-encodings & signal processing: Construct block-encodings of discretized operators; employ LCU, QSP/SVT, qubitization, and oblivious amplitude amplification for evolution, solvers, and transforms.
• Oracles & data access models: Formalize oracle definitions for operator sparsity/structure, boundary conditions, right-hand sides/initial data, and state preparation (incl. amplitude encoding and alternatives), articulating QRAM/streaming or other data assumptions.
• Complexity & lower bounds: Analyze query and gate complexity, robustness vs. conditioning and precision, and—where appropriate—connect to Hamiltonian complexity and reductions that inform feasibility or limitations.
• Fault-tolerant resource estimation: Produce qubit/T-count/depth estimates under plausible error-correction assumptions (e.g., LDPC code parameters, target logical error rates), identify bottlenecks, and pursue complexity and constant-factor improvements.
• Benchmarking & validation: Establish fair quantum-classical comparisons (accuracy targets, discretization choices, conditioning/preconditioning) and run prototype benchmarks in Python/Qiskit.
• Community engagement: Collaborate across quantum information, numerical algorithms, and dynamical systems communities; contribute to working groups, open-source artifacts, and publications. Why this role is compelling for Quantum Information scientists
• Direct impact of QI primitives: Your expertise in QSP/LCU/qubitization and block-encodings directly governs end-to-end performance for differential-equation workloads.
• Clear resource narratives: Tie query complexity to fault-tolerant resources (qubits, T-count, logical cycles), informing pathways from theory to practical advantage.
• Meaningful separations: Opportunities to articulate assumption-sensitive quantum-classical separations and to sharpen lower/upper bounds for PDE/ODE classes.
• Enrolled in a PhD during the full duration of the internship in Quantum Information/Quantum Computing, Quantum Algorithms, Theoretical Computer Science, or a closely related field.
• Demonstrated experience in at least two of:
• Hamiltonian simulation (s‑sparse or structured cases), LCU, QSP/SVT, qubitization, block-encodings
• Amplitude amplification/estimation, phase estimation, or quantum linear systems methods
• Query/gate complexity analysis and oracle modeling (data access assumptions, state prep)
• Proficiency prototyping in Python; experience with Qiskit or willingness to ramp quickly.
• Publications in quantum algorithms/quantum information or related top-tier venues.
• Experience formulating oracles for discretized operators and state-preparation pipelines (e.g., for PDE RHS/IC/BC).
• Familiarity with fault-tolerant costing (T-count/T-depth, surface-code logical qubits, target logical error budgets) and noise-aware considerations for earlier prototypes.
• Comfort engaging with numerical analysis concepts (conditioning, preconditioning, discretization error) sufficient to set fair benchmarks vs. classical methods.
• Hands-on with Git/GitHub and collaborative research workflows.
• Experience running quantum demonstrations or building precise classical baselines for comparison.
• Experience contributing to proposals and grants.