IWR Colloquium Summer Semester 2025 A “Super-Krylov” Approach to Diagonalization on a Quantum Computer

Quantum computing offers the promise of "exponential" increases in the speed of certain types of computations compared to classical computing machines.  However, such devices have limitations as to the type of operations that are admissible, and some operations carry with them levels of interference so great as to make the result unusable.  Thus, well-established (e.g., numerical linear algebraic) algorithms must be ported over and adapted to this setting, representing interesting, highly non-trivial new numerical analysis challenges.

In this talk, we discuss our first steps in exploring the adaption of Krylov subspace approaches to the quantum computing setting. The problem we consider is the estimation of the ground state eigenvalue of a Hamiltonian matrix (i.e., the left-most eigenvalue of a Hermitian matrix). This setting presents a number of challenges, including that we can only work with specifically prepared unitary matrices.  We overcome these challenges by constructing a specific "super Krylov" subspace using a unitary matrix and using a hybrid quantum-classical approach to take advantage of the strengths of both computational settings.  Theoretical analysis and practical numerical results demonstrate the utility of this approach.

Collaborators: Joint work with Adam Byrne (Trinity College Dublin/IBM Research) and Will Kirby, Sergiy Zhuk (IBM Research)