QLunch: Dmitry Grinko

Speaker: Dmitry Grinko, QuSoft, University of Amsterdam

Title: Linear programming with unitary equivariant constraints

Abstract: Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. Optimization problems with such symmetry can often be formulated as semidefinite programs for a d^{p+q}-dimensional matrix variable that commutes with U^{\otimes p} \otimes \bar{U}^{\otimes q}, for all U \in U(d). Solving such problems naively can be prohibitively expensive, especially when the local dimension d is large. We show that, under additional symmetry assumptions, this problem reduces to linear programming and we provide a general framework for solving such programs in time that does not scale in d. Our approach uses a compact parameterization of the solution space by diagrammatically expressed walled Brauer algebra idempotents. We expect our methods to extend to general unitary-equivariant semidefinite programs.