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 dp+q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d^{p+q}$$\\end{document}-dimensional matrix variable that commutes with U⊗p⊗U¯⊗q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U^{\\otimes p} \\otimes {\\bar{U}}^{\\otimes q}$$\\end{document}, for all U∈U(d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U \\in \ extrm{U}(d)$$\\end{document}. Solving such problems naively can be prohibitively expensive even if p+q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p+q$$\\end{document} is small but the local dimension d is large. We show that, under additional symmetry assumptions, this problem reduces to a linear program that can be solved in time that does not scale in d, and we provide a general framework to execute this reduction under different types of symmetries. The key ingredient of our method is a compact parametrization of the solution space by linear combinations of walled Brauer algebra diagrams. This parametrization requires the idempotents of a Gelfand–Tsetlin basis, which we obtain by adapting a general method inspired by the Okounkov–Vershik approach. To illustrate potential applications of our framework, we use several examples from quantum information: deciding the principal eigenvalue of a quantum state, quantum majority vote, asymmetric cloning and transformation of a black-box unitary. We also outline a possible route for extending our method to general unitary-equivariant semidefinite programs.