2024-11-06
Linear programming with unitary-equivariant constraints
Publication
Publication
Communications in Mathematical Physics , Volume 405 p. 278:1- 278:72
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 -dimensional matrix variable that commutes with , for all . Solving such problems naively can be prohibitively expensive even if 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.
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doi.org/10.1007/s00220-024-05108-1 | |
Communications in Mathematical Physics | |
Grinko, D., & Ozols, M. (2024). Linear programming with unitary-equivariant constraints. Communications in Mathematical Physics, 405, 278:1–278:72. doi:10.1007/s00220-024-05108-1 |