Linear programming with unitary-equivariant constraints

• URL: http://arxiv.org/abs/2207.05713v2
• Date: Tue, 4 Apr 2023 12:28:49 GMT
• Title: Linear programming with unitary-equivariant constraints
• Authors: Dmitry Grinko, Maris Ozols
• Abstract summary: Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. 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 $d$. We also outline a possible route for extending our method to general unitary-equivariant semidefinite programs.
• Score: 2.0305676256390934
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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 $U \in \mathrm{U}(d)$. Solving such problems naively can be prohibitively expensive even if $p+q$ 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 arXiv:1606.08900 inspired by the Okounkov-Vershik approach. To illustrate potential applications, 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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