# Linear programming with unitary-equivariant constraints

arXiv: https://arxiv.org/abs/2207.05713 · HTML: https://mlsanity.com/paper/2207.05713/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jul 2022.

## Built on by (most cited first, 3 of 3)

- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [Representation Theory for Geometric Quantum Machine Learning](https://mlsanity.com/paper/2210.07980/) (2022)
- [Toward Super-polynomial Quantum Speedup of Equivariant Quantum Algorithms with SU(d) Symmetry](https://mlsanity.com/paper/2207.07250/) (2022)

## Abstract (arXiv)

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-dimensional matrix variable that commutes with Uotimes p otimes barUotimes q, for all U ∈ 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.