Linear Programming with Unitary-Equivariant Constraints Linear Programming with Unitary-Equivariant Constraints

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 ⊗ p ⊗ U ¯ ⊗ q , for all U ∈ U ( d ) . Solving...

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Veröffentlicht in:Communications in mathematical physics Jg. 405; H. 12; S. 278
Hauptverfasser: Grinko, Dmitry, Ozols, Maris
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2024
Springer Nature B.V
Schlagworte:
Classical and Quantum Gravitation
Cloning
Complex Systems
Conservation laws
Eigenvalues
Error correction & detection
Linear programming
Machine learning
Mathematical and Computational Physics
Mathematical Physics
Neural networks
Optimization
Parameterization
Physics
Physics and Astronomy
Quantum computing
Quantum phenomena
Quantum Physics
Relativity Theory
Solution space
Symmetry
Theoretical
ISSN:0010-3616, 1432-0916
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Zusammenfassung: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 ⊗ p ⊗ U ¯ ⊗ 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 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.
Bibliographie:ObjectType-Article-1
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content type line 14
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-024-05108-1