{"title":"Linear Programming with Unitary-Equivariant Constraints","authors":"Dmitry Grinko, Maris Ozols","doi":"10.1007/s00220-024-05108-1","DOIUrl":null,"url":null,"abstract":"<div><p>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 <span>\\(d^{p+q}\\)</span>-dimensional matrix variable that commutes with <span>\\(U^{\\otimes p} \\otimes {\\bar{U}}^{\\otimes q}\\)</span>, for all <span>\\(U \\in \\textrm{U}(d)\\)</span>. Solving such problems naively can be prohibitively expensive even if <span>\\(p+q\\)</span> is small but the local dimension <i>d</i> 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 <i>d</i>, 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.\n</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"405 12","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00220-024-05108-1.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s00220-024-05108-1","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 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 \(U \in \textrm{U}(d)\). 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.
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.