Linear Programming with Unitary-Equivariant Constraints

IF 2.2 1区 物理与天体物理 Q1 PHYSICS, MATHEMATICAL Communications in Mathematical Physics Pub Date : 2024-11-06 DOI:10.1007/s00220-024-05108-1
Dmitry Grinko, Maris Ozols
{"title":"Linear Programming with Unitary-Equivariant Constraints","authors":"Dmitry Grinko,&nbsp;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.

查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
带单变量约束的线性规划
单元等差性是一种自然对称性,在物理学和数学的许多场合都会出现。对于所有 \(U \in \textrm{U}(d)\) 而言,具有这种对称性的优化问题通常可以表述为一个 \(d^{p+q}\) 维矩阵变量的半有限元程序,该矩阵变量与 \(U^{otimes p} \otimes {\{bar{U}}^{\otimes q}\) 相交。即使 \(p+q\) 很小,但局部维度 d 很大,以天真方式求解此类问题也会非常昂贵。我们的研究表明,在额外的对称性假设下,这个问题可以简化为一个线性程序,求解的时间不会随 d 的增大而增大,我们还提供了一个通用框架,用于在不同类型的对称性下执行这种简化。我们方法的关键要素是通过有墙布劳尔代数图的线性组合对解空间进行紧凑参数化。这种参数化需要格尔芬-策林基础的幂等子,我们通过调整受奥孔科夫-韦希克方法启发的一般方法获得了格尔芬-策林基础的幂等子。为了说明我们框架的潜在应用,我们使用了量子信息中的几个例子:决定量子态的主特征值、量子多数票、非对称克隆和黑箱单元变换。我们还概述了将我们的方法扩展到一般单元变量半inite 程序的可能途径。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
Communications in Mathematical Physics
Communications in Mathematical Physics 物理-物理:数学物理
CiteScore
4.70
自引率
8.30%
发文量
226
审稿时长
3-6 weeks
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.
期刊最新文献
The Fermionic Massless Modular Hamiltonian Topological Phases of Unitary Dynamics: Classification in Clifford Category Well-Posedness for Ohkitani Model and Long-Time Existence for Surface Quasi-geostrophic Equations On r-Neutralized Entropy: Entropy Formula and Existence of Measures Attaining the Supremum Effective Behaviour of Critical-Contrast PDEs: Micro-Resonances, Frequency Conversion, and Time Dispersive Properties. II
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1