{"title":"Homological Invariants of Pauli Stabilizer Codes","authors":"Blazej Ruba, Bowen Yang","doi":"10.1007/s00220-024-04991-y","DOIUrl":null,"url":null,"abstract":"<p>We study translationally invariant Pauli stabilizer codes with qudits of arbitrary, not necessarily uniform, dimensions. Using homological methods, we define a series of invariants called charge modules. We describe their properties and physical meaning. The most complete results are obtained for codes whose charge modules have Krull dimension zero. This condition is interpreted as mobility of excitations. We show that it is always satisfied for translation invariant 2D codes with unique ground state in infinite volume, which was previously known only in the case of uniform, prime qudit dimension. For codes all of whose excitations are mobile we construct a <i>p</i>-dimensional excitation and a <span>\\((D-p-1)\\)</span>-form symmetry for every element of the <i>p</i>-th charge module. Moreover, we define a braiding pairing between charge modules in complementary degrees. We discuss examples which illustrate how charge modules and braiding can be computed in practice.\n</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-024-04991-y","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
We study translationally invariant Pauli stabilizer codes with qudits of arbitrary, not necessarily uniform, dimensions. Using homological methods, we define a series of invariants called charge modules. We describe their properties and physical meaning. The most complete results are obtained for codes whose charge modules have Krull dimension zero. This condition is interpreted as mobility of excitations. We show that it is always satisfied for translation invariant 2D codes with unique ground state in infinite volume, which was previously known only in the case of uniform, prime qudit dimension. For codes all of whose excitations are mobile we construct a p-dimensional excitation and a \((D-p-1)\)-form symmetry for every element of the p-th charge module. Moreover, we define a braiding pairing between charge modules in complementary degrees. We discuss examples which illustrate how charge modules and braiding can be computed in practice.
我们研究具有任意维数(不一定是统一维数)的平移不变保利稳定器编码。利用同调方法,我们定义了一系列称为电荷模块的不变式。我们描述了它们的性质和物理意义。对于电荷模块的克鲁尔维度为零的码,我们得到了最完整的结果。这一条件被解释为激发的流动性。我们证明,对于在无限体积中具有唯一基态的平移不变二维密码来说,它总是满足的。对于所有激元都是流动的二维码,我们为第 p 个电荷模块的每个元素构建了一个 p 维激元和((D-p-1)\)形式对称性。此外,我们还定义了电荷模块之间的互补度编织配对。我们将举例说明如何在实践中计算电荷模块和辫状配对。
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.