Rhea Alexander, Si Gvirtz-Chen, Nikolaos Koukoulekidis, D. Jennings
{"title":"General Entropic Constraints on Calderbank-Shor-Steane Codes within Magic Distillation Protocols","authors":"Rhea Alexander, Si Gvirtz-Chen, Nikolaos Koukoulekidis, D. Jennings","doi":"10.1103/PRXQuantum.4.020359","DOIUrl":null,"url":null,"abstract":"Magic states are fundamental building blocks on the road to fault-tolerant quantum computing. CSS codes play a crucial role in the construction of magic distillation protocols. Previous work has cast quantum computing with magic states for odd dimension $d$ within a phase space setting in which universal quantum computing is described by the statistical mechanics of quasiprobability distributions. Here we extend this framework to the important $d=2$ qubit case and show that we can exploit common structures in CSS circuits to obtain distillation bounds capable of out-performing previous monotone bounds in regimes of practical interest. Moreover, in the case of CSS code projections, we arrive at a novel cut-off result on the code length $n$ of the CSS code in terms of parameters characterising a desired distillation, which implies that for fixed target error rate and acceptance probability, one needs only consider CSS codes below a threshold number of qubits. These entropic constraints are not due simply to the data-processing inequality but rely explicitly on the stochastic representation of such protocols.","PeriodicalId":74587,"journal":{"name":"PRX quantum : a Physical Review journal","volume":null,"pages":null},"PeriodicalIF":9.3000,"publicationDate":"2022-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"PRX quantum : a Physical Review journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1103/PRXQuantum.4.020359","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Magic states are fundamental building blocks on the road to fault-tolerant quantum computing. CSS codes play a crucial role in the construction of magic distillation protocols. Previous work has cast quantum computing with magic states for odd dimension $d$ within a phase space setting in which universal quantum computing is described by the statistical mechanics of quasiprobability distributions. Here we extend this framework to the important $d=2$ qubit case and show that we can exploit common structures in CSS circuits to obtain distillation bounds capable of out-performing previous monotone bounds in regimes of practical interest. Moreover, in the case of CSS code projections, we arrive at a novel cut-off result on the code length $n$ of the CSS code in terms of parameters characterising a desired distillation, which implies that for fixed target error rate and acceptance probability, one needs only consider CSS codes below a threshold number of qubits. These entropic constraints are not due simply to the data-processing inequality but rely explicitly on the stochastic representation of such protocols.