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- W2483163959 abstract "This paperdescribes a common mathematical framework forthedesign ofadditive andnon-additive Quantum ErrorCorrecting Codes. Itisbasedonacorrespondence between boolean functions andprojection operators. Thenewframework extends tooperator quantumerrorcorrecting codes. I.INTRODUCTION Theadditive orstabilizer construction ofQuantumError Correcting Codes(QECC)takes aclassical binary codethat isself-orthogonal withrespect toacertain symplectic inner product, andproduces aquantumcode, withminimumdis- tancedetermined bytheclassical code(Formoredetails see (6), (7)and(11)). In(16), Rainsetal.presented thefirst non-additive quantumerror-correcting code.Thiscodewas constructed numerically bybuilding aprojection operator with agiven weight distribution. Grassl andBeth(10) generalized this construction byintroducing unionquantum codes, where thecodes areformed bytaking thesumofsubspaces generated bytwoquantumcodes. Roychowdhury andVatan(18) gave somesufficient conditions fortheexistence ofnonadditive codes, andArvind etal.(4)developed atheory ofnon-additive codesbasedontheWeylcommutation relations. Followed bythis, Kribs etal.(13) introduced Operator Quantum Error Correction (OQEC)whichisaunifying approach that incor- porates thestandard errorcorrection model, themethodof decoherence-free subspaces, andthenoiseless subsystems as special cases. Wewill describe amathematical framework forcodedesign that encompasses bothadditive andnon-additive quantum error correcting codes. Itisbasedona correspondence between boolean functions andprojection operators inHilbert space thatisdescribed inSections II,IIIandV.Thistypeof correspondence wasfirst given in(2)wherethis relation was usedtoconstruct space-time codes. We willgivesufficient conditions fortheexistence ofQECCintermsoftheexistence oftheboolean function satisfying afewproperties inSection VII.Hence, we convert theproblem offinding a quantum codeintoaproblem offinding boolean function satisfying someproperties. Forsomeparameters ofQuantumcode, we havegiven examples oftheboolean functions satisfying these properties. We focus onnon-degenerate codeswhichis reasonable given that weknowofnoparameters k,M andd forwhichthere exists a((k, M,d))degenerate QECCbutnot" @default.
- W2483163959 created "2016-08-23" @default.
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- W2483163959 date "2007-01-01" @default.
- W2483163959 modified "2023-09-26" @default.
- W2483163959 title "Boolean Functions, Projection Operators and QuantumErrorCorrecting Codes" @default.
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