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QUANTUM CODES

 
Acknowledgements
 
Introduction
 
Constructing Quantum Codes
      Construction of General Quantum Codes
      Construction of Special Quantum Codes
      CSS Codes
      Cyclic Quantum Codes
      Quasi-Cyclic Quantum Codes
 
Access Functions
      Quantum Error Group
 
Inner Products and Duals
 
Weight Distribution and Minimum Weight
 
New Codes From Old
 
Best Known Quantum Codes
 
Best Known Bounds
 
Automorphism Group
 
Hilbert Spaces
      Creation of Quantum States
      Manipulation of Quantum States
      Inner Product and Probabilities of Quantum States
      Unitary Transformations on Quantum States
 
Bibliography







DETAILS

 
Introduction

 
Constructing Quantum Codes

      Construction of General Quantum Codes
            QuantumCode(S) : Code -> CodeQuantum
            Example QECC_SimpleQuantConstr (H157E1)
            Example QECC_SimpleQuantExtendedConstr (H157E2)
            Example QECC_SimpleQuantSelfDualConstr (H157E3)
            Example QECC_SimpleQuantAdditiveConstr (H157E4)
            QuantumCode(M) : ModMatRngElt -> CodeQuantum
            Example QECC_SimpleQuantConstrMat (H157E5)
            QuantumCode(G) : Grph -> QuantumCode
            Example QECC_HexacodeQuant (H157E6)
            Example QECC_DodecacodeQuant (H157E7)
            RandomQuantumCode(F, n, k) : FldFin, RngIntElt, RngIntElt -> CodeQuantum
            Example QECC_RandomQuantConstr (H157E8)
            Subcode(Q, k) : CodeQuantum, RngIntElt -> CodeQuantum

      Construction of Special Quantum Codes
            Hexacode() : -> CodeQuantum
            Dodecacode() : -> CodeQuantum

      CSS Codes
            CSSCode(C1, C2) : Code, Code -> CodeQuantum
            Example QECC_CSSQuantConstr (H157E9)

      Cyclic Quantum Codes
            QuantumCyclicCode(v) : ModTupFldElt -> CodeAdd
            Example QECC_CyclicQuantCodesimple (H157E10)
            QuantumCyclicCode(n, f) : RngIntElt, RngUPolElt -> CodeAdd
            Example QECC_CyclicQuantCodePoly (H157E11)
            QuantumCyclicCode(v4, v2) : ModTupFldElt, ModTupFldElt -> CodeAdd
            Example QECC_CyclicQuantCodeGF4GF2 (H157E12)

      Quasi-Cyclic Quantum Codes
            QuantumQuasiCyclicCode(n, Q) : RngIntElt, SeqEnum[RngUPolElt] -> CodeAdd
            QuantumQuasiCyclicCode(Q) : SeqEnum[ModTupFldElt] -> CodeAdd
            Example QECC_QuasiCyclicQuantCode (H157E13)

 
Access Functions
      QuantumBasisElement(F) : FldFin -> FldFinElt
      StabilizerCode(Q) : CodeQuantum -> CodeAdd
      StabilizerMatrix(Q) : CodeQuantum -> ModMatFldElt
      NormalizerCode(Q) : CodeQuantum -> CodeAdd
      NormalizerMatrix(Q) : CodeQuantum -> ModMatFldElt

      Quantum Error Group
            QuantumErrorGroup(p, n) : RngIntElt, RngIntElt -> GrpPC
            QuantumBinaryErrorGroup(n) : RngIntElt -> GrpPC
            Example QECC_quantum-error-group (H157E14)
            QuantumErrorGroup(Q) : CodeQuantum -> GrpPC
            StabilizerGroup(Q) : CodeQuantum -> GrpPC
            StabilizerGroup(Q, G) : CodeQuantum, GrpPC -> GrpPC
            Example QECC_quant-stab-group (H157E15)
            Example QECC_quant-stab-group-hack (H157E16)

 
Inner Products and Duals
      SymplecticInnerProduct(v1, v2) : ModTupFldElt, ModTupFldElt -> FldFinElt
      SymplecticDual(C) : CodeAdd -> CodeAdd
      IsSymplecticSelfDual(C) : CodeAdd -> BoolElt
      IsSymplecticSelfOrthogonal(C) : CodeAdd -> BoolElt
      Example QECC_SymplecticEg (H157E17)
      Example QECC_symplecticselforthog (H157E18)

 
Weight Distribution and Minimum Weight
      WeightDistribution(Q) : CodeQuantum -> SeqEnum, SeqEnum, SeqEnum
      Example QECC_QuantWeightDist (H157E19)
      MinimumWeight(Q) : CodeQuantum -> RngIntElt
      Example QECC_QuantMinWeight (H157E20)
      IsPure(Q) : CodeQuantum -> BoolElt
      Example QECC_QECC-IsPure (H157E21)

 
New Codes From Old
      DirectSum(Q1, Q2) : CodeQuantum, CodeQuantum -> CodeQuantum
      ExtendCode(Q) : CodeQuantum -> CodeQuantum
      ExtendCode(Q, m) : CodeQuantum, RngIntElt -> CodeQuantum
      PunctureCode(Q, i) : CodeQuantum, RngIntElt -> CodeQuantum
      PunctureCode(Q, I) : CodeQuantum, SetEnum -> CodeQuantum
      ShortenCode(Q, i) : CodeQuantum, RngIntElt -> CodeQuantum
      ShortenCode(Q, I) : CodeQuantum, SetEnum -> CodeQuantum
      Example QECC_NonQuantCombs (H157E22)

 
Best Known Quantum Codes
      QECC(F, n, k) : FldFin, RngIntElt, RngIntElt -> CodeQuantum, BoolElt
      Example QECC_QECC-egs (H157E23)
      Example QECC_QECC-self-dual (H157E24)
      Example QECC_QECC-verbose (H157E25)

 
Best Known Bounds
      QECCLowerBound(F, n, k) : FldFin, RngIntElt, RngIntElt -> RngIntElt
      QECCUpperBound(F, n, k) : FldFin, RngIntElt, RngIntElt -> RngIntElt
      Example QECC_QECCBounds (H157E26)

 
Automorphism Group
      AutomorphismGroup(Q) : CodeQuantum -> GrpPerm
      PermutationGroup(Q) : CodeQuantum -> GrpPerm
      Example QECC_QuantumAuto (H157E27)
      Example QECC_QECC-Auto-stabilizer (H157E28)

 
Hilbert Spaces
      HilbertSpace(F, n) : FldCom, RngIntElt -> HilbSpc
      Field(H) : HilbSpc -> FldCom
      NumberOfQubits(H) : HilbSpc -> RngIntElt
      Dimension(H) : HilbSpc -> RngIntElt
      IsDenselyRepresented(H) : HilbSpc -> RngIntElt
      H1 eq H2 : HilbSpc, HilbSpc -> BoolElt
      H1 ne H2 : HilbSpc, HilbSpc -> BoolElt
      Example QECC_HilbertSpaceCreate (H157E29)

      Creation of Quantum States
            QuantumState(H, v) : HilbSpc, ModTupFldElt -> HilbSpcElt
            H ! i : HilbSpc, RngIntElt -> HilbSpcElt
            H ! s : HilbSpc, [RngIntElt] -> HilbSpcElt
            SetPrintKetsInteger(b) : BoolElt ->
            Example QECC_QuantumStateCreate (H157E30)
            Example QECC_QuantumStateCreateCoerce (H157E31)

      Manipulation of Quantum States
            a * e : FldComElt , HilbSpcElt -> HilbSpcElt
            - e : HilbSpcElt -> HilbSpcElt
            e1 + e2 : HilbSpcElt, HilbSpcElt -> HilbSpcElt
            Normalisation(e) : HilbSpcElt -> HilbSpcElt
            NormalisationCoefficient(e) : HilbSpc -> FldComElt
            e1 eq e2 : HilbSpcElt, HilbSpcElt -> BoolElt
            e1 ne e2 : HilbSpcElt, HilbSpcElt -> BoolElt
            Example QECC_QuantumStateNormalisation (H157E32)

      Inner Product and Probabilities of Quantum States
            InnerProduct(e1, e2) : HilbSpcElt, HilbSpcElt -> HilbSpcElt
            ProbabilityDistribution(e) : HilbSpcElt -> Mtrx
            Probability(e, i) : HilbSpcElt, RngIntElt -> FldReElt
            Probability(e, v) : HilbSpcElt, RngIntElt -> FldReElt
            PrintProbabilityDistribution(e) : HilbSpcElt ->
            PrintSortedProbabilityDistribution(e) : HilbSpcElt ->
            Example QECC_QuantumStateProbabilities (H157E33)
            Example QECC_QuantumStateSortedProbabilities (H157E34)

      Unitary Transformations on Quantum States
            BitFlip(e, k) : HilbSpcElt,RngIntElt -> HilbSpcElt
            BitFlip(e, B) : HilbSpcElt, RngIntElt -> HilbSpcElt
            PhaseFlip(e, k) : HilbSpcElt,RngIntElt -> HilbSpcElt
            PhaseFlip(e, B) : HilbSpcElt, RngIntElt -> HilbSpcElt
            ControlledNot(e, B, k) : HilbSpcElt, RngIntElt, RngIntElt -> HilbSpcElt
            HadamardTrasformation(e) : HilbSpcElt -> HilbSpcElt
            Example QECC_QuantumStateUnitary (H157E35)

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013