Quantum Codes

• Introduction
• Constructing Quantum Codes
  • Construction of General Quantum Codes
    • QuantumCode(S): Code → CodeQuantum
    • Example: Simple Quant Constr
    • Example: Simple Quant Extended Constr
    • Example: Simple Quant Self Dual Constr
    • Example: Simple Quant Additive Constr
    • QuantumCode(M): ModMatRngElt → CodeQuantum
    • Example: Simple Quant Constr Mat
    • QuantumCode(G): Grph → CodeQuantum
    • Example: Hexacode Quant
    • Example: Dodecacode Quant
    • RandomQuantumCode(F, n, k): FldFin, RngIntElt, RngIntElt → CodeQuantum
    • Example: Random Quant Constr
    • Subcode(Q, k): CodeQuantum, RngIntElt → CodeQuantum
  • Construction of Special Quantum Codes
    • Hexacode() → CodeQuantum
    • Dodecacode() → CodeQuantum
  • CSS Codes
    • CSSCode(C1, C2): Code, Code → CodeQuantum
    • CalderbankShorSteaneCode(C1, C2): Code, Code → CodeQuantum
    • Example: CSS Quant Constr
  • Cyclic Quantum Codes
    • QuantumCyclicCode(v): ModTupFldElt → CodeAdd
    • QuantumCyclicCode(Q): [ModTupFldElt] → CodeAdd
    • Example: Cyclic Quant Codesimple
    • QuantumCyclicCode(n, f): RngIntElt, RngUPolElt → CodeAdd
    • QuantumCyclicCode(n, Q): RngIntElt, [RngUPolElt] → CodeAdd
    • Example: Cyclic Quant Code Poly
    • QuantumCyclicCode(v4, v2): ModTupFldElt, ModTupFldElt → CodeAdd
    • Example: Cyclic Quant Code GF4GF2
  • Quasi-Cyclic Quantum Codes
    • QuantumQuasiCyclicCode(n, Q): RngIntElt, SeqEnum[RngUPolElt] → CodeAdd
    • QuantumQuasiCyclicCode(Q): SeqEnum[ModTupFldElt] → CodeAdd
    • Example: Quasi Cyclic Quant Code
• Access Functions
  • QuantumBasisElement(F): FldFin → FldFinElt
  • Example: Quantum Basis Element
  • StabilizerCode(Q): CodeQuantum → CodeAdd
  • StabiliserCode(Q): CodeQuantum → CodeAdd
  • StabilizerMatrix(Q): CodeQuantum → ModMatFldElt
  • StabiliserMatrix(Q): CodeQuantum → ModMatFldElt
  • NormalizerCode(Q): CodeQuantum → CodeAdd
  • NormaliserCode(Q): CodeQuantum → CodeAdd
  • NormalizerMatrix(Q): CodeQuantum → ModMatFldElt
  • NormaliserMatrix(Q): CodeQuantum → ModMatFldElt
  • Quantum Error Group
    • QuantumErrorGroup(p, n): RngIntElt, RngIntElt → GrpPC
    • QuantumBinaryErrorGroup(n): RngIntElt → GrpPC
    • Example: Quantum Error Group
    • QuantumErrorGroup(Q): CodeQuantum → GrpPC
    • StabilizerGroup(Q): CodeQuantum → GrpPC
    • StabiliserGroup(Q): CodeQuantum → GrpPC
    • StabilizerGroup(Q, G): CodeQuantum, GrpPC → GrpPC
    • StabiliserGroup(Q, G): CodeQuantum, GrpPC → GrpPC
    • Example: Quant Stab Group
    • Example: Quant Stab Group Hack
• Inner Products and Duals
  • SymplecticInnerProduct(v1, v2): ModTupFldElt, ModTupFldElt → FldFinElt
  • Example: Symplectic Products Compared
  • Example: Symplectic Product Formats
  • SymplecticDual(C): CodeAdd → CodeAdd
  • IsSymplecticSelfDual(C): CodeAdd → BoolElt
  • IsSymplecticSelfOrthogonal(C): CodeAdd → BoolElt
  • Example: Symplectic Eg
  • Example: symplecticselforthog
• Weight Distribution and Minimum Weight
  • WeightDistribution(Q): CodeQuantum → SeqEnum, SeqEnum, SeqEnum
  • Example: Quant Weight Dist
  • MinimumWeight(Q): CodeQuantum → RngIntElt
  • Example: Quant Min Weight
  • IsPure(Q): CodeQuantum → BoolElt
  • Example: QECC IsPure
• 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
  • C ^ tau: CodeQuantum, GrpPermElt → CodeQuantum
  • Example: Non Quant Combs
• Best Known Quantum Codes
  • QECC(F, n, k): FldFin, RngIntElt, RngIntElt → CodeQuantum, BoolElt
  • BKQC(F, n, k): FldFin, RngIntElt, RngIntElt → CodeQuantum, BoolElt
  • BestKnownQuantumCode(F, n, k): FldFin, RngIntElt, RngIntElt → CodeQuantum, BoolElt
  • Example: QECC Egs
  • Example: QECC Self Dual
  • Example: QECC Verbose
• Best Known Bounds
  • QECCLowerBound(F, n, k): FldFin, RngIntElt, RngIntElt → RngIntElt
  • QECCUpperBound(F, n, k): FldFin, RngIntElt, RngIntElt → RngIntElt
  • Example: QECC Bounds
• Automorphism Group
  • AutomorphismGroup(Q): CodeQuantum → GrpPerm
  • PermutationGroup(Q): CodeQuantum → GrpPerm
  • Example: Quantum Auto
  • Example: QECC Auto Stabilizer
• Hilbert Spaces
  • HilbertSpace(F, n): FldCom, RngIntElt → HilbSpc
  • Field(H): HilbSpc → FldCom
  • NumberOfQubits(H): HilbSpc → RngIntElt
  • Nqubits(H): HilbSpc → RngIntElt
  • Dimension(H): HilbSpc → RngIntElt
  • IsDenselyRepresented(H): HilbSpc → RngIntElt
  • H1 eq H2: HilbSpc, HilbSpc → BoolElt
  • H1 ne H2: HilbSpc, HilbSpc → BoolElt
  • Example: Hilbert Space Create
  • Creation of Quantum States
    • QuantumState(H, v): HilbSpc, ModTupFldElt → HilbSpcElt
    • QuantumState(H, v): HilbSpc, MtrxSprs → HilbSpcElt
    • H ! i: HilbSpc, RngIntElt → HilbSpcElt
    • H ! s: HilbSpc, [RngIntElt] → HilbSpcElt
    • SetPrintKetsInteger(b): BoolElt
    • Example: Quantum State Create
    • Example: Quantum State Create Coerce
  • Manipulation of Quantum States
    • a * e: FldComElt, HilbSpcElt → HilbSpcElt
    • - e: HilbSpcElt → HilbSpcElt
    • e1 + e2: HilbSpcElt, HilbSpcElt → HilbSpcElt
    • e1 - e2: HilbSpcElt, HilbSpcElt → HilbSpcElt
    • Normalisation(e): HilbSpcElt → HilbSpcElt
    • Normalisation(~e): HilbSpcElt
    • Normalization(e): HilbSpcElt → HilbSpcElt
    • Normalization(~e): HilbSpcElt
    • NormalisationCoefficient(e): HilbSpc → FldComElt
    • NormalizationCoefficient(e): HilbSpc → FldComElt
    • e1 eq e2: HilbSpcElt, HilbSpcElt → BoolElt
    • e1 ne e2: HilbSpcElt, HilbSpcElt → BoolElt
    • Example: Quantum State Normalisation
  • 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: Quantum State Probabilities
    • Example: Quantum State Sorted Probabilities
  • Unitary Transformations on Quantum States
    • BitFlip(e, k): HilbSpcElt, RngIntElt → HilbSpcElt
    • BitFlip(~e, k): HilbSpcElt, RngIntElt
    • BitFlip(e, B): HilbSpcElt, {RngIntElt} → HilbSpcElt
    • BitFlip(~e, B): HilbSpcElt, {RngIntElt}
    • PhaseFlip(e, k): HilbSpcElt, RngIntElt → HilbSpcElt
    • PhaseFlip(~e, k): HilbSpcElt, RngIntElt
    • PhaseFlip(e, B): HilbSpcElt, {RngIntElt} → HilbSpcElt
    • PhaseFlip(~e, B): HilbSpcElt, {RngIntElt}
    • ControlledNot(e, B, k): HilbSpcElt, {RngIntElt}, RngIntElt → HilbSpcElt
    • ControlledNot(~e, B, k): HilbSpcElt, {RngIntElt}, RngIntElt
    • HadamardTrasformation(e): HilbSpcElt → HilbSpcElt
    • HadamardTrasformation(~e): HilbSpcElt
    • Example: Quantum State Unitary