Workshop on Quantum Algorithms, Computational Models and Foundations of Quantum Mechanics

Fast Decoders for Topological Quantum Codes Duclos-Cianci, Guillaume 2010

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Fast Decoders for Topological Quantum CodesFast Decoders for Topological Quantum CodesGuillaume Duclos-Cianci1 David Poulin11D epartement de Physique, Universit e de Sherbrooke, Qc, CaJuly 25th, 2010Workshop on Quantum Algorithms, Computational Models, and Foundations ofQuantum MechanicsUniversity of British Columbia, Vancouver, CaFast Decoders for Topological Quantum CodesMotivationTopological CodesLogical subspace ! linked to the topology of the systemOperators highly non-local ! tailored to resist local noiseError correction requires local measurements and operationsKitaev’s toric code ! useful toy modelQuantum error-correction (QEC) ! fast decoding algorithmsFast Decoders for Topological Quantum CodesMotivationTopological CodesLogical subspace ! linked to the topology of the systemOperators highly non-local ! tailored to resist local noiseError correction requires local measurements and operationsKitaev’s toric code ! useful toy modelQuantum error-correction (QEC) ! fast decoding algorithmsFast Decoders for Topological Quantum CodesMotivationTopological CodesLogical subspace ! linked to the topology of the systemOperators highly non-local ! tailored to resist local noiseError correction requires local measurements and operationsKitaev’s toric code ! useful toy modelQuantum error-correction (QEC) ! fast decoding algorithmsFast Decoders for Topological Quantum CodesMotivationTopological CodesLogical subspace ! linked to the topology of the systemOperators highly non-local ! tailored to resist local noiseError correction requires local measurements and operationsKitaev’s toric code ! useful toy modelQuantum error-correction (QEC) ! fast decoding algorithmsFast Decoders for Topological Quantum CodesMotivationTopological CodesLogical subspace ! linked to the topology of the systemOperators highly non-local ! tailored to resist local noiseError correction requires local measurements and operationsKitaev’s toric code ! useful toy modelQuantum error-correction (QEC) ! fast decoding algorithmsFast Decoders for Topological Quantum CodesMotivation1 Kitaev’s Toric Code2 Concatenation3 Topological Codes DecodingFast Decoders for Topological Quantum CodesKitaev’s Toric Code1 Kitaev’s Toric CodeStabilizer generatorsLogical OperatorsTopology2 Concatenation3 Topological Codes DecodingFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsStabilizer GeneratorsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsLatticeMM2D square latticePeriodic boundary conditionsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsLatticeMM2D square latticePeriodic boundary conditionsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsLattice + QubitsMM2D square latticePeriodic boundary conditionsA qubit per edge)2‘2 qubitsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsLattice + QubitsMM2D square latticePeriodic boundary conditionsA qubit per edge)2‘2 qubitsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsLattice + QubitsMM2D square latticePeriodic boundary conditionsA qubit per edge)2‘2 qubitsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsLattice + QubitsMM2D square latticePeriodic boundary conditionsA qubit per edge)2‘2 qubitsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsStabilizer GeneratorsXXXZZZZX Site (vertex) operator :As = Qi2v(s)XiPlaquette operator :Bp = Qi2v(p)Zi‘2 site and plaquetteoperatorsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsStabilizer GeneratorsXXXZZZZX Site (vertex) operator :As = Qi2v(s)XiPlaquette operator :Bp = Qi2v(p)Zi‘2 site and plaquetteoperatorsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsStabilizer GeneratorsXXXZZZZX Site (vertex) operator :As = Qi2v(s)XiPlaquette operator :Bp = Qi2v(p)Zi‘2 site and plaquetteoperatorsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsStabilizer GeneratorsXXXZZZZZZZXXXZXXZZZZXXXX [As;As0] = [Bp;Bp0] = 0[As;Bp] = 0The code is spanned by thesimultaneous +1 eigenstates of alltheseC =fj i: Asj i=j i;Bpj i=j i(8s;p)gFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsStabilizer GeneratorsXXXZZZZZZZXXXZXXZZZZXXXX [As;As0] = [Bp;Bp0] = 0[As;Bp] = 0The code is spanned by thesimultaneous +1 eigenstates of alltheseC =fj i: Asj i=j i;Bpj i=j i(8s;p)gFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsStabilizer GeneratorsXXXZZZZZZZXXXZXXZZZZXXXX [As;As0] = [Bp;Bp0] = 0[As;Bp] = 0The code is spanned by thesimultaneous +1 eigenstates of alltheseC =fj i: Asj i=j i;Bpj i=j i(8s;p)gFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsStabilizer GeneratorsXXXZZZZXZZZZXXXXQsAs = IQpBp = I)2‘2 2 independent generators)2 logical qubitsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsStabilizer GeneratorsXXXZZZZXZZZZXXXXQsAs = IQpBp = I)2‘2 2 independent generators)2 logical qubitsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsStabilizer GeneratorsXXXZZZZXZZZZXXXXQsAs = IQpBp = I)2‘2 2 independent generators)2 logical qubitsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeStabilizer generatorsStabilizer GeneratorsXXXZZZZXZZZZXXXXQsAs = IQpBp = I)2‘2 2 independent generators)2 logical qubitsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsLogical OperatorsFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsFirst Logical QubitZ1 = Qi2 1 Zi[Z1;Bp] = 0[Z1;As] = 08s2S [Z1;s] = 0Z1γ1Z Z Z Z Z Z Z Z Z ZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsFirst Logical QubitZ1 = Qi2 1 Zi[Z1;Bp] = 0[Z1;As] = 08s2S [Z1;s] = 0Z1γ1ZZZZZZZZZ Z Z Z Z Z Z Z Z ZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsFirst Logical QubitZ1 = Qi2 1 Zi[Z1;Bp] = 0[Z1;As] = 08s2S [Z1;s] = 0Z1γ1XXXXZ ZZZZ Z Z Z Z ZXXXXFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsFirst Logical QubitZ1 = Qi2 1 Zi[Z1;Bp] = 0[Z1;As] = 08s2S [Z1;s] = 0Z1γ1Z Z Z Z Z Z Z Z Z ZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsFirst Logical QubitX1 = Qi2 1 Xi[X1;Bp] = 0[X1;As] = 08s2S [X1;s] = 0fX1;Z1g= 0X1γprime1XXXXXXXXXXFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsFirst Logical QubitX1 = Qi2 1 Xi[X1;Bp] = 0[X1;As] = 08s2S [X1;s] = 0fX1;Z1g= 0X1γprime1XXXXXXXXXXZZZZZZZZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsFirst Logical QubitX1 = Qi2 1 Xi[X1;Bp] = 0[X1;As] = 08s2S [X1;s] = 0fX1;Z1g= 0X1γprime1XXXXXXXXXXXXXXXXXXFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsFirst Logical QubitX1 = Qi2 1 Xi[X1;Bp] = 0[X1;As] = 08s2S [X1;s] = 0fX1;Z1g= 0X1γprime1XXXXXXXXXXFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsFirst Logical QubitX1 = Qi2 1 Xi[X1;Bp] = 0[X1;As] = 08s2S [X1;s] = 0fX1;Z1g= 0Z1X1Z ZZ Z Z Z Z Z Z ZXXXXXXXXXXFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsSecond Logical QubitBy re ecting aroundthe diagonalfX2;Z2g= 0[X2;Z1] = 0[X1;Z2] = 0[X1;X2] = 0[Z1;Z2] = 0Z2X2γprime2γ2ZZZZZZZZZZXXXXXXXXXXFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsSecond Logical QubitBy re ecting aroundthe diagonalfX2;Z2g= 0[X2;Z1] = 0[X1;Z2] = 0[X1;X2] = 0[Z1;Z2] = 0Z2X2γprime2γ2ZZZZZZZZZZXXXXXXXXXXFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsSecond Logical QubitBy re ecting aroundthe diagonalfX2;Z2g= 0[X2;Z1] = 0[X1;Z2] = 0[X1;X2] = 0[Z1;Z2] = 0Z1X2Z Z Z Z Z Z Z Z Z ZXXXXXXXXXXFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsSecond Logical QubitBy re ecting aroundthe diagonalfX2;Z2g= 0[X2;Z1] = 0[X1;Z2] = 0[X1;X2] = 0[Z1;Z2] = 0Z1X2Z Z Z Z Z Z Z Z Z ZXXXXXXXXXXFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsSecond Logical QubitBy re ecting aroundthe diagonalfX2;Z2g= 0[X2;Z1] = 0[X1;Z2] = 0[X1;X2] = 0[Z1;Z2] = 0Z1X2Z Z Z Z Z Z Z Z Z ZXXXXXXXXXXFast Decoders for Topological Quantum CodesKitaev’s Toric CodeLogical OperatorsNew BasisA1; :::; An=2 1; B1; :::; Bn=2 1;  Z1;  Z2tA1; :::; tAn=2 1; tB1; :::; tBn=2 1;  X1;  X2Fast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyTopology ?Fast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyTrivial CyclesFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyTrivial CyclesFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyTrivial CyclesFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyTrivial CyclesFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyTrivial CyclesAll As, Bp are trivial cyclesThey act as the identity on thecode space :Asj i= Bpj i= +1j iTopologically and logically trivialXXXXZZZZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyTrivial CyclesAll As, Bp are trivial cyclesThey act as the identity on thecode space :Asj i= Bpj i= +1j iTopologically and logically trivialXXXXZZZZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyTrivial CyclesAll As, Bp are trivial cyclesThey act as the identity on thecode space :Asj i= Bpj i= +1j iTopologically and logically trivialXXXXZZZZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyTrivial CyclesfAs;Bpg span the set of trivialcycles) all trivial cycles are equivalentto the identity on the code spaceXXXXZZZZXXXXZZZZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyTrivial CyclesfAs;Bpg span the set of trivialcycles) all trivial cycles are equivalentto the identity on the code spaceXXXZZZXXXZZZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyTrivial CyclesfAs;Bpg span the set of trivialcycles) all trivial cycles are equivalentto the identity on the code spaceXX XXXXX XXXZZ Z ZZZZZZZZZZZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyNon-Trivial CyclesFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyNon-Trivial CyclesFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyNon-Trivial CyclesFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyNon-Trivial CyclesFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyNon-Trivial CyclesZ1 and Z2 wind around thetorus : non-trivial cyclesThey live on the latticeZ1Z2ZZZZZZZZZZZ Z Z Z Z Z Z Z Z ZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyNon-Trivial CyclesX1 and X2 are conjugate toZ1 and Z2They live on the dual latticeX1X2XXXXXXXXXXXXXXXXXXXXFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyNon-Trivial CyclesNon-trivial cycles have non-trivial e ects on the code spaceFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyHomological/Logical Classesj i= Bp0j iZ1j i= Z1Bp0j iZ1  Z1Bp0Z1  Z1Bp0Bp00Z1  Z1QpBpZ1pprimeZ Z Z Z Z Z Z Z Z ZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyHomological/Logical Classesj i= Bp0j iZ1j i= Z1Bp0j iZ1  Z1Bp0Z1  Z1Bp0Bp00Z1  Z1QpBpZ1BpprimeZZZZZ Z Z Z Z Z Z Z Z ZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyHomological/Logical Classesj i= Bp0j iZ1j i= Z1Bp0j iZ1  Z1Bp0Z1  Z1Bp0Bp00Z1  Z1QpBpZ1BpprimeZZZZ Z Z Z Z Z Z Z ZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyHomological/Logical Classesj i= Bp0j iZ1j i= Z1Bp0j iZ1  Z1Bp0Z1  Z1Bp0Bp00Z1  Z1QpBpZZZZZZZZ Z Z Z Z Z Z Z ZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyHomological/Logical Classesj i= Bp0j iZ1j i= Z1Bp0j iZ1  Z1Bp0Z1  Z1Bp0Bp00Z1  Z1QpBpZZZZ Z ZZZ Z Z Z ZFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologyHomological/Logical Classesj i= Bp0j iZ1j i= Z1Bp0j iZ1  Z1Bp0Z1  Z1Bp0Bp00Z1  Z1QpBpFast Decoders for Topological Quantum CodesKitaev’s Toric CodeTopologySummaryStabilizer$TopologyEvery element of the stabilizer is a trivial cycle and vice-versaEvery logical operator is a non-trivial cycle and vice-versa) Topological equivalence classesFast Decoders for Topological Quantum CodesConcatenation1 Kitaev’s Toric Code2 ConcatenationConcatenated codesE cient Optimal Decoder3 Topological Codes DecodingFast Decoders for Topological Quantum CodesConcatenationConcatenated codesConcatenated CodesFast Decoders for Topological Quantum CodesConcatenationConcatenated codesCodes[[n,k,d]]. . .|ψ 〉|ψ 〉}Fast Decoders for Topological Quantum CodesConcatenationConcatenated codesConcatenated Codes. . . . . . . . .. . .|ψ〉|ψ〉k layers of encoding !nk qubitsError rate decays doubly exponentially : k log Fast Decoders for Topological Quantum CodesConcatenationE cient Optimal DecoderE cient Optimal DecoderDavid Poulin, Phys. Rev. A 74, 052333 (2006)Fast Decoders for Topological Quantum CodesConcatenationE cient Optimal DecoderOptimal (Soft) Decodersyndrome. . .{P (I ), P (X ), P (Y ), P (Z )}Noise Mo delExponential in n, but n is constantDistillates error probability on the logical qubitsFast Decoders for Topological Quantum CodesConcatenationE cient Optimal DecoderRecursive Decodersyndrome. . .P(L)NoiseModelsyndrome. . .P(L)syndrome. . .P(L). . .syndromeP(L)k layers with at most nk codesComplexity : O(nkk)Fast Decoders for Topological Quantum CodesTopological Codes Decoding1 Kitaev’s Toric Code2 Concatenation3 Topological Codes DecodingFast Decoders for Topological Quantum CodesTopological Codes DecodingThreshold0.00010.0010.010.112 3 4 5 6 7 8 9 10Probabilite d’erreur du decodeurForce du canal Bit-Flip, p (%)l=8l=16l=32l=64l=128l=256l=512l=1024The threshold is the noise strength under which it is useful toencodeFast Decoders for Topological Quantum CodesTopological Codes DecodingPrevious MethodPMA : perfect matching algorithm (Preskill, Landahl et al.)Minimum distance decoderComplexity : O(‘6), in practice limited to ‘. 100Threshold of  15:5% under depolarizing noiseLimited to Kitaev’s toric codeFast Decoders for Topological Quantum CodesTopological Codes DecodingPrevious MethodPMA : perfect matching algorithm (Preskill, Landahl et al.)Minimum distance decoderComplexity : O(‘6), in practice limited to ‘. 100Threshold of  15:5% under depolarizing noiseLimited to Kitaev’s toric codeFast Decoders for Topological Quantum CodesTopological Codes DecodingPrevious MethodPMA : perfect matching algorithm (Preskill, Landahl et al.)Minimum distance decoderComplexity : O(‘6), in practice limited to ‘. 100Threshold of  15:5% under depolarizing noiseLimited to Kitaev’s toric codeFast Decoders for Topological Quantum CodesTopological Codes DecodingPrevious MethodPMA : perfect matching algorithm (Preskill, Landahl et al.)Minimum distance decoderComplexity : O(‘6), in practice limited to ‘. 100Threshold of  15:5% under depolarizing noiseLimited to Kitaev’s toric codeFast Decoders for Topological Quantum CodesTopological Codes DecodingPrevious MethodPMA : perfect matching algorithm (Preskill, Landahl et al.)Minimum distance decoderComplexity : O(‘6), in practice limited to ‘. 100Threshold of  15:5% under depolarizing noiseLimited to Kitaev’s toric codeFast Decoders for Topological Quantum CodesTopological Codes DecodingOur solution (Phys. Rev. Lett. 104, 050504 (2010))We designed an algorithm inspired by the concatenateddecoderComplexity : O(‘2 log‘) parallelizable to O(log‘) timeEnabled decoding of a ‘ = 1024 lattice without parallelizingMore resiliant to noise : threshold of  16:5% underdepolarizing noiseNot limited to toric code (e.g. color codes : triplet of defects)Fast Decoders for Topological Quantum CodesTopological Codes DecodingOur solution (Phys. Rev. Lett. 104, 050504 (2010))We designed an algorithm inspired by the concatenateddecoderComplexity : O(‘2 log‘) parallelizable to O(log‘) timeEnabled decoding of a ‘ = 1024 lattice without parallelizingMore resiliant to noise : threshold of  16:5% underdepolarizing noiseNot limited to toric code (e.g. color codes : triplet of defects)Fast Decoders for Topological Quantum CodesTopological Codes DecodingOur solution (Phys. Rev. Lett. 104, 050504 (2010))We designed an algorithm inspired by the concatenateddecoderComplexity : O(‘2 log‘) parallelizable to O(log‘) timeEnabled decoding of a ‘ = 1024 lattice without parallelizingMore resiliant to noise : threshold of  16:5% underdepolarizing noiseNot limited to toric code (e.g. color codes : triplet of defects)Fast Decoders for Topological Quantum CodesTopological Codes DecodingOur solution (Phys. Rev. Lett. 104, 050504 (2010))We designed an algorithm inspired by the concatenateddecoderComplexity : O(‘2 log‘) parallelizable to O(log‘) timeEnabled decoding of a ‘ = 1024 lattice without parallelizingMore resiliant to noise : threshold of  16:5% underdepolarizing noiseNot limited to toric code (e.g. color codes : triplet of defects)Fast Decoders for Topological Quantum CodesTopological Codes DecodingOur solution (Phys. Rev. Lett. 104, 050504 (2010))We designed an algorithm inspired by the concatenateddecoderComplexity : O(‘2 log‘) parallelizable to O(log‘) timeEnabled decoding of a ‘ = 1024 lattice without parallelizingMore resiliant to noise : threshold of  16:5% underdepolarizing noiseNot limited to toric code (e.g. color codes : triplet of defects)Fast Decoders for Topological Quantum CodesTopological Codes DecodingSubcodeImagine we had a surface encoding taking 2 qubits into 8Fast Decoders for Topological Quantum CodesTopological Codes DecodingToric Code : A concatenation?We could recurse on this encoding the build a bigger surfacecodeFast Decoders for Topological Quantum CodesTopological Codes DecodingToric Code : A concatenation?...We could recurse on this encoding the build a bigger surfacecodeFast Decoders for Topological Quantum CodesTopological Codes DecodingDecoding. . .If the toric code is just a concatenated code, then we knowhow to decode it e ciently !Fast Decoders for Topological Quantum CodesTopological Codes DecodingIncomplete StabilizersSome of the stabilizers are incompleteFast Decoders for Topological Quantum CodesTopological Codes DecodingClosing the StabilizersWe complete the stabilizer by adding qubits to the subcodeFast Decoders for Topological Quantum CodesTopological Codes DecodingClosing the StabilizersBy adding these qubits the construction is no more aconcatenationFast Decoders for Topological Quantum CodesTopological Codes DecodingConcatenated code decoderEven though shared qubits correspond to the same physicalentity, we are going to treat them as two di rent qubits withthe same noise modelMain approximation : Decode with the concatenated codedecoder anywayFast Decoders for Topological Quantum CodesTopological Codes DecodingCharacterizing the subcodeSubCode stabilizer generators : 10) 2 logical qubitsFast Decoders for Topological Quantum CodesTopological Codes DecodingCharacterizing the subcodeSubCode stabilizer generators : 10) 2 logical qubitsFast Decoders for Topological Quantum CodesTopological Codes DecodingResults0.16 6.5 7 7.5 8 8.5 9Probabilite d’erreur du decodeurForce du canal depolarizant, p (%)l=8l=16l=32Is there a threshold at all ? At best, these are size e ectsFast Decoders for Topological Quantum CodesTopological Codes DecodingInconsistencyBy treating shared qubits as independent ones, we introduceinconsistenciesA compromise between this and exact decoding would be toenforce consistencyFast Decoders for Topological Quantum CodesTopological Codes DecodingGeneralized Belief Propagation (Jonathan S. Yedidia)Self-consistency constraints on shared qubitsNeighboring unit cells exchange messages! Belief propagationCompromise on shared qubitsFast Decoders for Topological Quantum CodesTopological Codes DecodingIntuition about GBPX1DepolarizingChannel,p<<110 12 3Fast Decoders for Topological Quantum CodesTopological Codes DecodingIntuition about GBPX1DepolarizingChannelP(X)∼50%P(X)∼p2P(X)∼50%1P(X)∼50%0 12 3Fast Decoders for Topological Quantum CodesTopological Codes DecodingIntuition about GBPX1P(X)∼50%P(X)∼p2P(X)∼50%1P(X)∼50%0 12 3Fast Decoders for Topological Quantum CodesTopological Codes DecodingIntuition about GBPX1P(X)∼50%P(X)∼p2P(X)∼50%1P(X)∼50%0 12 3Fast Decoders for Topological Quantum CodesTopological Codes DecodingResults0.010.1112 13 14 15 16 17Probabilite d’erreur du decodeurForce du canal depolarizant, p (%)l=8l=16l=32l=64Fast Decoders for Topological Quantum CodesTopological Codes DecodingPreliminary Physical Decodings1s2s3P1(e) P2(e) Pi(e)....... . .BP on the bare stabilizers and qubitsAccounts correlations between X and Z introduced by YIts output is the input to the concatenated decoderFast Decoders for Topological Quantum CodesTopological Codes DecodingPreliminary Physical Decodings1s2s3P1(e) P2(e) Pi(e)....... . .BP on the bare stabilizers and qubitsAccounts correlations between X and Z introduced by YIts output is the input to the concatenated decoderFast Decoders for Topological Quantum CodesTopological Codes DecodingPreliminary Physical Decodings1s2s3P1(e) P2(e) Pi(e)....... . .BP on the bare stabilizers and qubitsAccounts correlations between X and Z introduced by YIts output is the input to the concatenated decoderFast Decoders for Topological Quantum CodesTopological Codes DecodingResults0.1115 15.5 16 16.5 17 17.5 18Probabilite d’erreur du decodeurForce du canal depolarizant, p (%)l=8l=16l=32l=64l=128PMAFast Decoders for Topological Quantum CodesTopological Codes DecodingResults0.00010.0010.010.112 3 4 5 6 7 8 9 10Probabilite d’erreur du decodeurForce du canal Bit-Flip, p (%)l=8l=16l=32l=64l=128l=256l=512l=1024Unit cell (2 1) + decoding the 2 types of defectsindependently )‘ = 1024 latticeFast Decoders for Topological Quantum CodesConclusionConclusionTopological codes use highly non-local operators to encodeinformationWe proposed an e cient (O(log‘) time) to decode them! Concatenated codes, GBPMore resiliant to noise under depolarizing noise than knownmethods (16:5% vs. 15:5%)It enabled decoding of color codes pth 8:7% (H ectorBombin)Fast Decoders for Topological Quantum CodesConclusionWork in progress : Color Codes                      Fast Decoders for Topological Quantum CodesConclusionColor Codes : MappingBBAAZiXiFast Decoders for Topological Quantum CodesConclusionColor Codes : Results0.00010.0010.010.117 7.5 8 8.5 9 9.5 10 10.5 11Decoding error probabilityBit-Flip channel strength p%l=16l=32l=64l=128l=256

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