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Algebraic Topology, Quantum Algorithmsand BIG DATA
Quantum Physics andGeometry" (2017) Paolo Zanardi,USC LosAngeles
ExtractingSmallPatternsoutBigData:TOPOLOGY
Classification ofvastsets ofcomplex objects intermsofsimple topological invariants
Topological Invariants:Euler’sCharacteristics
E=2(1-g),g=genus
Let’srefinethisconceptfortriangulable spaces(homeomorphictopolyhedra):SimplicialComplexes
BoundaryMap&ChainComplexes
Cn=n-th ChainGroup=Formallinearcombinationsofsimplices ofthecomplex
Boundary Map: sends asimplex toacombination ofitsfaces
Nilpotency=boundary ofboundary is0
HomologyGroups
βk=#ofgeneratorsofHk=Betti Number
β0=#ofconnected components,
β1=#of holes,
β2=#voids,…...
E.Betti,1823-1892Euler’s Characteristic
ComplexesfromPointCloudData(PCD)
Foreachscaleof𝞮 onebuildsasimplicialcomplexS𝞮 outofthePCDincreasing𝞮makesS𝞮 growingVarying𝞮 overarangeofscalesoneobtainsafamilyofnestedsimplicialcomplexesakaaFiltration
Datacanberepresented by ”clouds” ofpoints in ahigh dimensional space: howdowedotopologywiththat?!?
Čech complex Vietoris-Rips complex
ComplexFiltrationsandBarCodes
Tracking how Betti numbers change as function of the scale 𝞮 reveals how topological features come into existence and go away as the data is analyzed at different 𝞮
Atopologicalfeaturethatpersistsovermanylengthscalescanbeidentifiedwitha‘true’featureofthestructure:PersistentHomology
2)FindthekerneloftheLaplacian togettheBetti Numbers(QuantumPhaseestimation Algorithm)
0)Store(orcompute)distances betweendatapoints ina Q-RAM
1)Fix𝞮,constructaquantumstateencodingsimplicial complexatthescale𝞮 (GroverSearchAlgorithm)
3)Iterateoverthe𝞮 andlookforpersistent featuresacrossscales
QuantumAlgorithm forPersistentHomology:TheSketch
HowAboutcomputationalcomplexity?!?
ComputationalComplexity:Classical vsQuantum
OurQuantumalgorithmprovidesandexponentialspeed-upovertheclassicalone!
QUANTUM SUPREMACY…....
Grover’sSearchAlgorithm:
Spacegenerated bythek-simplex states inthe𝞮-Complex, --dimensional
Letsk ak-simplexwemapitontoaquantumstate|sk>=|j1,j2,…,jn>wherejp=1 iffp isinsk
TheGutsoftheQuantumAlgorithm I
Takes timewhereisfractionofsimplices actually present inthe𝞮-Complex; Classical time
QuantumPipeline1:Encodingthe𝞮-Complex
TheGutsoftheQuantumAlgorithm II
CombinatorialHodgeTheory:Betti numbersarethedimensions ofthekernelsofthe𝞮-complexLaplacian operators(0-eigenvectors=Harmonic forms ≅toHomology classes)
Ifthen ffisthe𝞮-complexDiracoperator
RuntheQuantumPhaseAlgorithmfor overtheuniformmixture ofallsimplicesDetermines thedimensions ofi.e.,theBetti’s numbers
Classically: Quantumly (n-sparsityà)
QuantumPipeline2
Summary&Conclusions
QuantumInformationProcessing inkickinginBIGtime intheBigDATAsceneweareexcited!
WeliveintheBIGDATAageANDintheQuantumInformationAge
BigQuantumDataalgorithmswithexponential speedups e.g.,Q-machinelearning
Wedescribed atopologicaldataanalysisquantumalgorithm forpersistenthomology