Algebraic Topology, Quantum Algorithms and BIG DATA Quantum Physics and Geometry" (2017) Paolo Zanardi, USC Los Angeles
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