Algebraic Topology, Quantum Algorithmsand BIG DATA

Quantum Physics andGeometry" (2017) Paolo Zanardi,USC LosAngeles

AnewBuzzword: BIGDATA

ZB=1,000,000,000,000,000,000,000 =10^21Bytes

What’sBIGDATA?

AndWe(=QuantumInformationFolks)makenodifference…..

Today’sTalkisaboutyetanotherBigQuantumDataapproach:

The Bad The Good The Ugly

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

Well,someone@MIT alittleover-excited,perhaps…�.

THANKSFORTHEATTENTION!!

Seth Lloyd

Universal PersistentQuantum theoryofEVERYTHING