UniversalHamiltoniansforExponentiallyLongSimulation:ExploringSusskind’sConjecture

ThomBohdanowiczInstituteforQuantumInformation&Matter

CaliforniaInstituteofTechnologyThursdayJune13,2019arXiv:1710.02625v2

JointworkwithFernandoBrandão

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WhatdoIhaveforyou?

•  AnewconstructionandresultinsimulationofHamiltoniandynamics

•  ProgresstowardsaconjecturebySusskind(Complexity+Holography)

•  ThemostcomplexHamiltonian

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HamiltonianSimulation

•  Whatdoesthismean?•  AnaloguesimulationreproducesallpossiblephysicsofaHamiltonian:eigenstates,spectrum,observables,thermalproperties,dynamics,etc.withintolerableerror

•  Cubittet.al.haveveryniceuniversalityresultsforanaloguesimulation:2DHeisenbergwithtunablecouplingscandoanything!(arXiv:1701.05182)

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HamiltonianSimulation

•  Inthiswork,weareconcernedwithuniversalityforaveryrestrictednotionofsimulation:thesimulationofHamiltoniandynamics

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Universality

•  Here,universalityofoursimulationschemereferstotheabilitytosimulatethedynamicsofanytime-independentHamiltonian

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StateoftheArt

•  Noknownsimulationschemescanfaithfullysimulatequantumdynamicsfortimesuptoexponentiallylargeinthesystemsize(withoutexponentialspaceresources)

•  Ourscan!

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CircuitComplexity

•  TheCircuitComplexityofastateistheminimumnumberoftwo-qubitgatesfromafixedgatesetthatisrequiredinordertobuildaquantumcircuitthatcreatesthatstatefromthetrivialreferencestate

•  Thecircuitcomplexityofaunitaryistheminimumnumberoftwo-qubitgatesfromafixedgatesetrequiredtobuildacircuitthatimplements

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| i

|0i⌦n

U

U

WhyMightYouCare:HolographyandComplexity

•  Consideranon-traversableAdSwormholeconnectingtwoblackholes,whosedual/boundarytheoryisapairofentangledCFTs

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|TDSi =2nX

i=1

|iiCFT1 ⌦ |iiCFT2

HolographyandComplexity

•  Classicalgravitydictatesthatthevolumeofthewormholeincreaseslinearlyintimeupuntilitsaturatesatatimeexponentiallylargeinsystemsize,andhitsrecurrencesatdoublyexponentialtimes

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HolographyandComplexity

•  AdS/CFTdualitysuggeststhatthereshouldbeananalogousphysicalquantityintheboundaryCFTthathassimilarqualitativebehavior

•  Dynamicalquantitiesinquantumfieldtheoriestendtosaturatequickly

•  So…whatkindofquantityintheCFTcouldbedualtotheever-growingAdSwormholevolume?

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Susskind’sProposal

•  SusskindhasproposedthatitshouldbethecircuitcomplexityoftheCFTthermofielddoublestatethatbehavesthisway!

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|TDS(t)i =2nX

i=0

eiHt|ii ⌦ eiHt|ii

Susskind’sProposal

•  StartingwithastandardmaximallyentangledTFDstate(whichhastrivialcomplexity),timeevolutionundertheCFT’sHamiltonianshouldgenerateastatewhosecomplexityisincreasinglinearlyintimeuptoexponentiallylongtimes

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C (|TDS(t)i) = ⇥(t)

t � 2n =) C (|TDS(t)i) ⇠ 2n

Susskind’sProposal

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Susskind’sProposal

•  AaronsonandSusskind(arXiv:1607.05256)haveprovedthefollowing:AssumingthatPSPACEisnotcontainedinPP/poly,thenthereexistsatimet=cnandapolynomialsizeunitaryUsuchthat

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C�U t|TDS(0)i

�⇠ 2n

Wishlist

•  WouldbebetterifitwereaphysicallyreasonabletimeevolutionfromaCFTHamiltonianthatgeneratedtheexponentiallycomplexstate

•  Wouldalsobebetteriflineargrowthwereexplicit

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TwoQuestions

•  Question1:IsthereaphysicallyreasonableHamiltonianswecouldwritedownwhosetimeevolutiongeneratesacircuitwhosecomplexityisexponentiallylargeafterexponentiallylongtimeevolutions?

•  Question2:Canonefaithfullysimulatethedynamicsofann-qubitsystemfortimesexponentialinnusingpolynomialresources?

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TwoBirdsWithOneStone

•  MotivatedbytheAaronson/Susskindproblem,webuiltafamilyofHamiltoniansthatactuallyaddressesboth!

•  Specifically:wehaveafamilyofgeometricallylocal,translationinvariant,timeindependentHamiltonianswhosedynamicscanfaithfullysimulatethedynamicsofanyHamiltonianfortimesuptoexponentialinthesystemsize

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And?

•  Wecanshowthatundersuitableconditions,itcangenerateastateofexponentiallylargecomplexityafteranexponentiallylongtimeevolution

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TechnicalStatementofMainResults

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UnpackingDefinition1

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How?

•  OurconstructionusestheconceptsofHamiltoniancomputation(asexploredbyNagaj)andcellularautomatatobuildaHamiltonianwhoselocaltermsareasetof54carefullychosenlocalcellularautomatontransitionrulesactingonaspinchainoflocaldimension14580

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ConstructionOverview

•  WebuildwhatiscalledaHamiltonianQuantumCellularAutomaton(HQCA)

•  Basically:takeaclassicalreversiblecellularautomaton(statespaceandreversibletransitionrules)

•  EncodethesetransitionrulesintolocalHamiltoniantermsforH

•  TimeevolutionunderHwillproducequantumsuperpositionsofstatesofyourclassicalCAstatespace!

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HQCA?

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WhatshouldourHQCAdo?

•  Well,whatIpromisedyouisasingleHamiltonianthatcansimulate*all*possibledynamics

•  Todothis,therehastobeawayofspecifying*which*dynamicsyouwanttosimulate.Thatis,whatistheunitaryUthatwewanttoapply?

•  Thisisspecifiedasinputtothesimulationprotocol

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But…

•  Ifwe’reinterestedinsimulatingdynamicsforalongandcomplicatedtimeevolution,thismeansweneedtodescribealongandcomplicatedcircuit!So,naively,thesimulatorwouldneedtobeexponentiallylargeforexponentiallylongtimeevolution

•  However,sincetheHamiltonianswe’resimulatingaretimeindependent…

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t ⇠ poly(n) =) C�eiHt

�= poly(n)

t ⇠ 2n =) eiHt = U t

C(U) = poly(n)

Sothen:

•  OursimulatorisanHQCAthattakesaninputstateforsomen-qubitsystem,adescriptionofapoly(n)circuitUwhoserepeatedapplicationgeneratesourdesiredtimeevolution,andthensimplygoesthroughthemotionsofapplyingUgatebygatetothesystemoverandover!

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Hereitis…

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Whydoesitwork?

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H =X

Hi

eitH | 0i = | 0i+ itH| 0i �

t2

2H

2| 0i � it3

6H

3| 0i+ ...

eiHt = I + itH � t

2

2H

2 � it3

6H

3 + ...

So?

•  ThankstocarefullyengineeredlocaltransitionrulesmakingupoursimulatorHamiltonian,theproblemendsuplookingthesameasaquantumparticlehoppingona1Dline

•  Justneedtowaitfortheparticletohopfarenough!

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TheSimulationinanutshell

•  Comeupwithapoly(n)Uthatwillgeneratethedynamicsyouwant

•  Feeditsdescriptionintothesimulator,waitlongenoughformostoftheamplitudesconcentrateontheparticlehavingdiffused“farenough”

•  Measurethecountertocollapsethestateoftheworkqubitstothedesiredonewithhighprobability

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(Overly)TechnicalDetails

•  I’mnotgoingtodescribethefullstatespaceandtransitionrules–readthepaper

•  Lengthofchain:m=poly(n,log(t))•  NumberofdiscretetimestepsbeforeUisappliedktimes:T=poly(n,k)

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ComplexityGrowthofDynamics

•  ThesimulationHamiltonianHistime-independent,translationinvariant,local

•  RunitwithUfromAaronsonandSusskind’sargument(UisthestepfunctionofauniversalclassicalcellularautomatonthatcansolvePSPACE-completeproblems)

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MostComplexHamiltonian

•  ThecircuitcomplexityofourHamiltonian’sevolutionmust(asymptotically)beascomplexasanyothertimeindependentHamiltonian

•  ThisisbecauseitgeneratesthetimeevolutionofanyotherTIHamiltonianwithonlypolynomialoverhead!

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InConclusion

•  Simulationschemethatallowsexponentiallylongsimulationtime

•  Hamiltoniansthatgeneratethemostcomplextimeevolutionspossible

•  AphysicalHamiltonianwhosetimeevolutionsupportsSusskindconjecture

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Thankyou!!

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