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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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