A general approach to achieving robust control of a quantum bit
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A general approach to achieving robust control of a quantum bit QMQT, Spetses, June 2016 Ed Barnes Virginia Tech
Transcript of A general approach to achieving robust control of a quantum bit
Ageneralapproachtoachievingrobustcontrolofaquantumbit
QMQT,Spetses,June2016
EdBarnes
VirginiaTech
Somequantumsystemswithvariableparametersandnoisyenvironments
Spinsinquantumdots
ee
nuclearspins
NVcentersindiamond
opticalormicrowavepulse
Superconductingcircuits
Controllingnon-idealquantumsystems
Uncertainties and arisefromenvironmentalnoiseornon-identicalsystems,alteringthesystemevolution.
Thisisasevereprobleminsolidstatequantumcomputingandnanoscaledevices/metamaterials.
e.g.,tunableenergysplitting,exchangecoupling,time-dependentexternalelectric/magneticfielde.g.,fixedenergysplitting,laser/microwavedetuning,constantexternalmagneticfield
Whatcanwedoaboutthenoise?
e.g.,
q Cleanermaterials,betterhardwaredesigns
q Improvesoftware:smartercontrolmethods
Ø Forslowfluctuations,canweachieveatargetstate/evolutionthroughacleverchoiceofdrivingfield?
Ø OldproblemfirstconsideredinNMR
Dynamicaldecoupling:Sequencesofdelta-functionpipulsescanproduceself-cancelationofnoiseerrors
Notalwaysapplicableorideal
Numericalrecipescanbeinsufficientq GRAPE(gradientascentpulseengineering)q Findlocalmaximaoffidelityq Canyieldcomplicateddrivingfieldwaveforms
q Unsuitableforsomesystemsduetophysicalconstraints:§ Singlet-tripletspinqubit 1-qubitgates§ Superconductingtransmon 2-qubitgates
Barends etal.Nature508,500(2014)(Martinisgroup)
D.EggerandF.Wilhelm,arXiv:1306.6894de
tuning
time
2-qubitCZgateforSCqubits
Squarepulses
à pulseduration~150ns
q Canrespectconstraintsusingsquarepulses
q Squarepulsesnotidealforrealexperiments
q Alternativestosquarepulses?Smoothpulses?
time(ht)
Pulse
J(t)/h
NatureCommunications3:997(2012)Phys.Rev.Lett.110,140502(2013)Phys.Rev.A89,022310(2014) AmirYacoby,
Harvard
Ageneralsolutiontothenoiseproblemrequiresanalyticalsolutionsforarbitrarytime-dependentHamiltonians
q Canweautomaticallycancelerrorsusingcarefullydesignedsmoothpulses?
q Canwefindallpossibledrivingfieldsthatimplementatargetevolutionwhilecancelingnoise?
Goalsforthistalk
Analyticalsolutions:1932-2012
• Squarepulse
• Landau-Majorana-Stueckelberg-Zener (1932)
• Rosen-Zener sech pulse(1932)
• Generalizationsofthesech (1980s)
• SolutionsbasedonHeun functions(2000s)
• Solutionsbasedonellipticfunctions(2010s)
Traditionalapproachtofindingnewsolutions
Schrödingerequation:
Canwefindnewsolutionsmoresystematically?
Single-axisdriving:
Evolutionoperator:
Evolutioninrotatingframe:
Oldstrategy:Pick toobtainafamiliarequatione.g. giveshypergeometric equation
Ourapproach:partialreverse-engineering
Schrödingerequation:
Insteadofguessing andsolvingfor,thinkofthisasanequationfor .
Thisnon-linearequationcanbesolvedexactly:
• Firstchoosetheevolution• Usetheformulatofindthe thatgeneratesthatevolution• Enforceunitarity
BarnesandDasSarma PRL109,060401(2012)
Reverseengineeringatwo-levelHamiltonian
BarnesPRA88,013818(2013)
BothHamiltonianandevolutiondeterminedfromauxiliaryfunction
mustobeytheinequality:
Somenewanalyticallysolubleexamples
Extensiontoageneraltwo-levelHamiltonian
mustobeytheinequality:
• Chooseandasdesiredbychoosing and• Thenchoose tofixdesired
BarnesPRA88,013818(2013)
evolution
driving
Nowtherearethreeauxiliaryfunctions:
•QSLcorrespondstosaturating,i.e.sincethisisthefastestwaytogofrom to
• TheQSLtimeisgivenbysolvingthefollowingforTQSL:
EvolutionatthequantumspeedlimitEnergy-timeuncertaintygivesalowerboundonthetimeittakestoevolvefromsomestatetoanorthogonalstate(Mandelstam&Tamm1945):
GeneralizationsbyBhattacharyya(1983),Margolus andLevitin (1998),Giovannetti,Lloyd,Maccone(2003)
• Forbx=constant andby=0,
• Choosewhere
Quantumgatesatthequantumspeedlimit
• Inthiscase,QSLmeans
•WecanconstructgatesthatoperateneartheQSLsimplybychoosingalmostlinearfunctions:
Multi-axispulseswitharbitrarilymanyparameters:
(a) bz fordifferentparameters
(b) bz andbx thatgiveHadamard gate
Landau-Majorana-Stueckelberg-Zenerinterferometryatthequantumspeedlimit
BarnesPRA88,013818(2013)
Exactformulaforinterferencepatterns:
Leakageinsuperconductingqubits
CZgate
CNOTgate
• Coupledtransmons haveadensespectrum
• Spectrallysharppulsesareslow
• Ourapproach:usesimpleanalyticalpulsesdesignedtospeedupevolution
SeetalkbySophiaEconomou tomorrowat12pm
Economou &Barnes,Phys.Rev.B91,161405(R) (2015)
Howcanwesuppressnoiseerrors?
q FluctuationsintheHamiltonian
canbethoughtofasfluctuationscomingfrom:
q determinesidealevolution
q anddependon
encoderesponseofsystemtonoiseand driving
Calculatingtheresponsefunctions
q Computingandishardbecausetheyaredeterminedbysolvingcomplicateddifferentialequations.
e.g.,
q Weneedageneralsolutiontotheseequationstomakeprogress.
q Adjustparametersin tocancelthesevariations()
Barnes,Wang,DasSarma arXiv:1409.7063(2014)
Remarkably,suchasolutioncanbefound:
Notquitethatsimple
q Ingeneral,wewanttoachieveatargetevolution(orstate)atsometime
q Cancelingrequires3variationstovanish:
q Wecanmakethesevanishbyadjustingparametersinbutwewillalterintheprocess!
Targetevolutionasawindingnumberq Thephaseinthetargetevolutionoperatorwillvarywith:
q Idea:interpretthisphaseasatopologicalwindingnumber,i.e.,
q Avoidnonlineardifferentialequationbywriting
Generalprocedure
q Problemreducestochoosingasinglefunction
q Fixtheboundaryconditionsoninaccordancewith:
q Findadrivingfieldthatgeneratesthisevolutionbyintegrating
andpluggingtheresultinto
Errorcancelationconstraints
Cancelingnoisein
Cancelingnoisein
Barnes,Wang,DasSarma arXiv:1409.7063(2014)
Errorsautomaticallycancelifobeys
Example:cancelingnoiseinthedrivingfield
geometricalpictureofdriving
Verifyingrobustness
time
error
energysplittingnoisestrength
drivingfield
time
error
driving fieldnoisestrength
drivingfield
Thingstodo
q Findthefastestpulses
q Higher-ordercancelation
q Non-staticnoise(includepowerspectrum)
q Similartechniquesinhigherdimensions?
