Gateway schemes of quantum control for spin networks

0.8 MB·

by  Koji Maruyama

Chapter 1
Gateway schemes of quantum control for spin
networks⋆
KojiMaruyama1andDanielBurgarth2
7
1
0
2
1Department of Chemistry and Materials Science, Osaka City University, Osaka

n 558-8585,Japan
a
2InstituteofMathematicsandPhysics,AberystwythUniversity,AberystwythSY23
J
3BZ,UK
1

]
h
p 1.1 Motivationand Overview
-
t
n
a Towardsthefull-fledgedquantumcomputing,whatdoweneed?Obviously,thefirst
u thingweneedisa(many-body)quantumsystem,whichisreasonablyisolatedfrom
q its environment in order to reduce the unwanted effect of noise, and the second
[
mightbeagoodtechniquetofullycontrolit.Althoughwewouldalsoneedawell-

1 designed quantumcode for informationprocessing for fault-tolerantcomputation,
v fromaphysicalpointofview,theprimaryrequisitesareasystemandafullcontrol
6
forit.Designingandfabricatingacontrollablequantumsystemisahardworkinthe
1
firstplace,however,weshallfocusonthesubsequentstepsthatcannotbeskipped
2
0 andarehighlynontrivial.
0 Typically,whenattemptingtocontrolamany-bodyquantumsystem,everysub-
. systemofithastobeasubjectofaccurateandindividualaccesstoapplyoperations
1
0 andtoperformmeasurements.Sucha(near-)fullaccessibilityleadstoaproblemof
7 notonlytechnicaldifficulties,butalsonoise(decoherence),asthesystemcanread-
1 ily interact with its surrounding environment. In a sense, we are wishing for two
:
v inconsistentdemands,namely,beingabletomanipulateaquantumsystemfullyby
i controllingthefieldparameterswhilesuppressingitsinteractionwiththefield.
X
r ⋆ Citeas:K.MaruyamaandD.Burgarth,Gatewayschemesofquantumcontrolforspinnetworks,
a
Chapter6inT.Takui,L.J.Berliner,andG.Hanson(eds.),ElectronSpinResonance(ESR)Based
QuantumComputing,SpringerNewYork,pp167–192(2016).
1
2 KojiMaruyama1andDanielBurgarth2
A good news is that the technological progress over the last decades has been
so great that we are now able to access and control quantum systems quite well,
provided they are not too large. The coherent manipulations of small quantum
systems, in addition to the observationsof quantum behaviours, have been repor-
ted forvarioussystems, e.g., NMR/ESR [1, 2, 3, 4], semiconductorquantumdots
[5, 6,7],superconductingquantumbits(qubits)[8, 9, 10], andNV-centresin dia-
monds[11,12].
Here,wediscussapossibleschemetobridgethegapbetweenwhatwewishto
achieveandwhatwecanrealisetoday.Namely,weaimatcontrollingagivenmany-
bodyquantumsystemandidentifyingitbyaccessingonlyasmallsubsystem,i.e.,
gateway.Restrictingthesize ofaccessiblegatewayandminimisingthenumberof
controlparametersshouldbeofhelpinsuppressingtheeffectsofnoise.
Thischapterconsistsoftwoparts,eachofwhichisdevotedtothesetwotopics,
fullquantumcontrolthroughagatewayandHamiltonianidentification,respectively.
Suchsituations,in whichonlyasubsystemisaccessible,ariseforexamplein net-
works of ‘dark spins’ in diamond and solid state quantum devices[12, 13, 14] as
wellasspinnetworksinNMRandESRsetups[1,4,15].
Inthefirstpart,we presenthowa systemcanbecontrolledthroughaccesstoa
smallgateway.Startingwithageneralargumentonthecontrollabilityofaquantum
system,weshowapossibleschemetocontrolspinnetworksunderlimitedaccess.
Thetwomajorissuesofourinterestintermsofthecontrollabilityconcernthealgeb-
raiccriterionfortheformofHamiltoniansandthetopological(orgraphtheoretical)
condition for the choice of gateway. While the consideration about these aspects
willleadtoclearinsightsintothecontrolofspin-1/2systems,thetheoryisgeneral
enoughtobeappliedtoothersystemsweencounterinthelab.Weshallalsodiscuss
a few issues related to efficiency,such as, can we computea pulse sequencefor a
certain unitary on the chain by a classical computer within polynomial time? Or
howmuchtimewouldaunitaryrequiretobeperformed?
All these discussions on the controllability assume the complete knowledge of
the system Hamiltonian. The second part of this chapter is devoted to the discus-
sionsonhowtheHamiltoniancanbeidentifieddespitethelimitedaccess.Without
theknowlegeofHamiltonian,wecannevercontrolaquantumsystematwill:itwill
belikegoingfortreasurehuntingwithoutamapandacompass.Havinglearnedthe
detailsofthesystemHamiltonian,wethenattempttofullycontrolit,enjoyingthe
quantumness of the dynamics. Nonetheless, both the full information acquisition
and the full control are still very hard. In addition, the operational complexity of
informationacquisition(state and processtomographies)growsrapidly(exponen-
tially)withrespecttothesystemsize.
Presumablythe moststraightforwardway to estimate the quantumdynamicsis
to apply quantum process tomography (QPT), which is a method to determine a
completelypositivemap onquantumstates.Themap onastateρcanbewritten
E E
as E(ρ) = iEiρEi†, where the operatorsEi satisfy iEi†Ei = I (if E occurs
withunitprobability)[16].ThecomplexityofQPTgrowsexponentiallywithrespect
tothesystemPsize;foraN qubitsystem,weneedtospPecify24N parametersfor
E
anditisanoverwhelmingtaskevenforsmallqubitsystems[17,18,19].Moreover,
1 Gatewayschemesofquantumcontrolforspinnetworks 3
QPT necessitates estimating all the matrix elements of ρ, the state of the whole
system,whichisimpossibleunderarestrictedaccesswithzeroorlittleknowledge
ontheHamiltonian.
The hardness of the task stems from our complete ignorance about the nature
ofthedynamics.However,herewewillconsiderthe casesinwhichsomeapriori
knowledgeorgoodplausibleassumptionsareavailabletous.Inreality,itisnatural
tohavesubstantialknowledgeonafabricatedphysicalsystem,whichisthesubject
ofourcontrol,duetotheunderlyingphysicsweintendtoexploit.Thus,herewewill
seehowsuchaprioriinformationonthesystemcanhelpreducethecomplexityof
Hamiltonian identification. We will primarily focus on the systems consisting of
spin-1/2particles.Thisislargelybecausetheyhavebeenattractingmuchattention
recentlyasapromisingcandidatefortheimplementationofquantumcomputers.
Yet, it would not make much sense if the size of the gateway is comparableto
thatoftheentiresystem.Fromtheviewpointofnoisesuppression,thesmallerthe
gatewaysize, thebetter. Thenhowcanwe find a minimalgatewaythatsufficesto
obtainfullknowledgeonthesystem?Aswewillseebelow,thesamegraphproperty
weintroduceinthefirstpart,i.e.,thestudyofspinnetworkcontrol,comesintothe
discussionasacriterionforestimabilityofthespinnetworkHamiltonian.
This Chapter is based on the results from [20, 21, 22, 23, 24] as well as some
newresults.

Part I
Indirect control of spin networks

7
1.2 Reachability inQuantum Control
A central question in control theory is provided a system, typically described by
states, interactions, and our influence on them, to characterize the operations that
canbeachievedbysuitablecontrols.In(unitary)quantumdynamics,theusualsetup
isatimedependentHamiltonianoftheform
H(t)=H + f (t)H , (1.1)
0 k k
k
X
where the time dependence f (t) can be chosen by the experimentator. While in
k
usual quantum mechanicswe solve the Schrödingerequation for a given f (t) to
k
obtain a time evolution unitary U, the question of control is exactly the inverse:
provideda unitary U, is there a controlf (t) which achievesit? The unitariesfor
k
whichthisistruearecalledreachable.
Given a system (1.1), how do we characterize the reachable unitaries? It turns
out that it is easier to include those unitaries which are reachable arbitrarily well
intoourconsideration,andto describethingsin termsofsimulableHamiltonians:
wecallaHamiltonianiH simulableifexp( iHt)isreachablearbitrarilywellfor
−
any t 0. Clearly, iH is effectivelyreachableby setting f 0 and letting the
0 k
≥ ≡
system evolve fora suitable time t. We couldalso set f 1 and allotherszero,
1
≡
andsimulateiH +iH ,andsoon.Letuscallthesimulableset andseewhich
0 1
L
rulesitobeys:
1. A,B A+B : this is a simple consequenceof Trotter’sformula,
∈ L ⇒ ∈ L
whichsaysthatbyswitchingquicklybetweenAandBthesystemevolvesunder
theaverageofAandB.
2. A ,α>0 αA :thisfollowssimplyfromlettingaweakerinteraction
∈L ⇒ ∈L
evolvelongertosimulateastrongerone,andviceversa.
3. A, A,B, B L [A,B] : this follows from a not so well-known
− − ∈ ⇒ ∈ L
variantofTrotter’sformulagivenby
lim eBt/neAt/ne−Bt/ne−At/n n2 =e−[A,B]t2 (1.2)
n
→∞(cid:16) (cid:17)
4. A A :Thisisapropertywhichheavilyreliesonfinitedimensions,
∈L⇒− ∈L
wherethequantumrecurrencetheoremholds,
ǫ,t>0 T >t: e AT 1 ǫ (1.3)
−
∀ ∃ || − ||≤
whichimpliese A(T t) e+At.
− −
≈
If we combine all the above properties we find that the simulable set obeys ex-
actlythepropertiesofaLiealgebraoverthereals.Thisisveryuseful;inparticular,
if throughrules 1-4 arbitrary Hamiltonianscan be simulated, then likewise arbit-
rary unitaries are reachable:the system is fully controllable [25, 26, 27] (in fact,
thisconditionisnecessaryandsufficient).ItwasshownbyLloydthatitisageneric
8
property:infacttworandomlychosenHamiltoniansareuniversalforquantumcom-
putingalmostsurely.Wewillnotprovethishereaswearegoingtoshowsomething
stronger:a randomlychosen pair of two-bodyqubitHamiltonians is universalfor
quantumcomputingalmostsurely.Thatis,Lloyd’sresultholdsevenwhenrestrict-
ingourselvestophysicalHamiltonians.
1.3 Indirect Control
Theaboveequationsdonotyettakeintoaccountthestructureofthecontrols.Asdis-
cussedintheintroduction,itisinterestingtoconsiderthecaseofcompositesystem
V =C C whereonlyapartC ofthesystemiscontrolled,whiletheremainderC
iscompletelyuntouched.InthelightofEq.(1.1)thismeansthatH = h(k) 1 .
S k C ⊗ C
Control is mediated to C only throughthe drift H = H , which acts on C and
0 V
C. If through H the whole system is controllable, it means that we have a case
V
ofweak controllability:thecontrolsH donotthemselvesgenerateallHamiltoni-
k
ans,thedriftevolutionisnecessary.ThisimpliesthatH setsatimelimitforhow
V
quicklythe system can be controlled.Italso revealsmany-bodypropertiesof H
V
andisthereforeinterestingfromafundamentalperspective.
Thequestionis,givenH andasplitofthesystemintoCC,howcanwedecide
V
if the system is controllable? Is the general result by Lloyd still correct when re-
strictingourselvestosuchasplit,andtoaphysicallyrealisticH ?Inthefollowing,
V
wewillaimtoanswerbothquestions.
Usingtheresultsfromthelastsection,V iscontrollableifandonlyif
iH , (C) = (V), (1.4)
V
h L i L
where, for the sake of simplicity, we have assumed the ih(k)’s to be generators
C
of the local Lie algebra (C) of C and where we use the symbol , to rep-
L hA Bi
resent the algebraic closure of the operator sets and . (V) denotes the full
A B L
LiealgebraofthecompositesystemV.Thecondition(1.4)canbetestednumeric-
allyonlyforrelativelysmallsystems.Itbecomesimpracticalinsteadwhenapplied
to large many-bodysystems where V is a collection of quantum sites (e.g. spins)
whoseHamiltonianis describedasa summationof two-sitesterms.Forsuchcon-
figurations,agraphtheoreticalapproachismorefruitful.
1.4 Graphinfection
Theproposedmethodexploitsthetopologicalpropertiesofthegraphdefinedbythe
couplingtermsenteringthemany-bodyHamiltonianH .Thisallowsustotranslate
V
the controllabilityprobleminto a simple graphproperty,infection [28, 29, 30]. In
many-bodyquantummechanicsthispropertyhasmanyinterestingconsequenceson
9
thecontrollabilityandonrelaxationpropertiesofthesystem[28,20].Also,thesame
property, also called zero-forcing, has been studied in fields of mathematics, e.g.,
graphtheory,inadifferentcontext[31].Letusstartreviewingthisinfectionproperty
for the most general setup, which will show more clearly where the topological
propertiescomefrom.
The infection process can be described as follows. Suppose that a subset C of
nodes of the graph is “infected” with some property. This property then spreads,
infecting other nodes, by the following rule: an infected node infects a “healthy”
(uninfected)neighbourifandonlyifitisitsuniquehealthyneighbour.Ifeventually
allnodesareinfected,theinitialsetCiscalledinfecting.Figure1.1wouldbehelpful
tograspthepicture.
(a) (b) (c) (d)
Fig. 1.1 An example of graph infection. (a) Initially, three coloured nodes inthe region C are
‘infected’.Asthenodelistheonlyoneuninfectednodeamongtheneighboursofk,itbecomes
infectedasin(b).(c)Similarly,l′becomesinfectedbyk′.(d)Eventuallyallnodeswillbeinfected
onebyone.
Note that the choice of C that infects V is not unique. Though we are inter-
estedinsmallC,findingthesmallestoneisanontrivial,andindeedhard,problem.
Nevertheless,fromapragmaticpointofview,thenumberofnodesweconsdierfor
the purposeofquantumcomputingwouldnotbe toolargeto dealwithas a graph
problem.
1.5 Controllabilityofspinnetworks
The link to quantum mechanics is that each node n of the graph has a quantum
degree of freedom associated with the Hilbert space , which describes the n-
n
H
th site ofthe many-bodysystem V we wish to control.The couplingHamiltonian
determinestheedgesthrough
H = H , (1.5)
V nm
(n,Xm)∈E
whereH = H aresomearbitraryHermitianoperatorsactingon .
nm mn n m
H ⊗H
WithinthiscontextwecalltheHamiltonian(1.5)algebraicallypropagatingifffor
alln V and(n,m) E onehas,
∈ ∈
10
[iH , (n)], (n) = (n,m), (1.6)
nm
h L L i L
whereforagenericsetofnodesP V, (P)istheLiealgebraassociatedwiththe
⊆ L
Hilbertspace 2.Thegraphcriterioncanthenbeexpressedasfollows:
n∈P Hn
Theorem: ANssumethattheHamiltonian(1.5)ofthecomposedsystemV isalgeb-
raicallypropagatingand thatC V infectsV. Then V is controllableacting
⊆
onitssubsetC.
Proof: ToprovethetheoremwehavetoshowthatEq.(1.4)holds,orequivalently
that (V) iH , (C) (the opposite inclusion being always verified). By
V
L ⊆ h L i
infectionthereexistsanorderedsequence P ;k = 1,2, ,K ofK subsets
k
{ ··· }
ofV
C =P P P P =V , (1.7)
1 2 k K
⊆ ⊆···⊆ ⊆···⊆
suchthateachsetisexactlyonenodelargerthanthepreviousone,
P P = m , (1.8)
k+1 k k
\ { }
andthereexistsann P suchthatm isitsuniqueneighboroutsideP :
k k k k
∈
N (n ) V P = m , (1.9)
G k k k
∩ \ { }
with N (n ) n V (n,n ) E beingthe set of nodesof V whichare
G k k
≡ { ∈ | ∈ }
connectedto n throughan elementof E. The sequenceP providesa natural
k k
structureonthegraphwhichallowsustotreatitalmostasachain.Inparticular,
it gives us an index k over which we will be able to perform inductive proofs
showingthat (P ) iH , (C) .
k V
L ⊆h L i
Basis:byEq.(1.7)wehave (P )= (C) iH , (C) .Inductivestep:assume
1 V
L L ⊆h L i
thatforsomek <K
(P ) iH , (C) . (1.10)
k V
L ⊆h L i
Wenowconsidern fromEq.(1.9).Wehave (n ) (P ) iH , (C) and
k k k V
L ⊂L ⊆h L i
[iH , (n )]=[iH , (n )] [iH , (n )],
nk,mk L k V L k − nk,m L k
m
X
wherethesumontherighthandsidecontainsonlynodesfromP byEq.(1.9).Itis
k
thereforeanelementof (P ).Thefirsttermontherighthandsideisacommutator
k
L
ofanelementof (P )andiH andthusanelementof iH , (C) byEq.(1.10).
k V V
L h L i
Therefore[iH , (n )] iH , (C) andbyalgebraicpropagationEq.(1.6)
nk,mk L k ⊆h V L i
wehave
[iH , (n )], (n ) = (n ,m ) iH , (C) .
h nk,mk L k L k i L k k ⊆h V L i
2 Note that thecondition (1.6) isa stronger property than thecondition of controlling n,mby
actingonn.According toEq.(1.4)thelatterinfact reads hiHnm,L(n)i = L(n,m), whichis
impliedbyEq.(1.6).