Prospects for measurement-based quantum computing with solid state spins

2 MB·English

by  Simon C. Benjamin

Laser&PhotonicsReviews,January21,2009 5
Abstract This article aims to review the developments, both
theoreticalandexperimental,thathaveinthepastdecadelaid
detector
the ground for a new approach to solid state quantum com-
array
puting. Measurement-based quantum computing (MBQC) re-
quiresneitherdirectinteractionbetweenqubitsnorevenwhat
would be considered controlled generation of entanglement.
Ratheritcanbeachievedusingentanglementthatisgenerated
probabilisticallybythecollapseofquantumstatesuponmea-
fast switched optical multiplexer
surement.Singleelectronicspinsinsolidsmakesuitablequbits
forsuchanapproach,offeringlongcoherencetimesandwell
definedroutestoopticalmeasurement.Wewillreviewthethe-
oretical basis of MBQC and experimental data for two fron-
trunner candidate qubits – nitrogen-vacancy (NV) centres in
qubits in
diamond and semiconductor quantum dots – and discuss the
cavities
prospectsandchallengesthatlieaheadinrealisingMBQCin
9 thesolidstate.
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Schematicarchitectureforameasurement-basedquantumcom-
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puter.
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Prospects for measurement-based quantum computing

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with solid state spins
p
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aSimonC.Benjamin,1,2BrendonW.Lovett,1andJasonM.Smith1
u
q1UniversityofOxford,DepartmentofMaterials,ParksRoad,OxfordOX13PH,UK
[2CentreforQuantumTechnologies,NationalUniversityofSingapore,3ScienceDrive2,117543Singapore

2
Received:...
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Publishedonline:...
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.Keywords: quantumcomputing,entanglement,graphstates,spinqubits,coherence,NVcentres,quantumdots
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XiContents glement between qubits in a deterministic and reversible
fashion. Early experimental architectures with electronic
r
a1. Introduction . . . . . . . . . . . . . . . . . . . . . . 5 qubits therefore focused on systems in which the qubits
2. Theoryofmeasurement-basedquantumcomputing . . 6 were spaced closely enough, within tens of nanometres
3. SolidstatearchitecturesforMBQC:Generalconsider- orless,tocommunicatewitheachotherthroughsomelo-
ations . . . . . . . . . . . . . . . . . . . . . . . . . 12 calinteractionthatcouldbegatedusingexternalcontrols
4. Candidatesolidstatequbits . . . . . . . . . . . . . . 15
[1,2]. Despite some impressive achievements and beau-
5. Outlook . . . . . . . . . . . . . . . . . . . . . . . . 21
tiful science resulting from this work, the need to locate
qubitssocloselytoeachotherwhilstretainingfullcontrol
onthemindividuallypresentsenormouschallenges,espe-
1. Introduction
ciallyinthedesignofscalablearchitectures.
Whenquantumcomputingwasfirstconceived,itwasgen- In the past decade, theoretical schemes for ‘measure-
erally assumed that to achieve a universal set of quan- ment-based’,or‘one-way’quantumcomputing(MBQC)
tum gates one would need to be able to generate entan- haveprovidedapromisingalternativetothecircuit-based
e-mail:[email protected]
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6 Benjamin,Lovett,andSmith:MBQCusingsolidstatespins
model. Rather than entanglement production being part otherneighbouringspins,andtherequirementsplacedon
of a quantum algorithm, an entangled state is generated thequbitmodulesbytheneedtoperformhighfidelityop-
in advance, and the algorithm is then executed via mea- ticalmeasurementofthespins.Section4thenfocuseson
surementswhichconsumethatentanglement.Suchanap- ourtwocandidatequbits,thepropertiesthatmakethemat-
proachbringsthemajoradvantagethaterrorsinentangle- tractiveforMBQC,andhighlightsthestrengthsandweak-
ment generation can be identified and corrected without nesses of each. We conclude by providing an outlook for
prejudicing the algorithm. This leads directly to the re- thechallengesthatremaininrealisingmeasurement-based
alisation that entanglement generation can be probabilis- quantumcomputinginthesolidstate.
tic(providedthatsuccessesandfailuresareheralded)and
thereforethatentanglementcanbegeneratedby‘measur-
ingout’termsintheproductstateoftwoormoreuncorre- 2. Theoryofmeasurement-basedquantum
latedsystems.Thequbitscanbespatiallyremote,provided
computing
weareabletoconstructanappropriatemeasurement.
The first demonstration of measurement-based entan-
In 1999 and 2001, two important theoretical insights into
glementofmacroscopicallyseparatesystemswasreported
thecreationandexploitationofentanglementwereattained.
byChouetalusingatomicensemblesin2005[3],anden-
The first was the discovery of an elegant method of en-
tanglement of two remote single atoms was reported by
tanglement generation by exploiting the act of measure-
Moehringetalin2007[4].Suchexperimentswillalmost
ment[6].Thesecondinsightwasthatcertainmany-qubit
certainly be extended to entanglement of multiple qubits
entangledstatesconstituteanenablingresourceforquan-
in these very promising systems and could pave the way
tum computing, in the sense that if one were given such
towards fully scalable quantum computing. Moreover, in
a resource of sufficient size then one could perform any
this review we shall argue that the ability to use remote
chosen algorithm just by measuring the qubits one at a
qubits is a particular boost to efforts to realize quantum
time [7]. Taken together, these two ideas allow us to see
computing in solid state systems. Not only does it solve
thatmeasurementcanbethesoledrivingforceofaquan-
the problem of single qubit manipulation outlined above,
tumcomputation,andweareleadtoconsidercomputerar-
italsoallowsustoaddresssuchissuesasinhomogeneities
chitecturesthatareverydifferenttotheearlycircuit-based
inthequbitlocalenvironmentsandlowyieldsofworking
conceptssuchasKane’sproposal[1].
qubits. To build a ten qubit computer it is no longer nec-
Here we will briefly review these two advances, and
essarytobeabletofabricatetenworkingqubitsonasin-
we will consider some schemes for computation that are
gledevice-onecanfabricateten‘devices’eachwithone
tenableinlightoftheseinsights.
workingqubit.Theideaofadistributedquantumcomputer
was thus born (see title figure), in which discrete mod-
ules containing individual qubits act as nodes connected
to an optical multiplexing system. The multiplexer and 2.1. Entanglementbymeasurement
qubit modules are all controlled by a classical computer
thatmakesrealtimedecisionsbasedonthepatternofde- Entanglement is generated naturally when physical sys-
tectioneventsandexecutessubsequentoperationsaccord- temsinteractcoherently.Forexampletwoelectronstrapped
ingly-aprocessknownas‘feedforwarddecisionmaking’. atnearbysitescanentanglewithoneanotherthroughtheir
The architecture is inherently scalable by the addition of mutual Heisenberg interaction. This is the basis of many
further nodes and a larger multiplexer, provided (as with schemes for QIP, especially in the solid state. However,
any quantum computation scheme) all the necessary op- there is another route to entanglement: rather than allow-
erationscanbeexecutedwithinthecoherencetimeofthe ingthephysicalqubitstointeractdirectly,insteadtheycan
qubits. bekeptatfixed,wellseparatedsitesandsimplysubjected
Even in this alternative measurement based paradigm toameasurement.
for performing quantum computing, DiVincenzo’s crite- Typically one would employ the following idea. The
ria [5] of long coherence times, controlled qubit initiali- basic unit is an optically active entity which we call the
sation and manipulation, and efficient optical readout are matter qubit: it could be an atom, or an atom-like struc-
principalconsiderations.Electronspinsinsolidscanoffer ture such as a quantum dot or an optically active crystal
allofthese,andinthisreviewwewilldiscusstwoleading defect. Following laser stimulation the matter qubit may
contenders, namely nitrogen vacancy defects in diamond emitaphoton;itsinternalstatethendependsonthepres-
andStranskiKrastanowquantumdots. ence/absenceofsuchaphoton,oronphotonpolarisation,
The paper is laid out as follows. In Section 2 we first etc. To generate entanglement between two matter qubits
reviewthebasicphysicsandmathematicsbehindmeasure- we would initialize them and then subject them both to
ment-basedquantumcomputing,highlightingtheaspects laser excitation simultaneously. We would monitor their
that lead to practical requirements that differ from those emissions in such a way that we infer characteristics of
ofcircuit-basedschemes.InSection3wethendiscussthe theirmutualstatewithoutlearningthesourceofanygiven
physics that is particular to electron spin qubits in solids, photon. The essential ‘trick’ in most schemes, including
suchasdecoherenceduetointeractionswiththelatticeand the original 1999 proposal of Cabrillo et al [6], is to use
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Laser&Photon.Rev.1,No.1(2008) 7
a beam splitter in theapparatus so that any detected pho- of the matter qubits in the event that the detectors regis-
ton could have originated from either source qubit. This teratotalofonephoton?Toanswerthiswehavetotrack
techniqueisreferredtoaspatherasure. theevolutionofthephoton.Forsimplicityweimaginethat
at a given moment the photon has been emitted from the
matterqubit(whetheritistheleftorrightsystem)buthas
notyetpassedthebeamsplitter.Atthistimethestatehas
evolvedasfollows:
(cid:16) (cid:17)
(|0e(cid:105)+|e0(cid:105))|vac(cid:105)→ |01(cid:105)a† +|10(cid:105)a† |vac(cid:105) (2)
R L
where |vac(cid:105) is the vacuum state of the electromagnetic
field.Weintroducethephotoncreationoperatora† torep-
L
resent a photon in the lower left channel in Fig. 1, and
similarly a† in the right channel. For simplicity we will
R
neglecttowritethe|vac(cid:105)explicitlyinsubsequentexpres-
Figure 1 An apparatus capable of creating entanglement sions; it should be understood to be present on the right
throughameasurement. side of the expression just as in (2). Now if the photons
inthetwoachannelsareformallyindistiguishable(inall
sensesexceptforthechanneltheyoccupy!),andtheyare
ConsiderthesimpleschematicdiagramshowninFig.1.
incidentonthebeamsplitterinsuchawayastomaponto
Suppose that each of the matter qubits A and B has the
thesametwooutputmodesb† andb†,thena† anda† are
“L” level structure shown on the right. For a moment let L R L R
transformedbythebeamsplitterasasfollows:
usalsoimaginethattheapparatusisideal,inthatnopho-
ton will ever be lost from the system. Any photon emit- 1 1
a† → √ (ib† +b†) a† → √ (b† +ib†) (3)
tedbythematterqubitswilleventuallyreachthedetectors L 2 L R R 2 L R
and register a ‘click’. Moreover we will assume that the
wherethephaseiisassociatedwithreflectionratherthan
detectors can count the number of incident photons (dis-
transmission. Using this transformation we can write the
tinguishing zero, one and two). Let us perform a thought
stateofthesystemoncethephotonhaspassedthesplitter:
experimentwiththefollowingsteps:(1)Weinitializeeach
matterqubittothestate|+(cid:105)≡ √12(|0(cid:105)+|1(cid:105)).(2)Wesub- |01(cid:105)(b†L+ib†R)+|10(cid:105)(ib†L+b†R) (4)
ject both matter qubits to a laser pulse, performing a π
=(|01(cid:105)+i|10(cid:105))b† +(i|01(cid:105)+|10(cid:105))b† (5)
rotationonthetransition|1(cid:105) → |e(cid:105).Now,atthisstagethe L R
stateofeachqubitis √12(|0(cid:105)+|e(cid:105)),sothatthestateofthe wherewehavesimplycollectedtermsinvolvingb† andb†
L R
entiresystemis andneglectedtheoverall √1 .Nowtheactionofthedetec-
2
1 tors is precisely to record either a photon corresponding
(|00(cid:105)+|0e(cid:105)+|e0(cid:105)+|ee(cid:105)) (1)
2 to b†, or one corresponding to b†. Suppose that in fact
L R
theleftdetectorclicks–thenwehaveprojectedthematter
wherewehavesimplymultipliedoutthestatesofthetwo
qubitsintothestate √1 (|01(cid:105)+i|10(cid:105)),insertingthecorrect
separate systems A and B, and have organized the qubit 2
values in each ‘ket’ as |AB(cid:105) (keeping track of the order- normalisation.Thus,ifweseethismeasurementoutcome
then we have projected the matter qubits onto an entan-
ing of the labels may appear inconsequential at present,
gledstate–infact,amaximallyentangledstate.Similarly
but our reason for doing so will become clear in the next
asingleclickintherightdetectorheraldsthematterqubit
section). We can now think about the future evolution of
eachterminthissuperposition.Firstly,weseethattheterm state √1 (i|01(cid:105)+|10(cid:105)).Noticethatitisessentialtoknow
2
|00(cid:105)hasnoexcitation,andthereforenophotonwillemerge whichdetectorclicked,becausethetwopossibleresulting
fromthematterqubitsandthedetectorswillnever‘click’. states have different phases, and in fact if we write down
We also note that state |ee(cid:105) will eventually produce two themixedstatecorrespondingtouncertaintyaboutwhich
photons, as the two excited matter systems relax to state detectorclicked,thenthismixedstatehasnoentanglement
|11(cid:105).Thesephotonswilleventuallybeseenbythedetec- atall.
tors (since we have no loss, and the capability to count).
In fact a final detector reading of zero indicates that the
matterqubitsareinstate|00(cid:105),whileafinaldetectorread- 2.1.1. Parityprojection
ingoftwoimpliesthatthematterqubitsareinstate|11(cid:105).
Neitheroftheseareinteresting–wecouldhaveprepared We could not achieve much of significance if we were
those states directly in the first place! However, what of limited to creating two-qubit entangled states using this
the other two terms? States |0e(cid:105) and |e0(cid:105) will both pro- method,butfortunatelyitturnsoutthatrepeatingthesame
duce one photon, and so a detector reading of one need processallowsustocreatemulti-qubitentangledstates.To
not differentiate between them. So what is the final state seethisimaginethatthelefthanddetectorintheprevious
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8 Benjamin,Lovett,andSmith:MBQCusingsolidstatespins
section has clicked, so that qubits A and B are entangled 18].Wecanruleoutthepossibilitythataphotonwaslost
in state |01(cid:105)+i|10(cid:105), and that we initialize a third matter (since we see one from each source). These schemes do
qubit‘C’intostate|+(cid:105).Thetotalthree-qubitstateisnow not necessarily require both photons to be present at the
sametime;theymaybeemittedintwosuccessive‘rounds’
|010(cid:105)+|011(cid:105)+i|100(cid:105)+i|101(cid:105) (6)
asin[16]forexample.Typicallyintheseapproaches,suc-
where each three qubit ‘ket’ is now ordered |ABC(cid:105). We cess is heralded by detector ‘clicks’ which correspond to
maynowsubjectqubitsBandCtoalaserpulseinducing single photons impinging on two different detectors, or
|1(cid:105) → |e(cid:105),andallowthesystemtoevolveusingasimilar on a given detector at two distinct times. Consequently
path erasure scheme as described previously. Once again there is no need for the detectors to be capable of count-
we will have ‘failure’ outcomes corresponding to seeing ingthenumberofincidentphotons.Moreover,two-photon
zero or two photons, this time with the added penalty of schemes are typically interferometrically stable, meaning
destroyingtheexistingentanglement,butintheeventthat thattheoverallprotocolisinsensitivetopathlengthvari-
we see exactly one photon then qubits B and C will have ations. The price to pay for these advantages is that en-
become entangled. The final state that will result in only tanglement only occurs when both emitted photons are
onedetectorclickis detected so the success rate falls quadratically with the
i|101(cid:105)(b† +ib†)+|010(cid:105)(ib† +b†) (7) probabilitythatagivenphotonisretained.However,even
L R L R
thisproblemcanbemitigatedifonecanexploitadditional
=i(|101(cid:105)+|010(cid:105))b† +(|010(cid:105)−|101(cid:105))b†. (8)
L R complexitywithineachnode(e.g.,othereigenstatesofthe
Byrecordingwhichdetectorclicks,wenowobtainathree- nanostructure)[19].
qubitmaximallyentangledstate.Thisprocessofaddinga Themajorityofmeasurement-induced-entanglementschemes
matterqubittoanexistingentangledstatecanberepeated fall into these categories, although there are several other
togrowanarbitrarilylargeentangledstate(althoughaswe approaches – an example is the idea of scattering a sin-
will see shortly, there are better ways to proceed than to glephoton(orstreamofsuchphotons)fromtwophysical
addonequbitatatime).Wecanexpresstheprocessofen- qubitspriortoitsmeasurement[20,21].
tanglementbuildingassubjectingthetwotargetqubitsto It is important to distinguish clearly between the suc-
aparityprojectionleavingthemintheoddparitysubspace cessrateofanentanglementoperation,andthefidelityof
–i.e.wehaveexcludedtheevenparitystates|00(cid:105)and|11(cid:105) that operation. Generally any given protocol will be able
from the |BC(cid:105) subsystem’s superposition. Formally, we to recognize that certain kinds of error have occurred; in
haveappliedtheoperatorPˆ = |10(cid:105)(cid:104)10|+p|01(cid:105)(cid:104)01|to particular,allpracticalprotocolsmustdistinguishwhena
BC
theseparablestateinequation6,wherepiseitherior−i photonhasbeenlost.Whenweknowthatanerrorhasoc-
depending on which detector clicked. Now this operation curredwecandesignatetheoperationasafailureandreset
isidealforbuildingupaparticularkindofmulti-qubitstate our qubits. Thus photon loss affects the success rate but
calledagraphstate(ofwhichtheclusterstateisaspecial neednotaffectthefidelityofthesuccessfulentanglement
case). Graph states are a resource that permits quantum operations,whentheyoccur.However,generallytherewill
computing,aswediscussbelow. beothererrorswhichagivenprotocolcannotdetect(orbe
immune to). One commonly vulnerability is that of dark
counts,wherea‘click’occursthoughnophotonwasemit-
2.1.2. Photonloss tedfromthematterqubits:onseeingsuchacount,wemay
wronglyconcludethatwehavecreatedentanglement.Ex-
As we move from considering an idealized apparatus to perimentaldefectsofthiskindwilldirectlyimpactthefi-
arealisticsystem,theprincipalissuetoaddressisphoton delity of the entanglement operations, and therefore it is
loss. Any photon lost from the apparatus in the scheme essentialtominimizethem.
described above will leave the matter qubits in an uncer- One variant of the two-photon approach was used by
tainstate.Wemustofcourseincludedetectorinefficiency Moehringetalinref.[4]toentangletwo171Yb+ionsata
as one form of ‘loss’. One solution is weak excitation: in success rate of about thirty per billion attempts. This low
thistypeofapproachwekeeptherateofphotongeneration success rate should improve by orders of magnitude with
verylow,sothatasingledetectorclickismuchmorelikely the use of improved apparatus, and in particular the use
to have resulted from a single photon being emitted than of cavities to direct emitted photons into the optical ap-
fromtwobeingemittedandonelost [6,11,12].Ofcourse paratus.Nevertheless,thelowsuccessrateachievedsofar
this assumption is imperfect; the fidelity of the final state doeshighlightthequestionofhowonecanperformacom-
canonlybeimprovedbyfurtherreducingtherateofentan- plexcomputationwhentheenablingoperation,i.e.entan-
glementgenerationandsotherewillalwaysbeafiniteer- glement generation, is prone to fail. There are essentially
ror.Nevertheless,theapproachisattractiveinitssimplicity twoclassesofsolution:onecanadoptasmartgrowthstrat-
andwasthemethodusedbyChouetaltoentangleremote egy which allows the construction of large states despite
atomic ensembles in ref. [3]. A more sophisticated alter- high failure probabilities on each individual operation, or
native would be to adopt a so-called two-photon scheme: onecanuseabrokeringprotocolwhichrequiresasecond
Herebothmatterqubitsemitaphoton,andsuccessfulen- physicalqubitateachlocation.Thesetwoapproacheswill
tanglementinvolvesdetectingthemboth[13,14,15,16,17, be addressed in the next section, but first we shall intro-
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Laser&Photon.Rev.1,No.1(2008) 9
duce the concepts behind the use of pre-prepared entan- Note that we only need a single-qubit rotation to trans-
gledstatesasaresourceforquantumcomputation. formbetweenthisstateandthetwo-qubitentangledstates
wefoundresultedfrommeasurementinourbeam-splitter
device.Anystatethatcanbeturnedintoagraphstatejust
bysinglequbitgates(unitarygates,notmeasurements)is
2.2. MBQCandgraphstates
saidtobelocalunitaryequivalent,orLUequivalent,tothe
graphstate[26].
In 2001, Briegel and Raussendorf reported a radical new
Beforereviewingtheutilityofgraphstates,letusex-
approachtoquantumcomputingbasedoncreatingalarge
tendthepictorialnotationoffigure2inawaythatwillbe
entangledstateandsubsequentlyconsumingitbyaseries
helpful. We will draw an open circle to represent a phys-
of measurements [7]. They were able to show that this
ical qubit that is connected to the rest of the graph state
approach permits one to accomplish the same tasks that
bytheusualcontrol-phasegate,butwhichwaspreparedin
can be performed by the earlier circuit model paradigm.
somegeneralqubitstate|ψ(cid:105)=α|0(cid:105)+β|1(cid:105)insteadofthe
Their new paradigm for quantum computation had clear
standard |+(cid:105) initialisation. Then of course the total state
practical potential in the context of optical lattice atom
will not be a graph state, because state |ψ(cid:105) is not part of
traps, where it may be easier to create many-qubit entan-
theproperdefinitionofagraphstate.
gledstatesthantoentanglespecificpairsofatoms.Butthe
Letusconsiderjustonesuchgeneralqubit,connected
applicabilityoftheideaisverywide.Inparticular,itoffers
byan‘edge’tooneregularqubit|+(cid:105),seeFig.3.Regarding
averynaturalwaytoproceedwhenonewishestoexploit
thisdiagramasaprescription,itistellingusto(a)taketwo
measurementinducedentanglement.
qubits,(b)putoneinsomestate|ψ(cid:105)andoneinstate|+(cid:105),
and then (c) do a control-phase gate between them. Then
wehave
1
Verticies Edges |G(cid:48)(cid:105)= √ (α|00(cid:105)+α|01(cid:105)+β|10(cid:105)−β|11(cid:105)). (9)
2
Nowlet’smeasureoutthefirstqubit,theonethatwas
preparedinstate|ψ(cid:105).Wemeasureitinthex-basis,sothat
theoutcomescanbe|+(cid:105)and|−(cid:105).Thetwopossiblestates
of the remaining qubit are found from (cid:104)+|G(cid:48)(cid:105) or (cid:104)−|G(cid:48)(cid:105)
withrenormalisation,i.e.
α+β α−β
√ |0(cid:105)+ √ |1(cid:105) foroutcome|+(cid:105) (10)
2 2
Figure2 Left:agraphstate.Right:aclusterstate[22].
α−β α+β
√ |0(cid:105)+ √ |1(cid:105) foroutcome|−(cid:105) (11)
2 2
ThekindofstatethatBriegelandRaussendorfconsid-
We see that the state |ψ(cid:105) has rotated, and hopped or tele-
eredcanberepresentedbyadiagramthatmathematicians
portedfromonephysicalqubittotheother(Fig.3b).This
callagraph:Asetofpointsorverticesconnectedbylines
connectionbetweentheideaofquantumteleportationand
oredges.SeeFig.2foranexample.Graphsareahelpful
graph states is a deep one; indeed teleportation was ex-
constructioninunderstandingvariousaspectsofquantum
ploitedtoachievecomputationinthelinear-opticalscheme
information,forexampleerror-correctingcodes[23].Here
of Knill, Laflamme and Milburn (KLM) [27] which was
wewishtoassociateaquantumstatewithagraphtoobtain
proposedaroundthesametimethatRef.[7]waspublished.
whatiscalledagraphstate[24,25].Themoststraightfor-
Theformerapproachcanbeseenasaninstanceofthelat-
ward way to understand the nature of this state is to use
ter[28],althoughtypicallyonethinksofadifferentsetof
theso-calledconstructivedefinition:Foreachnodeinthe
allowedprimitiveoperationsinthetwoparadigms.
graph,prepareaqubitinstate|+(cid:105),andforeachedgecon-
Thesame‘hopping’worksinothercases–wecanuse
necting two nodes in the graph, perform a control-phase
measurements to drive a state from one end of a linear
gate between the corresponding qubits. Now the control-
graph state to the other as in Fig. 3(c). There is an ele-
phaseoperationissimplytheunitarythatputsaphaseof
gantwaytoverifythatthisworks,usingthetime-ordering
−1 on the |11(cid:105) component of the system. We shall see
argumentdepictedasaflowdiagraminFig.3(d).Thisdia-
presently that this can be replaced with the parity projec-
gramillustratesthatwhenwemeasureoutqubitsfromour
tionthatwemetearlier.Figure2alsoshowsaclusterstate:
completegraphstate,thesamefinalstateoccursaswould
agraphstatewitharegularsquarelattice,suchasthe2D
occurifwesimplyentangleeachqubitwithitssuccessor
squarearray.
and immediately measure it. And since we already know
The simplest non-trivial graph state is just two nodes
theeffectofmeasuringanentangledpair,consequentlyit
joinedbyanedge.Thestaterepresentedbythisgraphis
must work completely! This argument is an instance of a
1 usefulgeneralrule:ifwewant,wecanbuildagraphstate
|G(cid:105)= (|00(cid:105)+|01(cid:105)+|10(cid:105)−|11(cid:105)) atthesametimeaswearemeasuringit.Thebehaviourof
2
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10 Benjamin,Lovett,andSmith:MBQCusingsolidstatespins
(b) state ψ alone but also another rotation Uˆz(φ) which incorporates
‘hops’ ourchosenparameterφ.Andthisisinteresting,becauseit
(a) (c) R1ψ showsthatwecanadjusttherotationbyourchoiceofmea-
surementdirection. Infact,itturns outthatwe canstring
ψ + Measure Rψ R2R1ψ together three successive measurements in a linear graph
tocreateanysinglequbitrotationwewish(uptotheusual
unwanted cumulative rotation, which we need only track
(d) Entangle Measure byrecordingmeasurementoutcomes).
all all three RememberingthatallweneedforuniversalQIPis(a)
Initial Final the capacity to perform single qubit gates plus (b) an en-
state state
tanglinggate,itisnownaturaltoask“canwedoanentan-
Measure Measure
glinggate,too?”
Entangle Measure Entangle
2.2.1. Two-dimensionalgraphstates.
Figure 3 (a) A diagram corresponding to a qubit in a general Wehavetheideaofalineargraphstateasakindofwire.
state|ψ(cid:105)(opencircle)entangledwithaqubitinthestandard|+(cid:105) It is natural to consider two such wires, and then putting
state(filledcircle).(b)Asimple‘hop’ofonestep,resultingfrom abridgebetweenthem.Thegraphstatethatwewouldtry
measuringtheleft-handqubit.(c)Anextendedlineargraphstate wouldbeasshowninFig.4a.
can act as a wire, in that a state will hop along it as we mea-
sureoutqubits.(d)Anillustrationofwhytheprocessof(c)must
work:Supposethat,fromtheinitialstate,weentangleallthere-
mainingqubitsandthenmeasurethemout,asshownbythered
arrows.Infactthefinalstatemustbethesameasifwehadper-
formedthealternativetimeorderingofoperationsshownbythe Entangle Measure
green arrows. This follows from the fact that the measurement all all six
operationscommutewithentanglementoperationson3rd party a
qubits–beinglocaltodifferentqubits,theyoperateondifferent Initial Final
subspacesofthesystem’sfullHilbertspace. state state
Measure Measure
b1 b3
thewholesystemwillbejustasifwehadbuiltthewhole
thingfirst,providedthatwheneverwemeasureaqubit,its
Entangle Entangle
entanglement‘edges’arealreadyinplacetoreachitsnear-
estneighbors[24].
b2
Thus a linear graph state can act as a wire for con-
ducting an unknown quantum state – one can drive the
state along by making measurements. The state will get
rotated each time it ‘hops’, but as long as we record all Figure4 Verifyingthatthe2Dgraphstateformedfromconnect-
the measurement outcomes we can keep track of the cu- ingtwolineargraphstatescanimplementanentanglinggate.
mulative rotation so that we will always be able to ‘fix’
thestatewhenitgetstotheotherend.Theseminalpaper
ofRaussendorfandBriegel[7]containedacrucialfurther Infactthiswillindeedsufficetoentangletwounknown
observation: Suppose we try our same trick of measuring states.Onceagain,wecanjustuseourruleaboutbuilding
a qubit to make it ‘hop’ along a linear graph state, but agraphstateaswego:weareallowedtomeasureanodeas
this time we don’t measure it in the x-basis, but instead soonasallthe‘edges’outofthatparticularnodearecom-
in some more general basis for which the states |A(cid:105) = plete.Someasuringallthequbitsinthestate(a)ofFig.4
√1 (cid:0)|0(cid:105)+eiφ|1(cid:105)(cid:1) and |B(cid:105) = √1 (cid:0)|0(cid:105)−eiφ|1(cid:105)(cid:1) are the must be equivalent to the step-by-step process shown by
2 2 the green arrows in the lower half of the Figure. We see
possiblemeasurementoutcomes.Byevaluating(cid:104)A|G(cid:48)(cid:105)we
thattheactionislikethatoftwoparallelwires,exceptthat
findthataftermeasurementoutcome|A(cid:105)weareleftwith
atpoint(b2)thequbitsstoredonaparticularpairofnodes
(α+e−iφβ)|0(cid:105)+(α−e−iφβ)|1(cid:105) get a phase gate between them. This final observation al-
√ lows us to see the full power of graph states: if we can
2
make a graph state with the right topology then we can
afternormalisation.Thisissimilartobefore,thestatehops doanyquantumalgorithmwewant,justbymakingsingle
androtates.ButthistimeitrotatesnotsimplybyaHadamard qubitmeasurements.
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Laser&Photon.Rev.1,No.1(2008) 11
If there is a particular algorithm that we want to per- tic parity operation can indeed suffice to efficiently grow
form,thenweshouldbeabletodrawitasaquantumcir- graphstates.
cuit.Thenit’seasytoworkoutatleastoneparticulargraph
statethatiscapableofimplementingthatcircuitoperation,
asshowninFig.5.Onefurtherinterestingpointisthatwe
canremoveanyqubitfromwithinagraphstatesimplyby
measuring it in the z-basis. If the outcome is |1(cid:105), we ap-
plyafurther‘fix’:subjecteachneighbourtoasinglequbit
phase gates σ . The result is a new graph state with the
Z
nodecorrespondingtothemeasuredqubitremoved.Simi-
larly,measuringintheybasiswillremoveaqubitandcon-
nectitsneighbours.Usingthisprinciplewecanseethata
(sufficientlylarge)clusterstateisauniversalresourcefor
any algorithm: we simply prune out unwanted qubits un-
til our cluster state becomes the correct graph state for a
giventask(seeFig.5d).
(a) (c)
U1
U2
Figure 6 A simple example of how one might ‘grow’ a graph
state ‘just in time’ for it to be consumed, as described in the
(b) (d) text. The regular graph state structure on the left is a universal
U1 UH E UH resource,liketheclusterstatesseenearlier.Theinsetsshowthe
S
HA effectofsuccessfulandunsuccessfulparityprojectionsonqubits
P
E C- U2 (red)thathavepriorentanglement[29].
S
A
H
P
UH C- UH Figure 6 shows one approach. The main graph state
consists of ‘branches’ which grow when successful par-
ityprojectionsaddBellpairstothe‘tips’;afailureresults
Figure 5 Finding a graph state that contains all the entangle-
inthebranchshortening,howeversincesuccessaddstwo
mentneededforaparticularquantumalgorithm.Takesomede-
qubitswhereasfailureonlyremovesone,wewillhaveav-
siredquantumcircuit(a)andtranslateallthetwo-qubitgatesinto
eragegrowthprovidedtheprobabilityofsuccesspexceeds
control-phasegates(b).Thenwritedownagraphstatewiththe
1/3.Wemakethebrancheslongenoughtoabsorbanun-
sametopology,allowingthreeverticesinalinearchainforeach
lucky string of failures. More sophisticated strategies can
arbitrary single qubit gate. Also note that one could obtain the
samegraphstatefromaregular2Dclusterstate(d),simplyby handlearbitrarilylowp[30,31,32,33]althoughinpractice
measuringoutunwantedqubitsineitherthezbasisortheybasis thedecoherencetimeofthesystemwillbecomeanissue.
(greenandorangerespectively). Conversely,veryrapidgraphstategrowthcanoccurwhen
pisaboveapercolationthreshold[34,35].Therehasalso
beenworkongraphstatesynthesiswhenthereareexper-
imental imperfections such as systematic asymmetries in
theapparatus[36,37].
2.2.2. Growinggraphstates When p is very low, for example due to high photon
loss,thenitbecomespracticallyessentialtofindawayof
We have seen that a graph state with a suitable topology preventingfailuresfromdamagingthenascentgraphstate.
canallowonetoperformaquantumalgorithmsimplyby Thesolutionisbrokering[38].Thisrequiresthatourbasic
making measurements. But we have not explicitly shown physicalsystemhasatleasttwoqubitswithinit,forexam-
howtocreatesuchagraphstate;theconstructivedefinition pleanelectronspin(withassociatedopticaltransitionasin
of a graph state is expressed in terms of a control-phase Fig.1)andanuclearspin.Weassumethatthetwoqubitsin
gate,whereasmostschemesformeasurementinduceden- eachlocationcancontrollablyinteract,sothatwithineach
tanglement produce a parity projection (with at least one elementarynodeofourdevicewearefreetoperformone-
notable exception [17]). Moreover, all realistic schemes and two-qubit operations deterministically and with high
haveahighprobabilityoffailinganygivenentanglement fidelity.Giventhislevelofresource,wecanusetheopti-
operation. In this section we note that even a probabilis- callyactivequbitstoachieveentanglementbetweennodes
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12 Benjamin,Lovett,andSmith:MBQCusingsolidstatespins
(aprocessthatmayfailmanytimesbeforesucceeding)and spatially remote qubits for measurement-based protocols
thentransferthatentanglementtothesecondqubitswhich means that many of the aspects of solid state qubits that
areactuallystoringthelargescalegraphstate.Thustheop- present major headaches for circuit-based schemes, such
ticallyactivequbitsactasentanglement‘brokers’,insulat- as low qubit yield and the need to fully characterise each
ingtheirpartner‘client’qubitsfromthemanyfailuresthey qubit,canbeaccommodated.
mayencounterbeforesuccessfullybecomingentangled.A Some drawbacks remain, however. The most serious
simplifiedformofthisprocessisdepictedinFig.7.Sub- of these is that there are many different uncontrolled in-
sequent work has shown that two qubits also suffice for teractions between the qubit and the various degrees of
certain kinds of entanglement distillation, including pro- freedomthatexistinitsenvironment.Thesecanleadtoa
cedurestocombatphasenoise[39]andcorruptiondueto numberofproblems.First,thecouplingbetweenthequbit
photonloss[19].Ifthephysicalsystemissuchthatmore spin state and its surroundings cause a leaking of quan-
thantwoqubitsexistateachsite,thentherearepotentially tum information known as decoherence. Second, optical
further advantages such as general entanglement distilla- transitionscanfluctuateinenergyonawiderangeoftime
tion[40,41]. scales as a result of noise in the local environment and
causeanuncertainphaserelationbetweenremotecentres.
Third, a loss of fidelity in single qubit rotations which,
like decoherence, occurs due to fluctuations in spin tran-
a initial state b c sitionenergiesresultingfrommagneticnoise,butpresents
anadditionalsourceoferrors.Controlofphotonsisalsoa
challenge.Anopticalcavitywouldbeusedtoimprovethe
photoncollectionefficiencybutabsorption,scatteringand
interfaceeffectsinsolidscanlimittheirfinesse.Thispro-
vides an additional source of inhomogeneity between the
qubitcontrolmodules,whichcanaffectthesuccessofthe
e final state d path erasure protocol. Spurious ‘dark’ counts in the pho-
todetectorsintroduceerrorsbyfalselyheraldingsuccessin
entanglementoperations.However,existingsinglephoton
counting technology, particularly for wavelengths in the
range400nm<λ<1µmusingsilicondevices,provides
darkcountprobabilitiesoforder10−8 inatypicalsponta-
neous emittion lifetime, whilst simultaneously benefiting
from the highest detection efficiencies of up to 70%, and
Figure 7 Creating a graph state through brokering [38]. Each
sothissourceoferrorscanprobablybeneglectedrelative
physicalnodehastwoqubits,theopticallyactivebroker(green)
totheothersmentionedabove.Theaggregateofallthese
and the passive client (blue). The role of the client qubits is to
errorsmustbekeptwithinthelimitsimposedbytheover-
holdthelargescalegraphstate.Herewewishtoreachfinalstate
allfaulttoleranceofthescheme,typicallyapercentorless
(e)frominitialstate(a)withoutriskingthelossoftheexistingen-
[42,43].
tanglement.Thebrokerqubitsareprojectedintoentangledpairs
Wenextdiscussthemostimportantenvironmentalin-
(b)sothatthedesirednewentanglement‘edges’existinthebro-
ker space. Now the brokers are entangled with their client by teractionmechanismswhichmayoccurforopticallymea-
deterministiclocaloperations(c),andfinally(d)thebrokersare sured spin qubits, before introducing the physics relevant
measuredoutinthey-basisinordertoprojecttheirentanglement toopticalcavitiessuitableforsolidstatequbitsemittingin
ontotheclients. thevisibleornearinfrared.Weshallfocusinparticularon
twophysicalexamples,NVcentresindiamondandIII-V
semiconductorquantumdots,whichwillbethemaintopic
ofsection4.
3. SolidstatearchitecturesforMBQC:
Generalconsiderations 3.1. Decoherenceandrelatedeffects
Inmanyways,solidstatesystemsofferanidealrealization The discussion of decoherence is broken down into three
ofmeasurementbasedquantumcomputing.Whereasfree subsections.Thefirsttwoinvolvespindecoherence,thatis
ionsneedacomplexarrayofelectromagnetsandultra-low tosaydecoherenceinthespinbasis,resultingfrominter-
temperaturesinordertobepositionedforaccuratecontrol, actionswithlatticevibrationsandwithotherneighbouring
solidstatequbitsarestationaryandcanbeindividuallyad- spins. Such decoherence mechanisms are important not
dressedevenatroomtemperature.Nano-fabricationtech- onlytoMBQCbuttoanyspin-basedquantumcomputing
niques also allow for the creation of arrays of qubits, or architecture,andhavebeenthoroughlyreviewedinrecent
theisolationofasinglequbit,architecturesthatmightboth worksbyHansonetal.[44,45])soweshallnotdwelllong
beusedinameasurement-baseddevice.Thesuitabilityof onthemhere.Thethirdsubsectionconcernsdecoherence
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Laser&Photon.Rev.1,No.1(2008) 13
intheopticallyexcitedstate|e(cid:105),whichisofparticularrel- butions give pure dephasing. The T and T are then re-
1 2
evance to measurement-based quantum computing archi- lated:
tectures. 1 1 1
= + , (12)
T2 2T1 T2pdp
where 1 is the pure dephasing contribution. Interest-
Tpdp
3.1.1. Latticevibrations ingly,for2electronspinsinquantumdots,thepuredephas-
ingcontributionispredictedtobezero,tolowestorderin
thespin-orbitcoupling[48].
Latticevibrations(phonons)arenotmagneticparticlesand
thereforedonotdirectlycoupletospin.Rather,theycause
small electric field fluctuations that would not couple to 3.1.2. Otherspins
spinatall,ifitwerenotforthespin-orbitinteraction.This
The other major headache for the designer of spin-based
linkstogetherthespinandspatialdegreesoffreedomsuch
quantumcomputersistheexistenceofotherspinsthatdo
thatachangingaspineigenstatewilltypicallyalsomean
notformpartofthequantumprocessor.Eventhoughthey
changing the orbital wavefunction slightly. The coupling
arenotpartofthecomputerarchitecture,spinsinthesur-
originatesfromtheeffectofanelectron,whichischarged,
roundingmatrixcannonethelessinteractwithspinqubits
movingthroughtheelectricfieldofthesurroundingcrystal
anddegradetheinformationcontainedwithinthem.There
environment. This motion bends the orbit of an electron,
are two types: electronic and nuclear. Of course, the nu-
leading to a magnetic moment that can interact with the
clear spin has a much smaller magnetic moment than the
spin magnetic moment. The size of the interaction varies
electronspin–buteachtypecanplayanimportantrole.
considerably;itcanbeverysmallincertainmaterials(e.g.
Let us first consider nuclear spins. In many candidate
in diamond, spin-orbit interaction is weak enough to be
solid state systems, particular examples being the III-V
neglibleformostpurposes)-butquitelargeinothers(e.g.
andII-VIsemiconductors,almostallthenucleihaveamag-
inGaAsweshallseethatitisveryimportant).
netic moment. The dominant interaction with an electron
There are two different classes of spin decoherence:
is the hyperfine coupling, which occurs when the elec-
relaxation and dephasing; any error process that occurs
tron wavefunction has a significant amplitude at the po-
withinasinglequbitHilbertspacecanbewritteninterms
sition of the atomic nucleus. In the solid state, an elec-
of these two classes. Relaxation is spin depolarization -
tron is often quite delocalized: for example in a quantum
i.e. spin up states change to spin down, with a change in
dot it can overlap with tens of thousands of nuclei. Such
energy,andviceversa–untilathermalequilibriumisset
a system of many nuclei and just a single electron spin
up.Dephasingdescribesthelossofphasecoherenceofthe
is a highly complex many-body problem, but happily the
spinstate–i.e.therelativephaseofspinupandspindown
physics can often be described well using a huge simpli-
componentsofthewavefunctionbecomesscrambled;this
fication:thatalloftheelectron-nuclearhyperfineinterac-
type of decoherence does not involve a change in energy
tionscanbethoughtofasasingleeffectivemagneticfield,
andthereforeusuallyoccursonashortertimescale.
theOverhauserfield,actingontheelectron.Thevalueand
Therelaxationrate1/T1canbecalculatedusingFermi’s direction of this field is not normally known in an exper-
GoldenRule,whichtellsusthattheratewilldependonthe iment and the Zeeman splitting of electron spin in an ap-
electron-phonon matrix element that couples the spin up plied magnetic field is changed. This change in the field
andspindownlevels,aswellasonthedensityofphonon causes an unwanted extra phase to accumulate in a spin
states.Therearetwowaysinwhichphononscanproduce superposition, and if the phase is unknown this is equiv-
electric field fluctuations, and each has a different matrix alent to dephasing. However, some experimental groups
element. First, deformation of the crystal lattice can lead have found several ways round this problem. First, a ‘re-
directly to band gap modification, and second the strain focusing’ pulse sequence can be applied to the electron
caused by phonons can give rise to electric fields in po- spin, causing it to flip its direction by 180 degrees, and
larsystemsthroughpiezoelectriccoupling.Pluggingallof theextraphasetounwind.Thiscausesa‘spinecho’once
thefactorsintoFermi’sGoldenRulegivesarelaxationrate the phase has fully reversed, and the phase coherence is
thatdependsontheZeemansplittingofthespinsublevels, restored. This kind of inhomogeneous dephasing is not
and so on the applied magnetic field B. The dependence then a true decoherence, but it is often a limiting factor
is B5 for piezo-electric coupling and as B7 for deforma- andthetimescaleoverwhichisacts,calledT∗,canbeas
2
tion potential coupling. Hence this decoherence mecha- shortasafewnanoseconds[57].Anotherwayaroundthe
nism is much more important at high fields, where it can Overhauserfieldproblemistocontrolitbypolarizingthe
reducetherelaxationtimetolessthanatenthofamillisec- nuclear spin bath; recent impressive measurements have
ond [46,47]. shownthatthiscanindeedbeachievedbyclevermanipu-
Any T process automatically creates a dephasing, or lationoftheelectronspinqubititself[58,59].
1
T , channel. However, on top of this there are additional Evenwiththeseingenioustricks,nuclearhyperfinecou-
2
ways in which 1/T can become larger, and these contri- pling can still be a problem, since the nuclei themselves
2
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14 Benjamin,Lovett,andSmith:MBQCusingsolidstatespins
fluctuate,albeitonaratherlongertimescale.Thistypically Thoughthis isreally a T typeprocess, itcan looklike a
1
limitsthephasecoherenceofspinqubitstotimesoforder T process since the bare energy levels of the qubit sys-
2
microseconds [57]. For this reason, it is a very attractive temare‘dressed’byphotonsunderlaserexcitation.Ittyp-
proposition to turn to materials where the most common icallygivesadephasingonthesub-nanosecondtimescale,
isotopes have zero nuclear spin, for example silicon and though this can be increased by using adiabatic methods
carbon. If this can not be achieved, substantial improve- foreigenstatefollowing [49,50,51].Thedecoherencetime
mentsinT maybeobservedbyusingelectronswithvery ofcoursesetsanupperlimitonthetimethatalasercanbe
2
small wavefunction amplitude at the nuclei to reduce the applied to the qubit, if quantum coherence is to be pre-
coupling;thiswouldbetypicalofthep-symmetryofhole served.
statesinIII-Vsemiconductors. A particularly strong phonon interaction can occur in
Ifnuclearspindecoherencecanbeovercomeorelimi- certain nanosystems, such as crystal defects, that are ac-
nated,itcanbetheelectronspinsassociatedwithimpuri- companied by local distortion of the crystal lattice. Such
tiesthatarethemainspindecoherencechannel.Thenum- distortionmayleadtolocalphononmodes,asopposedto
berofdefectelectronspinsinasampleistypicallymany the more commonly discussed bulk modes, which have a
orders of magnitude smaller than the number of nuclei. muchlargeramplitudeatthequbitthantheirbulkcousins,
However, this does not make electron spin defects harm- sothatcouplingisincreased.Theyarealsomorestrongly
less; their dipole moment is two thousand times that of a confined,whichdiscretizestheirexcitationspectrum.Itis
typicalnucleus,andthereforethedipole-dipoleinteraction then more profitable to consider these quantized levels to
withanelectronspinqubitcanbequitesignificant.Unlike bepartofthelevelstructureofthequbit[52],andspecial
withnuclearspins,thereisalsothepossibilityofdirectres- measures must be taken to avoid populating those levels
onant‘flip-flop’transitionsofadefectspinwiththequbit whichhavesignificantphononcharacter.
electron spin itself. Together, these effects can cause de- Iftheenergiesoftheelectricdipoletransitionenergies
phasingtimesonthenanosecondscale.Fortunately,there in two nanostructures are not equal, then this will intro-
are again ways around this. In particular, polarization of duceanextraphasetermintotheresultingentangledspin
thedefectspins,whichforelectronscanbeachievedsim- state. However if the arrival of the photon can be mea-
plybyapplyingalargemagneticfieldatlowtemperature, sured with a timing resolution faster than that in which
can increase the decoherence time by several orders of the phase changes significantly, this error source can be
magnitude [60]. suppressed [53]. The best timing resolution for high effi-
ciency single photon detectors is currently of order 50 ps
suggestingthatthemismatchshouldnotbeanymorethan
3.1.3. Decoherenceandenergyfluctuationsinthe a few tens of micro-electron volts. The additional infor-
opticallyexcitedstate mationrecordedusingfastdetectionisalsoabletoremove
theproblemsofothermismatchedparameters,forexample
Wehavesofarconsidereddecoherenceprocessesthatoc- transitiondipolestrength[36,37].
curdirectlyonthespinqubits.However,inmeasurement Electricdipoletransitionenergiescanalsobeaffected
-basedschemestheoperationsthatgeneratespinentangle- byrandomlyfluctuatinglocalelectricfieldscausedbythe
ment often involve excitation outside of the qubit Hilbert movement of carriers. In well designed experiments with
spaceandthentheprincipaldecoherencemechanismsare high quality samples and minimal extraneous excitation,
likelytobedifferent. the time scale of such fluctuations can be greater than T
1
Forexample,manyideasrelyonphotonemissionand fortheopticaltransitionandsodonottoaffectthecoher-
a higher energy state must be excited optically as part of enceofemittedphotons,asdemonstratedbySantorietal
the measurement process. We must take account of any in their 2002 report of indistinguishable photons emitted
new mechanisms that occur whilst such an optical exci- from a single quantum dot [54]. However optical transi-
tation is being performed. To lowest order, such a tran- tion line widths measured in absorption and PLE experi-
sition would need to be electric dipole allowed - mean- ments are often somewhat larger than the relaxation rate
ing that a direct coupling to the electric field fluctuations 1/T [55,56]suggestingthatovermillisecondtimescales
1
caused by phonons is possible. A detailed calculation of some‘spectraldrift’isencountered.Thismustbeavoided
the open system dynamics can be done using a density in order to permit the sustained energetic resonance be-
matrix master equation technique [49], and this reveals tweentheopticaltransitionsofindividualqubitsthatwill
thatthedecoherenceratedependsonseveralfactors.First, bevitalto‘patherasure’andthustohighfidelityentangle-
thespectraldensityofthephononcoupling,whichcharac- mentgeneration.
terizes the density of states and coupling strength of the
phonons at a particular energy is crucial, with the rele-
vantenergycorrespondingtotheRabifrequencythatchar-
3.2. Photoncontrolinthesolidstate
acterizes the rate of excitation of the higher level. Sec-
ond,thenumberofphonons,whichofcoursedependson
temperature through the Bose-Einstein distribution func- Theopticalcavitiesdepictedinthetitlefigurearenotes-
tion,affectsbothphononabsorptionandphononemission. sential to the measurement-based entanglement schemes
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