Global operations for protected quantum memories in atomic spin lattices

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by  G. K. Brennen

Globaloperationsforprotectedquantummemoriesinatomicspinlattices
G. K. Brennen1, K. Hammerer3, L. Jiang2, M. D. Lukin2, and P. Zoller3
1 CentreforQuantumInformationScienceandSecurity,MacquarieUniversity,2109, NSWAustralia
2Physics Department, Harvard University, Cambridge, MA 02138, USA and
3Institute for Theoretical Physics, University of Innsbruck,
andInstituteforQuantumOpticsandQuantumInformationoftheAustrianAcademyofScience,6020Innsbruck,Austria
Quantuminformationprocessedinstronglycorrelatedstatesofmattercanprovidebuiltinhardwareprotec-
tionagainsterrors. Wemayencodeinformationinhighlynonlocaldegreesoffreedom, suchasusingthree
dimensionalspinlatticesforsubsystemcodesortwodimensionalspinlatticesfortopologicallyorderedsurface
codesandmeasurementbasedcodes. Recently,in[L.Jiangetal.,NaturePhysics4,482(2008)]theauthors
showedhowtomanipulatetheseglobaldegreesoffreedomusingopticallatticescoupledtoabosonicdegree
offreedomviaacavity. Weelaborateontheseideasandrecapitulatetwoapproachestoimplementmanybody
gatesnecessaryforquantuminformationprocessing,bothrelyingoncontrolledinteractionsofanancillarycav-
9 itymodewiththespinsystemandsingleancillaparticles.Themainfocusofthepresentpaperistoanalyzethe
0 effectofimperfectionssuchacavitydecayandcollectiveandindividualspindecoherence.Wepresentstrategies
0 tofightdecoherencebymonitoringcavitydecayandshowthathighgatefidelitiescanbeachievedinthestrong
2 couplingregimeofcavity-QEDwithstateoftheartparameters.

n
a PACSnumbers:03.67.Lx,75.10.Jm,37.10.Jk,37.30.+i
J

5
I. INTRODUCTION andinvolvessinglephotonexcitationofthecavitymode. The
2
otherisageometricphasegateandrequirescoherentexcita-

] tionsofthecavitymodealongwithconditionalphaserotation
h One way to store quantum information is to prepare it in
controlledbytheancillaqubit.
p thedegenerategroundsubspaceofamanybodyHamiltonian.
While all interactions of atomic (or molecular) spins and
- The preparation will be robust if the logical states are stable
t thecavityfieldareobtainedinthedispersivelimit,cavityde-
n underenvironmentalnoise. Thiscanbeestablishedifthereis
cayandspontaneousemissionwillnecessarilyaffectthegate
a agapδEbetweenthecodespaceanditsorthocomplementand
u fidelities. Inthepresentpaperwecomplementourrecentpro-
ifthelogicaloperatorsthemselvesaresufficientlynonlocalin
q posal[8]andprovideadetailedanalysisofthefidelityofgate
nature. Insuchacase,atsufficientlylowtemperatureT,there
[ willbeasuppressionofindividualerrorsbyafactore−δE/kBT operations in the presence of decoherence. We identify the
requirementsonphysicalparametersofthesystemandshow
1 and an accumulation of errors that leads to an unrecoverable
v logical error will be unlikely. Models for quantum memo- thathighgatefidelitiescanbeachievedinthestrongcoupling
0 regime of cavity-QED with state of the art parameters. Sig-
riesandcomputationsbasedonsuchcodeshavebeenputfor-
2 nificantexperimentalprogresshasbeenmadetowardcoherent
ward in [1, 2, 3]. Means to realize the necessary spin lattice
9 controlinthisregime[9,10,11].Wealsoidentifysomeadap-
3 Hamiltonians, carryingtheprotecteddegenerategroundsub-
tiveprotocols,basedoncountingofphotonsleakingoutofthe
. space,existinsystemsofinteractingatomsorpolarmolecules
1 cavity,whichcanimprovethegatefidelityinthepresenceof
trapped in optical lattices [4, 5, 6]. Yet manipulating such
0 cavitydecay.
codesisnontrivialfortheveryreasonthatthestatesareonly
9
The paper is organized as follows. In Section II (supple-
0 coupledbyglobaloperations. Thatis,gatesonsinglelogical
mented by Appendix ) we introduce examples of protected
: qubitsarerealizedbymulti-qubitgates,whicharenotoriously
v quantum memories and describe their possible implementa-
hardtoimplementphysically. Otherstrategiesforadigitized
i
X simulation of many body Hamiltonians use quantum circuits tions in optical lattices. In Section III we present the two
gateoperations,thesinglephotonprotocolandthegeometric
r or entanglement assistance for teleporation of gates (see e.g.
a Ref. [7]). However,thequantumcircuitapproachoftentakes phasegate.SectionIV(supplementedbyAppendix)provides
a comprehensive treatment of the impact of cavity decay on
the system outside of the code space mid-circuit and the en-
bothprotocolsalongwithadiscussionofstrategiesbasedon
tanglementassistedgatesrequirempartiteGHZstatestosim-
monitoringthecavitydecay. SectionVfinallytreatsotherde-
ulateasinglesummandofanmbodyHamiltonian.
coherenceeffects, inparticularindividualandcollectivespin
InRef. [8]theauthorsaddressedthischallengesuggesting
decay. Weconcludewithasummaryoftheresults.
amethodtoimplementtherequiredmulti-qubitgateswithul-
tracoldatomsormoleculesinanopticallatticeembeddedina
high-qualityopticalresonator(seeFig. 1). Thekeyideaisto
achieve controlled, collective interactionsof selected subsets II. PROTECTEDQUANTUMMEMORIES
ofspinswiththecavitymodeinterspersedwithinteractionsof
asingleancillaparticlecoupledtothesamecavitymodealong We begin with a brief review of the basic properties of
withmeasurementsoftheancillaqubit. Usingthesetoolstwo ground state quantum memories in three models: subsystem
types of gates can be constructed: one is based on the idea codes, surface codes, and spin−1 chains used for measure-
to teleport a gate from the ancilla qubit to the encoded qubit mentbasedcomputation.
2
B. Surfacecodes
Considerthe2Dlatticewhereeachedgeofthelatticerepre-
sentsthelocationofaspin-1/2particlewithacouplinggraph
such that particles on edges which meet a vertex interact via
H =Π σxandedgeswhichsurroundafaceinteractvia
v j∈star(v) j
H =Π σz. The operators H and H always collide on
f j∈∂f j v f
an even number of edges and hence commute. Furthermore,
theyassumeeigenvalues±1andinformationcanbeencoded
in the +1 coeigenspace of these operators. By choosing the
so-calledsurface-codeHamiltonian[1,2]:
H =−U∑H −J∑H , (2)
surf v f
v f
with U,J > 0 the code space corresponds to the ground
subspace with degeneracy that depends on the topology:
dimH =22g+h where g is the genus of the surface and h is
gr
FIG. 1: Schematic depicting a step in implementing a many body thenumberofholes[2]. Designinglatticeswithgenusg>0,
operationontrappedatomsormolecularspinsmediatedbyacavity. e.g., on the surface of a torus, would be challenging, but al-
Thelasersforopticaltrappingareindicatedasistheregionofmax- ternativelyitwouldbepossibletocreateseveralholes(h>0)
imum cavity field strength. Planes of spins can be moved into the inaplanarlatticewithboundarybyforinstancedeactivating
strongcavitycouplingregionbybystretchingthelatticealongone regionsofthelatticewithfocusedfardetunedlasers. Further-
dimension(herealongzˆ)bye.g. rotatingthezˆtrappingbeamsaway
more,theplanarcodewithroughboundariescanalsoprovide
fromthezˆaxisbyanangleζ. Thelatticebeamsareassumedtuned
atwofoldgroundstatedegeneracyof[13]. Sincetheyarein-
farfromcavityresonancesothatthemirrorsaretransparentatthat
sensitive to local perturbations, the degenerate ground states
frequency. Onecanalsouseadditionallaserswithshapedintensity
provide a good quantum memory. A caveat is that thermal
profiles[21,22]tomakeaparticularregionC ofthelatticeoptically
fragility of the topological order implies a threshold temper-
activesuchthatonlyspinsinthatregioncaninteractwiththecavity
mode. ature Tthres above which logical information decoheres. This
temperaturescalesasT ∼δ /lnn[15]wherenisthelin-
thresh E
eardimension.
A. Subsystemcodes The code states are coupled by the logical operators: Z =
Π σz,andX =Π σx wheretheconfigurationsC (C )
j∈CZ j j∈CX j Z X
We focus on a specific subsystem code, the three dimen- arestringsonthelattice(duallattice)asillustratedinFig.3.A
sionalspinlatticemodelstudiedbyBacon[3]alsoknownas morerealistictwobodyHamiltoniandescribedbyanisotropic
the 3D compass model. The system is comprised of qubits nearest neighbor Ising like interactions on honeycomb cou-
residingontheverticesofann×n×nsimplecubiclattice(n pling graph was proposed by Kitaev [14] which in a certain
odd),asillustratedinFig. 2. TheinteractionHamiltonianis: parameter regime yields an effective Hamiltonian unitarily
equivalenttoH .
H =−J(cid:0) ∑ σxσx+ ∑ σyσy+ ∑ σzσz(cid:1), (1) surf
cp
x−links y−links z−links
where J >0. We label the single qubit operators σα for C. Groundcodecomputing
i,j,k
σα at site x = i,y = j,z = k. It is unclear how the gap of
thissystemscaleswithsizethoughnumericalevidenceforthe Another means of processing information in many body
2D compass model suggests that the gap scales as 2−n [12]. statesisviagroundcodemeasurementbasedquantumcompu-
Nevertheless, one could encode in a small system to get the tation(GMQC).Heretheinformationisprocessedindegener-
benefitofgroundstateprotection. ategroundstatesofagappedHamiltonianoveratwodimen-
Stabilizeroperatorsaregeneratedby2(n−1)operatorsas- sional lattice of spins, and computation flows by sequential
sociated with adjacent planes, including σx operators in the measurementontheconstituentspins. ItwasshowninRefs.
xˆ−yˆplaneandσzoperatorsintheyˆ−zˆplane: [16,17]thatitsufficestohaveanearestneighboronlyinter-
actionbetweenspin−1particlestorealizeGMQCwithsingle
n n
andtwospinmeasurements. Intheprotocolof[17],eachlog-
VX = ∏ σx σx , VZ = ∏ σz σz .
i i,j,k i+1,j,k k i,j,k i,j,k+1 icalqubitisstoredandprocesssinginthegroundstatesofspin
j,k=1 i,j=1
chainwithspin−1particlesinthebulkandspin−1/2particles
Wecanencodeonequbitofinformationinthegroundstates ontheboundariesinteractingviatheso-calledAKLTHamil-
of Hcp, with logical operators X =∏nj,k=1σx1,j,k (product of tonian[18]
operatorsintheyˆ−zˆplane)andZ=∏n σz , productof
i,j=1 i,j,1 N−1
operators in the xˆ−yˆ plane (see Fig. 2). Properties of this H =J[∑ P2 +P3/2+P3/2 ]
AKLT j,j+1 0,1 N,N+1
subsystemcodearediscussedinAppendixA.
j=1
3
with J >0. Here PS is the projector onto the spin-S ir-
j,j+1
reducible representation of the total spin for particles j and
j+1,i.e.P2 =1(S ·S +1(S ·S )2)+19 andP3/2=
j,j+1 2 j j+1 3 j j+1 3 j,j(cid:48)
23(16+sj·Sj(cid:48)), where S,s are spin-1, 1/2 representations of
su(2). ThegroundstateofH isnon-degeneratebutafter
AKLT
3/2
turningofftheinteractionP andmeasuringthequbitatlo-
0,1
cation 0, the chain is initialized into a logical state of a two
fold degenerate ground subspace of H(cid:48) =H −P3/2.
AKLT AKLT 0,1
Computation flows by measuring spins along the the chain
andpairsofneighboringspinsinparallelchainswiththefinal
outputstateonthelastqubitlocatedatN+1. Suchamodel
is protected by the hardware from errors but for fault toler-
ant computation it may be useful to employ multiple chains
for each logical qubit using a quantum error correction pro-
tocol where the state of the last qubit of one chain is tele-
portedintothegroundstatesofafreshlypreparedchain. The FIG.2: Acubiclatticewithanisotropicnearestneighborinteraction
groundstatesofH(cid:48) areconnectedbythestringoperators encodingonelogicalqubitinn3 physicalqubits. Spinsinteractac-
AKLT
Σµ=eiπ∑Nk=1Skµ⊗σµN+1. Hence, giventhestringinitializedin cpoarirdionfgsptoinsaiHnadmireilcttoinoinasnαH=cpxw,yi,tzh.Tσhαiσsαsyitnetmeraccatniobnesimampolenmgeenatcehd
thestateinducedbymeasuringthequbitatposition0 in|↓ (cid:105)
B B0 e.g.usingmicrowaveinduceddipole-dipoleinteractionsbetweenpo-
wecanteleportthelogicalstateofqubitAintoBviatheuni- larmoleculesasshownin[5].TheplanaroperatorsLX andLZwhich
taryU=|0(cid:105)(cid:104)0|⊗1+|1(cid:105)(cid:104)1|⊗Σx(j). WedescribeinSec. IIIC involveaproductofσx andσz operationsalongthe1respecti1veyˆ−zˆ
howthiscouldbedonebymappingthestateofthelastqubit andxˆ−yˆplanesactaslogicalX andZoperationsonthecode.
ofonechaintoaphoton,thenallowingthephotontointeract
withthenewchaintogeneratethemanybodyoperatorU.
E. Physicalimplementation
D. Universaloperations
Methodsforanaloguesimulationofthethreemodelsabove
have been proposed using trapped atoms [4, 19, 20] or
Intheremainderofthispaperweconsiderhowtogenerate molecules [5, 6] in optical lattices. For trapping of polar
unitaryevolutionbythemanybodyoperators molecules with lattice spacings of ∼200 nm, the interaction
strength using microwave induced dipole-dipole interactions
can be as high as J ∼2π×10 kHz with decoherence dom-
Sζ = ∏σζ, (3) inated by spontaneous emission of optical photons at a rate
C j
j∈C γ∼2π×1 Hz. Such small lattice spacings are possible us-
ing trapping lasers tuned to the properly chosen molecular
transitions [6]. The implementation H will not be exact.
where C is some set of spins and |C|=m. We may be in- imp
There will be spurious longer ranger interactions and devia-
terestedinperformingstringoperatorsonsurfacecodes[8]in
tionsfromtherequiredsymmetryofthenearestneighborin-
whichcaseC isasetwhichdefinesaconnectedstringofspins
teraction. Yet such deviations need not break the code. For
in a 2D lattice (see Fig. 3), or perhaps we wish wish to per-
example, the latter two models H and H have a gap
formencodedoperationsonthe[[n3,1,n]]subsystemcodein surf AKLT
whichprovidesforsomeresiliencetoimperfectimplementa-
whichcaseC isaplaneofspins(seeFig.2).Onemethoduses
tion. Furthermore, for an implementation of H with polar
asinglephotontogeneratethemanybodyinteractionandthe cp
molecules, the microwave fields that induce the interactions
otherusesageometricphasegate.
arelinearlypolarizedandtheerrorsaretheformofproducts
GiventheabilitytogeneratearbitraryrotationseiφSCX,eiφSCZ ofpairsofPaulioperatorswhichhencepreservethetimere-
andtheCNOTgate,andmeasurementsofZ,universalquan- versalsymmetryofthemodel.Forthisreason,thedegeneracy
tum computation can be achieved. For both surface codes ofthecodestatesispreservedandthegapconditionforsuffi-
andsubsystemcodesdiscussedabove,thelogicalCNOTop- cientlysmallsystems,canbemaintainedprovidedthedevia-
eration can be done transversally between two code blocks. tionsaresmall. Justhowsmalldependsonadetailedcompu-
The logical operator Z (or X) is a global operator that cou- tation of the energy gap to excited states. By optimizing the
ples multiple spins over the encoding block. The measure- microwavebeamsthatinducethedipole-dipoleinteraction,its
ment of Z can be achieved by measuring individual physical found that for a 4 spin configuration on a 3D trine, the devi-
spins. However, itisanon-trivialtasktoperformthelogical ation of the implemented Hamiltonian H to the target H
imp cp
rotationseiφSCZ. Thecenteraimofthispaperistoprovidetwo is: ||Himp−Hcp||<10−4J, where the norm is defined as the
approachestoperformsuchlogicalrotations. supremumnormofthetracelesspartoftheoperator[5].
4
excited states could be electronic excited states in the case
C of an optical cavity or rotational excited states in the case of
X
a microwave cavity. If, for example, the photonic mode is
σ+ polarized then there will be a differential shift on the |0(cid:105)
v and|1(cid:105)spinstatesofthepolarmoleculesduetothedifferent
f angular momentum coupling coefficients c0,1 for the ground
C to excited state transitions [6]. Up to a constant this inter-
Z action is equivalent to V where g ∼(c2−c2)d22πωc with
Z Z 0 1 2V∆
V the effective mode volume of the cavity and d the optical
dipolemomentofthepolarmolecule. Theancillaryparticle,
withadifferentstatespacesuchthatonlyonestate|1(cid:105) inter-
A
acted with the cavity mode could be brought into and out of
the cavity with optical tweezers to generate evolution byV .
A
Recently,adispersiveinteractionbetweenatomsinanoptical
latticeandanopticalcavitywasproposedasawaytomeasure
quantumphases[24].Therethecavityfrequencyischosenfar
FIG.3:Aplanarcodewhichencodesonelogicalqubitintheground offatomicresonancesuchthattheinteractionisofthetypein
states.Thereisaspin-1/2particle(filleddot)foreachedgeofthelat-
V except that there is no spin dependence of the atoms so
Z
tice. TheinteractionsofthelocalHamiltonianHsurfarealongedges that the atomic number operator rather than σz is measured.
thatboundaface f, andedgesthatmeetatavertexv. Thestrings
When coupling via an optical cavity it will be important to
CX,Z indicatepathsofproductsofσx,zoperatorsthatarelogicalop-
choosethelatticespacingalongthecavityaxiscommensurate
eratorsonthecode.
withthecavityspatialmodespacinginordertoensureequal
couplingtoallspins.
III. IMPLEMENTATIONS Itwouldbeadvantageoustohaveawaytoturnonandoff
the coupling between the spins and the cavity. This is pos-
Wefocusonphysicalsystemsinwhichanopticallatticeis sibleusingcavityassistedRamanpulses. Heretheideaisto
placed within a high-finesse optical cavity. We assume that introduce an auxillary classical field Ω detuned by ∆ from a
we can control the coupling between the cavity mode and a transition g→e and have the cavity field detuned by ∆+δ
selectedsetofspinsintheopticallattice. Suchselectivema- from a different transition g(cid:48) →e. For ∆(cid:29)Ω,γ,g where γ
nipulationcanbeachievedusingacontrollaserbeamwithap- isthelinewidthoftheexcitedstate,thenthereisanACstark
propriatelyshapedintensityprofile[8,21,22]. Alternatively, |geff|2/δ with geff =Ωg(1/∆+1/(∆+δ))/2, and the auxil-
ifselectedspinsformasimplepattern,suchasoccupyingall iary field can turn the coupling on and off. It is difficult to
thesitesinastraightlineoraplane,wecanstretchthelattice find such a closed optical transition in polar molecules ow-
[23]andcouplethecavitymodewiththespinsonlylocatedat ingtothecomplexstatespacewhichtendstocoupleladders
certainlineorplane. ofvibrationallevels;althoughnearlyclosedtransitionsdoex-
Weassumeaspindependentdispersivecouplingofourlat- ist(see[25]). Itispossibletofindmicrowavecavityassisted
ticetoacavitymode: Ramanprocessesforexamplebychoosingthecavityfieldto
betunedneartheN=1→N=2transitionandtheauxiliary
V =g a†a∑σz, (4) classicalfieldontheN=0→N=2.Thelatterwouldhaveto
Z Z j
j∈C beastrongfieldtocoupleviaquadrapoletransitions.Depend-
ingontheintermolecularspacing,itmaybenecessarytouse
whereg isthedispersivecouplingstrength,andC istheset
Z atleasttwocavityfrequencestoobtainonlysinglespininter-
ofselectedspinsthatcanbeassociatedwithstringoperatorfor
actions without inducing spurious dipole-dipole interactions
thetoriccodeorAKLTmodel,orplanaroperatorforthesub-
(seemethodsof[5]).
systemcode. Inaddition,itwillbeconvenienttohaveaspin
dependentcouplingofasingleancillaryspintothecavity:
V =g a†a|1(cid:105) (cid:104)1| , (5) A. Singlephotonprotocol
A A A A
withcouplingstrengthg ,whichcanbeactivatedbybringing Manybodygatescanbegeneratedwithasinglecavityex-
A
anancillaryparticleintocontactwiththecavitymode,allow- citationcouplingtothelattice.Thebasicideaistoteleportthe
ingthemtointeractfromsometime(withoutinteractingwith quantumgatefromtheprobequbittotheencodedqubitofthe
thespindegreesoffreedomofthesystemparticles),andthen latticespins.Onechoicefortheprobequbitcanbethephoton
de-activatedbyremovingtheancillaryparticle. numberstatesofthecavitymode,withzerooroneexcitation.
The dispersive coupling of the atoms or molecules to the Alternatively, we may introduce an ancilla spin as our probe
cavity can be achieved via a state dependent AC stark shift qubit,whichcouplestothelatticespinsviathecommoncav-
[Fig. 4(a)]. Whenthecavityfrequencyω isdetunedby∆far itymode. Intherestofthediscussion,wewillusethesecond
c
off resonant from the excited states, then there will be a dis- choice, with the advantage that it is more convenient to ma-
persive interaction that is photon number conserving. These nipulatetheancillaspincomparedtothephotonnumberstates
5
(a) (b)
e A vac |+(cid:105) • Xθ (cid:13)(cid:13)
'
e
|Ψ(cid:105) / X X˜ |Ψ(cid:105)
g' θ
g A t (cid:62)(cid:61)
(cid:23)(cid:16)(cid:22)(cid:17)(cid:21)(cid:18)(cid:20)(cid:19)
1 a† vac FIG.5: Gateteleportationcircuitforarbitraryx-rotationX˜θ=eiθSCX
0 1 0 vac 1 vac A on the memory. The circuit represents the following procedure:
A A (1) use the CNOT gate Λ(cid:2)X˜(cid:3) to entangle the probe qubit (upper
(c) line) and the memory (lower line with a slash), (2) projectively
measure the probe qubit in a rotated basis, and (3) perform an en-
coded Pauli X gate over the memory conditioned on the measure-
ment outcome. An analogous circuit (using Λ(cid:2)Z˜(cid:3) and classically
controlledPauliZ gate)implementsZ˜θ=eiθSCZ,hencebyEulerde-
(1) (2) (3) (4) composition, any gate can be formed on the memory using three
controlled operations. For the geometric phase gate scheme, we
can actually implement rotations of the encoded qubit without the
probe qubit, e.g., x-rotation of the encoded qubit can be decom-
(cid:16) (cid:17) (cid:16) (cid:17)
posedaseiθX˜ =D(−β)D −αeiπ2X˜ D(β)D αeiπ2X˜ ,andchoosing
|αβ|=θ.
FIG. 4: Cavity-assisted gate based on single photon approach. (a)
Theenergylevelsofaselectedmemoryspin(|0(cid:105)and|1(cid:105))interacting
dispersively with the cavity mode, which implementsthe Hamilto- (cid:113)
nianwithspindependentdispersiveinteraction.Thecouplingcoeffi- (cid:0)g(cid:48)|1(cid:105) ⊗|vac(cid:105)−Ω(t)|1(cid:48)(cid:105) ⊗a†|vac(cid:105)(cid:1)/ |g(cid:48)|2+|Ω(t)|2,
A A
cientisgZ=g2/∆,withsingle-photonRabifrequencyganddetun- where g(cid:48) is the single-photon Rabi frequency. Since |Λ(t)(cid:105)
ing∆fromtheexcitedstate|e(cid:105). (b)Theenergylevelsoftheancilla approaches |1(cid:48)(cid:105) ⊗a†|vac(cid:105) for Ω(t) (cid:29) g(cid:48), we effectively
spin(differentfrommemoryspins)andthecavitymodeforthesin- A
transfer the state |1(cid:105) ⊗|vac(cid:105) to −|1(cid:48)(cid:105) ⊗a†|vac(cid:105). Mean-
glephotonapproach. AdifferentcontrollaserwithRabifrequency A A
ΩA(t)connectsthestates|1(cid:105)A⊗|vac(cid:105)and|1(cid:48)(cid:105)A⊗a†|vac(cid:105),anden- while, nothing happens if the initial state is |0(cid:105)A⊗|vac(cid:105).
ablescoherentcreationandabsorptionofacavityphotonconditioned Sinceadiabaticstatetransferisacoherentprocess,therelative
ontheancillaspin. (c)Cartoonillustrationoftheprocedureforthe phase between |0(cid:105) ⊗|vac(cid:105) and |Λ(t)(cid:105) is maintained. Thus,
A
implementationofsingle-photonapproachforcontrolledmanybody wecreateanentangledstatebewteentheancillaspinandthe
gate: (1)Initializetheancillaspin(thelefthighlightedspin)inasu- cavitymode
perpositionstateα|0(cid:105) +β|1(cid:105) (bluefor|0(cid:105) andredfor|1(cid:105) ),with
A A A A
nophotoninthecavityandstate|ψ(cid:105)S forthetopologicalmemory. √1 (|0(cid:105) ⊗|vac(cid:105)−(cid:12)(cid:12)1(cid:48)(cid:11) ⊗a†|vac(cid:105)).
(2)Coherentlycreateacavityphoton(orangeshade)forancillaspin 2 A A
state|1(cid:105) (upperbranch);nophotoniscreatedforancillaspinstate
A
|0(cid:105) (lowerbranch). (3)Switchontheinteractionbetweenthecav- Similarly, we can reverse the state transfer (from −|1(cid:48)(cid:105) ⊗
A A
ityphotonandtheselectedspins. Ifthereisacavityphoton(orange a†|vac(cid:105) back to |1 (cid:105)⊗|vac(cid:105)) by adiabatically decreasing the
A
shade), a non-trivial evolution SCz (pink dots) is implemented. (4) RabifrequencyΩ(t).
Turnofftheinteractionandcoherentlyabsorbthecavityphotoninto
Usingthetechniquedescribedabove,wecanimplementthe
theancillaspin. Finallythestateα|0(cid:105) ⊗|ψ(cid:105) +β|1(cid:105) ⊗Sz |ψ(cid:105) is
A S A C S controlledmany-bodyoperations[Fig. 4(c)]asthefollowing:
prepared.
• Theancillaspin(probequbit)startswithstateα|0(cid:105) +
A
β|1(cid:105) , the cavity mode has no photon |vac(cid:105), and the
A
topologicalmemoryisinstate|ψ(cid:105) .
ofthecavitymode. S
The procedure of teleporting the quantum gate is summa- • Then we adiabatically turn on the control laser Rabi
rizedinFig.5. First,wecoupletheprobequbitandthelattice frequency Ω(t) and coherently create a cavity photon
spins. Then, we perform the quantum gate over the probe if the ancilla spin starts in |1(cid:105) , so we obtain the state
A
qubit. Finally,wemeasuretheprobequbit,whichdetermines α|0(cid:105) ⊗|vac(cid:105)−β|1(cid:48)(cid:105) ⊗a†|vac(cid:105).
A A
thePauliframeoftheencodedqubitofthelatticespins. Af-
tertheseoperations,themanybodygatehasbeeneffectively • Next we switch on the interaction between the cavity
appliedtothelatticespins. photonandselectedspinsfortimeτ, duringwhichthe
topologicalmemoryundergoesanevolutionSz ifthere
Before we give the procedure of coupling the ancilla C
spin and the lattice spins, we first describe a technique isonephotoninthecavity(i.e. a†|vac(cid:105)).
that allows use to entangle the ancilla spin and the cav-
• Finally, weswitchofftheinteractionandadiabatically
ity mode, using the energy levels shown in Fig. 4(b). transfer−|1(cid:48)(cid:105) ⊗a†|vac(cid:105)backto|1(cid:105) ⊗|vac(cid:105).
Suppose the ancilla spin and cavity mode starts with A A
state |1(cid:105) ⊗|vac(cid:105). When we slowly increase the Rabi After these four steps, the cavity mode restores the initial
A
frequency of the control laser Ω(t), the ancilla spin and state |vac(cid:105), while the ancilla spin and the topological mem-
cavity mode adiabatically follow the dark state |Λ(t)(cid:105) = ory evolve from (α|0(cid:105) +β|1(cid:105) )⊗|ψ(cid:105) to α|0(cid:105) ⊗|ψ(cid:105) +
A A S A S
6
β|1(cid:105) ⊗(−i)mSz |ψ(cid:105) , which is the controlled operation up • Performthesequenceofsteps:
A C S
toaknownphase.
By choosing Sz as the encoded Z operator, we implement U =D(−β/2)e−it7VAD(β/2)e−it6VAe−it5VZ
C
thecontrolled-Zgatebetweentheancillaandthelatticespins, ×D(−α)e−it4VZD(β/2)e−it3VAD(−β/2)
whichcanbeconvertedintotheCNOTgatebyconjugatingall
latticespins withthe Hadamardgate. Therefore, we canim- ×e−it2VAe−it1VZD(α),
plementallthequantumgatesappearedinthecircuitinFig.5
which returns the cavity to the vacuum state. Choose
andachieveunitaryevolutionofmanybodyoperators.
parameterssatisfying: −g t =g t =−g t =g t =
A 2 A 3 A 6 A 7
π,−g t =g t =−g t =π/2,and|αβ|= π mod2π.
Z 1 Z 4 Z 5 2
ThedisplacementoperatorsaregeneratedbyV andwe
B
B. Geometricphasegate pick the relative phase φ=argα−argβ=π/2(0) for
thenumberofspinsm=|C|even(odd). Reversedevo-
AnalternativetousingasingleFockexcitationofthecavity lutionduringtheinteractiontimestepst ,t ,t ,t canbe
1 2 5 6
mode is to use coherent state control to perform a geometric achievedbysettingg →−g ,e.g. bychangingthe
Z,A Z,A
phase gate [26]. Since these operations use only linear opti- signofthedetuning.
cal elements, they may be much easier to realize in experi-
Assuming the system state was initially |ψ(cid:105) the joint
S
ment. Themechanismmakesuseoftwobasicoperators, the
stateofsystemandancillaisnow
displacement operator D(α)=eαa†−α∗a and the rotation op-
eratorR(θ)=eiθa†a whichsatisfytherelations: D(β)D(α)= U|ψ(cid:105) |+ (cid:105) = |ψ(cid:105)S|0(cid:105)A+f√(m)SCz|ψ(cid:105)S|1(cid:105)A,
S y A
eiℑ(βα∗)/2D(α+β), andR(θ)D(α)R(−θ)=D(αeiθ). Putting 2
theseprimitivestogether,onecanrealizeanevolution
m m−1
where f(m)=(−1)2 formevenand f(m)=(−1) 2
formodd.
e−iHintt =D(−βB)R(θC)D(−α)R(−θC)
×D(βB)R(θC)D(α)R(−θC), (6) • Measu√re the ancilla in the basis |±x(cid:105)A = (|0(cid:105)A ±
|1(cid:105) )/ 2. The probabilities for the measurement out-
A
accordingtoaneffectiveHamiltonian comesx=±1are
1
Hintt=|αβ|sin(θC+φ), (7) P(sx=±1)= 2(1±f(m)(cid:104)SCz(cid:105)).
whereφ=arg(α)−arg(β)andCisanarbitraryoperatorcom-
Thisprotocolcanbeusedtomeasuretheproductoperators
muting with cavity operators a and a†. Picking in particular Sx Sz,whereC,C(cid:48)arepossiblyoverlappingconfigurationsof
θ=π/2andC=∑j∈Cσζj,whichrequiresdrivingofthecav- spCi(cid:48)nsC,viatheite√ratedsequence: [∏i∈C(cid:48)Hi]U[∏i∈C(cid:48)Hi]U with
itywithcoherentfieldsandinteractionsofthecavitywiththe Hi =eiπ2(σxi+σzi)/ 2) the Hadamard gate on qubit i. Measure-
spinsystemaccordingtoEq. 4,thesimulatedHamiltonianis
ment errors can be ameliorated by redundifying the state of
the ancilla before measurement using many, possible faulty,
Hintt=sin(φ+mπ/2)|αβ|∏σζj. (8) CNOTgatesofflinebetweenthatancillaandmanyotherspre-
j∈C paredinstate|0(cid:105).
Hence by choosingC=∑m σζ and φ=0, for the case
j,k=1 1,j,k
m odd, we can simulate evolution generated by Sζ. What is C. Gatesonspin−1particles
C
requiredisaspindependentcouplingbetweeneachqubiton
theplaneandthebosonicchannel. Weconcludethissectionwithabriefdiscussionofgenerat-
ingmanybodygatesforteleportationofquantuminformation
A non destructive measurement of the many body opera-
torSz ispossibleusingtheassistanceofanancillaryparticle in spin−1 chains as prefaced in Sec. IIC above. This could
C be done in the context of a lattice embedded in a cavity by
A that also couples the bosonic channel in a spin dependent
mapping the state of the qubit located at position N+1 of
manner. Controlleddisplacementscanbeimplementedbythe
logical chain A to the state of a photon and using the same
sequence
procedure as described above. The goal is to then gener-
ate the unitary operatorU =|0(cid:105)(cid:104)0|⊗1+|1(cid:105)(cid:104)1|⊗Σz, where
|0(cid:105)A(cid:104)0|⊗1+|1(cid:105)A(cid:104)1|⊗D×(Dβ)(−=βD/2(β)R/2(−)Rπ(aπ†aa†|a1|(cid:105)1A(cid:105)(cid:104)A1(cid:104)|1).|) (9) Σ|↓µB0=(cid:105).eTiπh∑iNks=1isSkµg⊗enσeµNra+te1d, obnyaanneidwencthicaainlpBroicneitdiaulriezetdotihnasttfaoter
controlledstringoperatorsonqubitsbutinsteadofEq. 4,we
Theprotocolformeasurementof(cid:104)SCz(cid:105)isasfollows use
N
• Prepare the an√cillary particle in the state |+y(cid:105)A = V =g a†a(∑|Sx=0(cid:105) (cid:104)Sx=0|+σx ), (10)
X Z j N+1
(|0(cid:105) +i|1(cid:105) )/ 2,andthefieldstateinthevacuum.
A A j=1
7
whereg isthedispersivecouplingstrength.Thiskindofstate In order to evaluate the performance of many body oper-
Z
dependentinteractionbetweenaphotonandspin−1particles ations mediated by the cavity we will calculate the process
canberealizedusinge.g. polarizationtodifferentiallycouple fidelityforimplementingamanybodygateU=e−iχSCz where
theinternalstatesoftheparticles. Sz =∏ σz on a configuration C of m=|C| spins. Many
C j∈C j
body measurements of the type (cid:104)Sz(cid:105) are obtained using the
C
many body gates with rotation angle χ=π/2 as a primitive.
IV. GATEFIDELITYWITHCAVITYDECAY
The details are given in the following subsections with the
mainresultthattheexpectederrorscaleslikeκ/|g |.
Z
Cavityfielddecayatarateκactsasasourceoferrorforthe
many body interactions which it mediates. For the protocols
abovewherethesystemofspinsinteractwiththecavityfield, A. Singlephotonmediatedgate
thejointstatecanbedecomposedas
Consider a protocol where we begin with the separable
(cid:16)
ρ(t)= ∑ σΛ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48) |Λ,J,MJ(cid:105)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)| state:
Λ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48) ρ(0)=ρ (0)⊗ρ (0)
(cid:17) S F
⊗|αMJ(cid:105)(cid:104)βMJ(cid:48)| (t), (11) = ∑ σΛ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48)AMJ,MJ(cid:48)(0)
Λ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48)
where Λ,J,M are labels corresponding to Λ labeled irreps
J
with total angular momentum J and Jz projection M (see where
J
[fi2e7ld]),staantdesσ|Λα,JM,MJ(cid:105)J,,Λ|β(cid:48),JM(cid:48),JM(cid:105)J(cid:48),=maTyr[dρe(p0e)n|Λd(cid:48),oJn(cid:48),tMheJ(cid:48)(cid:105)a(cid:104)nΛgu,Jla,rMmJ|o].mTehne- AMJ,MJ(cid:48)(0)=|Λ,J,MJ(cid:105)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|⊗|yθ+=π4(cid:105)(cid:104)yθ+=π4|,
tumprojections. Wedescribeevolutionforthecasewherethe andthefieldstatesaredefined
fieldstatesareFockstatesforthesinglephotonprobeproto-
colandwhentheyarecoherentstatesforthegeometricphase |yθ+(cid:105)=cosθ+|0(cid:105)+sinθ+|1(cid:105), |yθ−(cid:105)=|yθ++π/2(cid:105).
gate. Depending on the protocol used, the field states them-
Herethestates|0(cid:105),|1(cid:105)coulddenotephotonnumberinagiven
selvesmaybeentangledwiththestateofanancillaryspinbut
mode or a single photon in two orthonormal modes where
wefocusoncomputingfidelitiesforevolutionstepswherethe
onlymode|1(cid:105)interactswiththespins.
ancillaisnoninteracting.
Thediscussionherecanbegeneralizedtothecasethatthe
Considertheevolutionduringanatomfieldcouplingstage.
cavity mode is also entangled with an ancilla spin, with the
Theequationofmotionforthejointstateis mappingfrom{|0(cid:105),|1(cid:105)}to(cid:8)|0(cid:105) ⊗|vac(cid:105),−|1(cid:48)(cid:105) ⊗a†|vac(cid:105)(cid:9).
A A
Inprinciple, themappingdoesnotholdforthecavitydecay,
ρ˙(t)=L(ρ(t))
because the decay process without the ancilla spin (from |1(cid:105)
κ
=−i[V ,ρ(t)]+ (2aρ(t)a†−a†aρ(t)−ρ(t)a†a). to|0(cid:105))cannotbemappedtothedecayprocesswithanentan-
Z
2 gledancilla(from|1(cid:48)(cid:105) ⊗a†|vac(cid:105)to|1(cid:48)(cid:105) ⊗|vac(cid:105)). However,
(12) A A
ifweareonlyinterestedinthefidelityofthelatticespins,the
analysishereisstillvalid,becausethereduceddensitymatri-
TheevolutionconservesthequantumnumbersΛ,J hencewe
cancomputetheactioneLtAMJ,MJ(cid:48)(0)where: cesforthelatticespins(aftertracingoutthecavitymodeand
theancillaspin)arethesameforbothcases.
AMJ,MJ(cid:48)(t)≡|Λ,J,MJ(cid:105)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|(t)⊗|αMJ(cid:105)(cid:104)βMJ(cid:48)|(t). 13Foratomfieldcouplingoveraperiodt, wehavefromEq.
Thesolutionsareeasilyverifiedtobegivenby AMJ,MJ(cid:48)(t) = |Λ,J,MJ(cid:105)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|⊗21(cid:16)(1+bMJ,MJ(cid:48)(t))|0(cid:105)(cid:104)0|
AMJ,MJ(cid:48)(t)= ∑∞ bnMJMJ(cid:48)(t)e−(i2gZMJ+κ/2)a†at +e(i2gZMJ(cid:48)−κ/2)t|0(cid:105)(cid:104)1|+e(−(cid:17)i2gZMJ−κ/2)t|1(cid:105)(cid:104)0|
n=0 n! +e(−i2gZ(MJ−MJ(cid:48))−κ)t|1(cid:105)(cid:104)1| .
×anAMJ,MJ(cid:48)(0)(a†)ne(i2gZMJ(cid:48)−κ/2)a†at (13)
Ifwechoosetheinteractiontimeg τ= π,thentheactionon
Z 2
thestateis
where
(cid:16) (cid:17) ρ(τ) = eLτρ(0)
κ 1−e−[κ+i2gZ(MJ−MJ(cid:48))]t
1(cid:16)
bMJMJ(cid:48)(t)= κ+i2gZ(MJ−MJ(cid:48)) . (14) = 2 ρS(0)⊗|0(cid:105)(cid:104)0|+P(ρS(0))⊗|0(cid:105)(cid:104)0|
+e−κτ/2ρ (0)eiπJz⊗|0(cid:105)(cid:104)1|
Theevolvedstateisthen S
+e−κτ/2e−iπJzρ (0)⊗|1(cid:105)(cid:104)0|
ρ(t)=eLtρ(0)= ∑ σΛ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48)AMJ,MJ(cid:48)(t). +e−κτe−iπJzρ (S0)eiπJz⊗|1(cid:105)(cid:104)1|(cid:17), (15)
Λ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48) S
8
where with
P(ρS(0))= ∑ σΛ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48)bMJ,MJ(cid:48)(t) VMJ,MJ(cid:48) =(cosχ+sinχe−iπMJ)(cosχ+sinχeiπMJ(cid:48))).
Λ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48)
×|Λ,J,M (cid:105)(cid:104)Λ(cid:48),J(cid:48),M(cid:48)|. WhereastheactionofthemapE onthesamebasisbymulti-
J J plicationbyacomplexnumber:
Anidealmanybodygateresultsifκ=0andwemeasurethe
photoninarotatedbasis.Considerthecasewherethenumber E(ρS(0))= ∑ σΛ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48)QMJ,MJ(cid:48)
ofspins,misodd.Ifwemeasurethephotoninthebasis|yθ+(cid:105), Λ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48)
then we obtain outcome ±1 with equal probabilities p± = 21 ×|Λ,J,MJ(cid:105)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|,
andtheresultantstateis
whereifwefixθ =χsuchthatweapproximatetheunitary
Tr[ρ(τ)|θ (cid:105)(cid:104)θ |] +
ρ±(τ) = ± ± U then
S Tr[·]
= e−iθ±(−i)m−1SCzρS(0)eiθ±(−i)m−1SCz. QMJ,MJ(cid:48) = 12(cid:104)cos2χ(1+bMJ,MJ(cid:48)(τ))+sinχcosχe−πκ/4|gZ|
Saythetargetevolutionoperatorise−iθ+(−i)m−1SCz butweob- ×(e−iπMJ+eiπMJ(cid:48))+sin2χe−πκ/2|gZ|e−iπ(MJ−MJ(cid:48))
tfaoirnatphpelymineagstuhreemloecnatllryesguelnte−ra1t.edSuucnhitaanryoSuztc.oTmheeicsacsoerrwehcteerde +e−iπ(MJ−MJ(cid:48))(cid:0)sin2χ(1+bMJ,MJ(cid:48)(τ))−sinχcosχ
misevenishandledinasimilarwaybutweCmeasurethepho- ×e−πκ/4|gZ|(e−iπMJ+eiπMJ(cid:48))+cos2χ
toninthebasis ×e−πκ/2|gZ|e−iπ(MJ−MJ(cid:48))(cid:1)(cid:1)(cid:105).
|x (cid:105)=cosθ |0(cid:105)+isinθ |1(cid:105), |x (cid:105)=−i|x (cid:105).
θ+ + + θ− θ++π/2
Tthheecmonedaistuiorenmalesnttatoeuitscoρm±Se(τs)±=1ea−rieθ±aigmaSCzinρSe(q0u)iepirθo±bimaSbCzl.eTanhde VOMfTJc,hMoeJ(cid:48)u.prsreocfeosrstfihdeelciatyseisorfeandoildyeccoamy,pκute=d0u,sitnhgenthQefMaJc,MttJ(cid:48)h=at
correctionproceduregiventheoutcome−1isasbefore. thenoisemapE(ρ (0))commuteswiththetargetunitaryU.
S
WedenoteE themapforimperfectimplementationofthe Hence,wecancomputethefidelitywhichmeasureshowclose
gateU =e−iχSCz inthecaseofnonzeroκ. Forourconditional the noisy map E(cid:48)(ρS(0))=U†E(ρS(0))U is to the ideal op-
protocolthiscanberepresentedas: eration,i.e. theidentityoperation:
E(ρS(0))=p+S+(ρS(0))+p−SCzS−(ρS(0))SCz, (16) Fpro(E,U)=Fpro(E(cid:48),I)=S,S(cid:48) (cid:104)Φ+|ρE(cid:48)|Φ+(cid:105)S,S(cid:48).
where
HerewearecomputingtheoverlapoftheJamiołkowski-Choi
1 (cid:16) (cid:17) representations of the maps as states in the Hilbert space
S (ρ (0))= X(θ )ρ (0)X(θ )†+P(ρ (0) .
± S 2p± ± S ± S HS⊗HS(cid:48) containing our system space and a copy each with
dimensionD:
We assume the product of local unitaries Sz can be imple-
C 1
mentedwithperfectfidelity. |Φ+(cid:105)S,S(cid:48) = √ ∑ |Λ,J,MJ(cid:105)S⊗|Λ,J,MJ(cid:105)S(cid:48),
For the remainder of this subsection, we fix m odd and D
Λ,J,MJ
mmamnnoedr.4F=rom1.ETq.he15owtheerhacvaeses follow in a straightforward ρE(cid:48) =IS⊗ES(cid:48)(cid:48)(|Φ+(cid:105)S,S(cid:48))
1
X(θ±)=cosθ±1+sinθ±e−κτ/2e−iπJz. = DΛ,J,MJ∑,Λ(cid:48),J(cid:48),MJ(cid:48)QMJ,MJ(cid:48)VM∗J,MJ(cid:48)
Noticethatnowtheprobabilitiesformeasuringthephotonin ×|Λ,J,MJ(cid:105)S(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|⊗|Λ,J,MJ(cid:105)S(cid:48)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|.
|θ (cid:105)arenotnecessarilyequal,ratheronefinds
±
Hence
p −p =(1−e−κτ)cos(2θ ).
+ − +
1 J J(cid:48)
TheprocessfidelityFpro(E,U)measureshowcloseaquantum Fpro(E,U)= D2 ∑ ∑ ∑ ∑ RMJ,MJ(cid:48),
operationE istotheidealoperationU asmeasuredbysome Λ,Λ(cid:48)J,J(cid:48)MJ=−JMJ(cid:48)=−J(cid:48)
suitable metric. The fidelity measure we use is the overlap
where
betweentheinducedJamiołkowski-Choistaterepresentations
oftheoperations[28].Theactionoftheunitaryonacomplete
operator basis {|Λ,J,MJ(cid:105)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|} is multiplication by a RMJ,MJ(cid:48) =QMJ,MJ(cid:48)VM∗J,MJ(cid:48).
unimodularnumber:
Now R = 0, and the for off diagonal elements, for
MJ,MJ
U|Λ,J,MJ(cid:105)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|U†=VMJ,MJ(cid:48)|Λ,J,MJ(cid:105)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|, κ/|gZ|(cid:28)1wefind,
9

ℜ[RMJ,MJ(cid:48)]≈11−−(cid:16)π2π22|gκ+Z|(+−1π)4(222M((MJ2+gκJ−Z1))M/22J(cid:48)s)in(4χ)(cid:17)2|gκZ|+(cid:16)(−1)(42M(MJ+J−1)M/2J(cid:48)s)in(4χ)+2c(oMs(J4−χM)+J(cid:48))12 +π2(3+1c6os(4χ))(cid:17)(2gκZ)2 MMJJ−−MMJJ(cid:48)(cid:48) oevdedn.
(17)
The above expression is made a bit simpler by counting cm, • Ifnophotonisdetected,thenmeasurethephotoninside
J
thenumberofinequivalentspin−J irrepsofthemfoldsym- the cavity in the rotated basis |χ (cid:105) and if the outcome
±
metrized direct product of SU(2), where m is the number of is|χ (cid:105)applylocalgateSz asinSec. IIIA. End.
− C
spin−1/2particlesinthesystem. Thedimensioncanbecom-
putingusingYoungtableau[29]: • Ifaphotonisdetectedattimet applythelocalcorrec-
tiongateW (t). Repeat.
corr
(cid:18) (cid:19)
2J+1 m
cm= .
J m+J+1 m−J Intheaboveprotocol,theunitaryU isapproximatedaftera
2 2 sequenceofm−1clicksandafinalnullcount,aneventwhich
MakinguseofthefactthatQMJ(cid:48),MJ =Q∗MJ,MJ(cid:48),thefidelityis occurswithprobability:
Z τ Z τ Z τ
1 m/2 J J(cid:48) pm= dtmpnull(tm) ··· dtm−1...dt1pdet(tm)
Fpro(E,U)= 22mJ,J∑(cid:48)=1/2cmJcmJ(cid:48)MJ∑=−JMJ(cid:48)∑=−J(cid:48)ℜ[RMJ,MJ(cid:48)]. =(|0gZ|)m( πκ −0eπκ/2|g0Z|+1)( πκ +eπκ/2|gZ|−1)m−1,
πκ 2|g | 2|g |
Z Z
FromEq. 17wefindalowerboundforthefidelity:
wheretheintegrationmeasureisdt =1/τand
π κ k
F (E,U)>1− . (18)
pro
22|gZ| 1+e−κtk
p (t )= , p (t )=1−p (t ).
null k det k null k
2
1. Protocolwithadetector Assuming for simplicity that the total time to detect a pho-
ton, apply the correction gate and re-prepare a probe photon
The above computation of fidelity may be overly pes- isroughlyτ,thenthemeant¯ andvariance∆t ofthetime
gate gate
simistic because one could adopt a strategy where the out- toperformthegateis
put of the cavity is continuously monitored for leakage of a
∞
photon during the coupling and altering the protocol accord-
t¯ =τ ∑ mp
ingly. Consider the situation where we have a perfect pho- gate m
m=1
ton detector outside the cavity that measures the presence
of a leaked photon. In the case of a null result, the sys- = τeπκ/2|gZ|πκ/|gZ| ≤2τ,
temevolvesviaanon-HermitianHamiltonian: ρ(t)= eLtρ(0) eπκ/2|gZ|(πκ/2|gZ|+1)−1
Tr[·] (cid:113)
whereL[ρ(t)]=−i(VZ−iκ2a†a)ρ(t)+iρ(t)(VZ+iκ2a†a).The ∆tgate= t¯2gate−t¯g2ate
operatorbasiselementsthenevolveas (cid:112)
τ eπκ/2|gZ|(eπκ/2|gZ|(πκ/2|gZ|−1)+1)πκ/|gZ|
=
AMJ,MJ(cid:48)(t) = |Λ,J,M (cid:105)(cid:104)Λ(cid:48),J(cid:48),M(cid:48)|⊗ 1 (cid:16)|0(cid:105)(cid:104)0| √ eπκ/2|gZ|(πκ/2|gZ|+1)−1
null J J 1+e−κt ≤ 2τ.
+e(i2gZMJ(cid:48)−κ/2)t|0(cid:105)(cid:104)1|+e(−i2gZMJ−κ/2)t|1(cid:105)(cid:104)0|
(cid:17) Noticethatthesevaluesareboundedabove.
+e(−i2gZ(MJ−MJ(cid:48))−κ)t|1(cid:105)(cid:104)1| , Thefidelityiseasilycalculatedbynotingthattheactionof
each map associated with a detector click is a unitary opera-
whereasinthecaseofdetectedphotonattimet wehave tion which is undone by a correction step. So the only step
in the protocol that acts non trivially is the step with the fi-
AMdeJt,MJ(cid:48)(t) = |Λ,J,MJ(cid:105)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|e−i2gZ(MJ−MJ(cid:48)))t. (19) nalnullcount. Evaluatingthefidelityasbeforebutusingthe
expressionAMJ,MJ(cid:48)(t)wefind
The evolution in the later case is unitary and we can restore null
thesystemtoitspriorstatebyapplyingthelocallygenerated (cid:32) (cid:33)
unitary operator Wcorr(t)=ei2gZtJz. We adopt the following F (E,U)= 1 1+cos2(2χ)+ 2eπκ/4gZ sin2(2χ)
protocoltoimplementU =e−iχSCz: pro 2 1+eπκ/2gZ
π2κ2
• Prepare the ancilla as in Sec. IIIA and let the photon =1− sin2(2χ)+O((κ/2g )4),
interactwiththesystemforatimeτ=π/2gZ. 64g2Z Z
10
wheretheexpansionisvalidforκ/|g |(cid:28)1. field. Using,Eq. 13,thesumbecomesanexponentialandthe
Z
Infactwecandobetter. Usingourknowledgeofthenec- evolvedstateis
essarygatetimeτintheeventofanulldetectionitisadvan-
tageoustopreparetheinitialphotonprobestate ρ(t)=eLtρ(0)
(|0(cid:105)+eπκ/4|gZ||1(cid:105))/(cid:112)1+eπκ/2|gZ|. =Λ,J,MJ∑,Λ(cid:48),J(cid:48),MJ(cid:48)ed(t)σΛ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48)|Λ,J,MJ(cid:105)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|
Evolutionofthebasisstatesis ⊗|e−(igM+κ/2)tαMJ(cid:105)(cid:104)e−(igM(cid:48)+κ/2)tβMJ(cid:48)|, (20)
AMJ,MJ(cid:48)(t) = |Λ,J,M (cid:105)(cid:104)Λ(cid:48),J(cid:48),M(cid:48)|⊗ 1 (cid:16)|0(cid:105)(cid:104)0| where
null J J 1+eπκ/2|gZ|−κt
+e(i2gZMJ(cid:48)t+πκ/4|gZ|−κt/2)|0(cid:105)(cid:104)1| d(t)=αMJβ∗MJ(cid:48)bMJ,MJ(cid:48)(t)−(|αMJ|2+|βMJ(cid:48)|2)1−e2−κt. (21)
+e(−i2gZMJt+πκ/4|gZ|−κt/2)|1(cid:105)(cid:104)0|
Forcompleteness,analternatederivationoftheevolutionus-
(cid:17)
+e(−i2gZ(MJ−MJ(cid:48))t+πκ/2|gZ|−κt)|1(cid:105)(cid:104)1| , ingthecharacteristicequationforthejointstateisgiveinAp-
pendixB.
andAMJ,MJ(cid:48)(t)isasinEq. 19. Afteratimeτanulldetection Weignoredecayduringthedisplacementstagesoftheevo-
det lution (i.e. we assume these are done quickly relative to the
givesthetargetevolutionandthefidelityisone. Asbefore,in
decay rate), and we assume that the system particles do not
the case of a photodetection event, the system evolution can
interactwiththefieldduringthesesteps. Inordertoperform
bereversedandtheprotocolrepeated.
logical operations on the protected momory, we do not need
Nowtheprobabilitiesforanullcountordetectionofapho-
an ancilla and there are seven time steps beginning with the
tonaremodifiedto:
cavityinthevacuumstate:
1+eπκ/2|gZ|−κtk
pnull(tk)= 1+eπκ/2|gZ| , pdet(tk)=1−pnull(tk). D(−β)e−iτ5VZD(−α)eiτ3VZD(β)e−iτ1VZD(α),
The meant¯gate and variance ∆tgate of the time to perform the asdescribedinSec. IIIB. Letτ5=τ3=τ1sothattheperiods
gateis of spin field coupling are all equal in duration. Notice the
change in sign of the evolution during the period τ . This
3
∞ can be accommodated by changing the sign of the coupling
t¯ = τ ∑ mp
gate m parameter g by e.g. changing the sign of field detuning. In
Z
m=1
orderthatthefieldstatereturntothevacuumattheendofthe
= τ(1+eπκ/2|gZ|)πκ/2|gZ|, sequence, we choose α(cid:48)−κτ1,β(cid:48)−κτ1. The total sequence then
πκ/2|gZ|+eπκ/2|gZ|−1 yieldstheoutputstate:
(cid:113)
∆t = t¯2 −t¯2
gate gate gate
= τ(cid:112)(1+eπκ/2|gZ|)(eπκ/2|gZ|(πκ/2gZ−1)+1)πκ/2|gZ|. ρout= ∑ σΛ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48)RMJ,MJ(cid:48)
πκ/2|gZ|+eπκ/2|gZ|−1 Λ,J,MJ,Λ(cid:48),J(cid:48),MJ(cid:48)
×eiχsin(φ+2gZτ1MJ)|Λ,J,MJ(cid:105)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|
The mean and variance of the gate time is now unbounded
with increasing κ but for κ/|g | (cid:28) 1 the values are com- ×e−iχsin(φ+2gZτ1MJ(cid:48))⊗|0(cid:105)(cid:104)0|,
Z
parable to t√he prior case with the photon probe prepared in
(|0(cid:105)+|1(cid:105))/ 2. where we defined RMJ,MJ(cid:48) = ed(t2)+d(t4)+d(t6) and χ =
Forasitutationwithfinitedetectorefficiencyηwhichcan |αβ|(e−3κτ1/2+e−κτ1/2)/2. Thiscanbeinterpretedascoher-
be modeled as a rank 2 projector on the photon, the fidelity
entevolutionwithaneffectiveevolutionoperator
will degrade ultimately to the case of no detector as derived
above. e−iHintT =eiχsin(φ+2gZτ1Jz),
where T is an effective time for the gate, followed by de-
B. Geometricphasegate
phasinginthe{Λ,J,M }basis. Matrixelementsdiagonalin
J
M are invariant. For m even, the parameters g τ = π/2,
J Z 1
In order to evaluate the effect of cavity decay during the |α|=|β|,φ=±π/2,generate
the geometric phase gate, we are particularly interested in
the case where initially AMJ,MJ(cid:48)(0)=|Λ,J,MJ(cid:105)(cid:104)Λ(cid:48),J(cid:48),MJ(cid:48)|⊗ U =exp[∓i(−1)m2|α|2SCz(e−3κτ1/2+e−κτ1/2)/2].
|αMJ(cid:105)(cid:104)βMJ(cid:48)|, with |αMJ(cid:105), |βMJ(cid:48)(cid:105) coherent states. This kind
of factorization is true at any stage of spin coupling to the Thestrengthofthedephasinganddecayisthen,