Heralded quantum gates with integrated error detection in optical cavities

2.2 MB·English

by  J. Borregaard

Heralded quantum gates with integrated error detection in optical cavities
J. Borregaard,1 P. K´om´ar,2 E. M. Kessler,2,3 A. S. Sørensen,1 and M. D. Lukin2
1The Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
2Department of Physics, Harvard University, Cambridge, MA 02138, USA
3ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA
(Dated: March 2, 2015)
Weproposeandanalyzeheraldedquantumgatesbetweenqubitsinopticalcavities. Theyemploy
an auxiliary qubit to report if a successful gate occurred. In this manner, the errors, which would
have corrupted a deterministic gate, are converted into a non-unity probability of success: once
successful the gate has a much higher fidelity than a similar deterministic gate. Specifically, we
describe that a heralded , near-deterministic controlled phase gate (CZ-gate) with the conditional
errorarbitrarilyclosetozeroandthesuccessprobabilitythatapproachesunityasthecooperativity
of the system, C, becomes large. Furthermore, we describe an extension to near-deterministic N-
qubit Toffoli gate with a favorable error scaling. These gates can be directly employed in quantum
repeater networks to facilitate near-ideal entanglement swapping, thus greatly speeding up the
entanglement distribution.
5
1 PACSnumbers: 03.67.Pp,03.67.Hk,03.67.Bg,32.80.Qk,42.50.Pq
0
2
b Exploiting quantum systems for information process- gates can, in principle, also be directly implemented in
e ing offers many potential advantages over classical in- optical cavities [14], but the experimental requirements
F formation processing like highly secure quantum net- forthisareverychallengingduetospontaneousemission

7 works[1–3]andpowerfulquantumcomputers[4–6]. One andcavityloss. Theessentialparameterquantifyingthis
2 of the main challenges for the realization of functional is the cooperativity of the atom-cavity system, C. It has
quantum computers is to perform gates with sufficiently been argued that directly implementing gates in op√tical
]
h high quality so that the remaining errors can be sup- cavities leads to a poor error scaling 1 − F ∝ 1/ C,
p pressed by error correction codes, which makes the com- where F is the fidelity of the gate [15, 16]. However, as
- putation fault tolerant [7]. At the same time, applica- a result of the integrated error detection, the heralded
t
n tions to long distance quantum communication can be gates that we propose exhibit high fidelities when suc-
a
enabled by quantum repeaters, which combine proba- cessful. Thisenablesefficiententanglementswappingand
u
bilisticentanglementgenerationovershortdistanceswith removes the necessity of intermediate entanglement pu-
q
[ subsequent entanglement connection steps [3]. For these rificationinquantumrepeatersthusincreasingthedistri-

protocols, the probabilistic nature of the entanglement butionratesignificantly. Comparedtousingotherdeter-
3
generation is acceptable, but it is essential that high fi- ministic,cavitybasedgates,anincreaseintherateofup
v
6 delity entanglement is achieved conditioned on a herald- to two orders of magnitude can be achieved for modest
5 ing measurement. Experimentally, such high fidelity en- cooperativities (<100) and a distance of 1000 km [17].
9 tanglement is often much easier to implement and may
0
be realized in situations where it is impossible to per- The basic idea is to use a heralding auxiliary atom in
0
form any quantum operations deterministically. Here we addition to qubit atoms in the same cavity. One of the
.
1 introduce a similar concept for gate operations and de- atomic qubit states, e.g., state |1(cid:105) couples to the cav-
0
velop the concept of heralded quantum gates with inte- ity mode while |0(cid:105) is completely uncoupled (see Fig. 1a).
5
grated error detection. In the resulting gate, the infi- Such a system has previously been considered for two-
1
: delity, which would be present for a deterministic gate qubit gates [16, 18–21], multi-qubit gates [18, 22] and
v
is converted into a failure probability, which is heralded photon routing [23]. If any of the qubit atoms is in state
i
X byanauxiliaryatom. Oncesuccessful,theresultinggate |1(cid:105) the cavity resonance is shifted compared to the bare
r can have an arbitrarily small error. Such heralded gates cavity mode, which can be exploited to make a gate be-
a
couldfacilitatefaulttolerantquantumcomputationsince tween two or more qubits by reflecting single photons off
detectable errors may be easier to correct than unde- the cavity [18]. The efficiency of such schemes, however,
tectableerrors[8–10]. Alternativelyitcanbedirectlyin- islimitedbyphotonlosses, inefficientdetectorsandnon-
corporated into quantum repeater architectures for long idealsinglephotonsources[21,23]. Wecircumventthese
distance quantum communication. problems by introducing an auxiliary atom in the cav-
ity to serve as both an intra-cavity photon source and a
Optical cavities are ideal for conversion between the detector. As opposed to previous heralded gates in opti-
stationarygatequbitsandflyingqubits(photons),which calcavities,whichreliedonthenulldetectionofphotons
is fundamental for quantum networks [11–13]. Quantum leaving the cavity [24–26], the final heralding measure-
2
mentontheatomcanthenbeperformedveryefficiently.
Inourapproach,theauxiliaryatomhastwometastabl!e ! ! !
states |g(cid:105),|f(cid:105), which can be coupled throug h an excited
state |E(cid:105) (see Fig. 1a). We assume the |E(cid:105) ↔ |f(cid:105) tran-
!! ! ! ! !! ! ! !
sition to be energetically close to the cavity frequency
g g

and to be a nearly closed transition, so that we needgto g
drivethe|g(cid:105)→|E(cid:105)transition,e.gwithatwo-phoΩto nprfo - ! Ω f !
cess, (see below). The gate can be understood through
the phase evolution imposed on the atoms. We con- ! ! ! !
sider adiabatic excitation of the auxilia(rby) control !atom ! (a) (b) ! ! (a)
viaStimulatedRamanAdiabaticPassage[27,28],driven
by an external driving pulse with Rabi frequency Ω(t) FIG.1. (Coloronline)(a)Levelstructureofthequbitatoms.
and a coupling to the cavity photon g . In the case Only state |1(cid:105) couples to the cavity and we assume that the
f
when all the qubit atoms are in the non-coupled states excited level decays to some level |o˜(cid:105), possible identical to
|f(cid:105) or |0(cid:105). (b) Level structure of the auxiliary atom and the
|00..0(cid:105), an adiabatic excitation will result in a dark state
transitions driven by the weak laser (Ω) and the cavity (g ).
∼g |0,g(cid:105)−Ω|1,f(cid:105)withzeroenergyandvanishingphase. f
f We assume that |E(cid:105)↔|f(cid:105) is a closed transition, i.e. γ =0.
Here the number refers to the number of cavity photons. g
However,thequbitstatesΨwithatleastoneofthequbit
atoms in the coupled state, results in a cavity-induced
decay of the excited state of the auxiliary atom and
shift of the state |1,f,Ψ(cid:105), which in turn, causes an AC Lˆ = √γ|o˜(cid:105) (cid:104)e| describes the decay of the excited qubit
k k
Stark shift and dynamical phase to be imprinted into
states to some arbitrary ground state |o˜(cid:105). The nature of
the |g,Ψ(cid:105) state after the driving pulse is turned off. All
|o˜(cid:105) is not important for the dynamics of the gates and it
states but the completely uncoupled qubit state |00...0(cid:105)
may or may not coincide with |0(cid:105) or |1(cid:105).
will thus acquire a phase, the magnitude of which de-
Weassumeaweakdrivingpulsejustifyingforapertur-
pendsonthelengthofthedrivingpulse. Withanappro- bative treatment of Vˆ using the formalism of Ref. [30].
priate pulse length and simple single qubit rotations, we
In the perturbative description we adiabatically elimi-
can use this to realize a general N-qubit Toffoli gate or
natethecoupledexcitedstatesoftheatomsandthecav-
a control-phase (CZ) gate.
ity (assuming Ω2/∆ (cid:28) ∆ and Ω (cid:28) g), which leads
Naively, the gates will be limited by errors originating E E
to an energy shift of the ground states but otherwise
from cavity decay and spontaneous emission from the
conserves them since the Hamiltonian cannot connect
atoms, which carry away information about the qubit
different unexcited states without decay. The dynam-
state. These errors are, however, detectable since the
ics are therefore described by an effective Hamiltonian,
auxiliary atom will be trapped in state |f(cid:105) if either a Hˆ =|g(cid:105)(cid:104)g|(cid:80) ∆ Pˆ where
cavity excitation or an atomic excitation is lost. Condi- eff n n n
tioningondetectingtheauxiliaryatominstate|g(cid:105)atthe (cid:40) −Ω2((∆e−i/2)i+2nC) (cid:41)
endofthegatethusrulesoutthepossibilityofanydissi- ∆ =Re 4γ γ (2)
pative quantum jumps having occurred during the gate. n (2∆e−i)((2∆E−i)i/4+C)+(2∆E−i)nC
γ γ γ
As a result, the conditional fidelity of the gate is greatly
enhanced at the modest cost of a finite but potentially and Pˆ projects on the states with n qubits in state |1(cid:105).
n
low failure probability. For simplicity, we have assumed that the auxiliary atom
We now analyze the performance of the gates and de- is identical to the qubit atoms such that g = g and
f
rive the success probabilities, gate times and gate errors γ = γ (see [29] for a more general treatment) and we
f
(see Tab. S1 in [29]) The Hamiltonian in a proper rotat- have defined the cooperativity C = g2/γκ. We consider
ing frame is (see Fig. 1) the limit C (cid:29) 1 and from Eq. (2) we find that the en-
Hˆ =∆ |E(cid:105)(cid:104)E|+g (aˆ|E(cid:105)(cid:104)f|+H.c)+Vˆ +Vˆ† ergy shift, in the case when all qubit atoms are in |0(cid:105),
E f
(cid:88) becomes very small ∆0 ∼ ∆EΩ2/(16γ2C2) → 0, i.e., we
+ ∆ |e(cid:105) (cid:104)e|+g(aˆ|e(cid:105) (cid:104)1|+H.c), (1)
e k k drive into a zero energy dark state as mentioned in the
k description above. On the contrary, for n > 0, the C
where k labels the qubit atoms ((cid:126) = 1), 2Vˆ = Ω|E(cid:105)(cid:104)g| in the nominator of ∆ reflects that the coupling of the
n
and we have assumed that all couplings (g,Ω) are real. qubitatomsshiftsthecavityresonanceandasaresultan
We have defined ∆ = ω − ω − ω , and ∆ = AC stark shift of ∼ Ω2/∆ is introduced. Furthermore,
E E g L e E
ω −ω −ω +ω −ω , where ω is the laser frequency we find that in the effective evolution, errors caused by
e g L f 1 L
and otherwise ω is the frequency associated with level spontaneousemissionorcavitydecay(Lˆ ,Lˆ ,Lˆ )project
x 0 f k
x. Wedescribethecavitydecayandatomicspontaneous thesystemoutoftheeffectivespaceintoorthogonalsub-
√
emission with Lindblad operators so that Lˆ = κaˆ cor- spaces, which allows for an efficient error detection by
√ 0
responds to the cavity decay, Lˆ = γ |f(cid:105)(cid:104)E| to the measuring the ancilla atom.
f f
3
The dynamics described by Hˆ can be used to im-
eff
plement a Toffoli gate. Assuming the qubit√atoms to be 1 103 10−3
eorngyresshoinftasn∆cen(>∆0e∼=Ω02)/(a4nγd√hCa)viwnghi∆leE∆∼0 ∼γOC(Ωg2i/vCeds3a/tea2n1)-. 0.8 1tdC−aZPtsa1 10−4 CCC===251000
aHnedncwe,e|0c0a.n..0c(cid:105)hiosotsheeaonglaytesttaitme,ewohficthr∼em4aπin√sCuγn/shΩdia2fttaet3do 1−Ps00..46 γtCZError10−5 C=100
T
data5
make a Toffoli gate. By conditioning on measurindgattah6e 0.2 (a) 10−6 (b)
auxiliaryatominstate|g(cid:105)attheendofthegate, thede-
1
tseiocntaobnlelyerrerdoruscefrtohmescuavcciteyssdpercoabyaabnildityspionnstteaandeooufsre1e20dmuics-- 100 0 101 C 1021100000 10130 10−17 02 ΔE2/γ 103
ing the fidelity. Consequently, the fidelity becomes lim-
FIG.2. (Coloronline)(a)Failureprobability(1−P -leftaxis)
ited by more subtle, undetectable errors (see Ref. [29]). s
andgatetime(t -rightaxis)asafunctionofthecooperativ-
The dominant error originates from the qubit dependent CZ
ity(C)fortheCZgate. Thegatetimeisinunitsoftheinverse
decay rate, Γ , of |g(cid:105) → |f(cid:105). As we demonstrate in
n linewidth1/γ ofthequbitatoms. Wehaveassumedadriving
Ref.[29],thisleadstoafidelitylowerboundedby1−F (cid:46) √
√ of Ω = Cγ/4. (b) Gate error as a function of the detun-
0.3/C, with a success probability of Ps ∼ 1 − 3/ C. ing∆E2 inthetwo-photon-drivenCZ-gateforC =10,20,50,
Thus is a substantial improvement over the leading er- and 100. We have assumed that Ω = 4γC1/4 and that
MW
ror in the case of deterministic cavity-assisted gates. For γ =γ. The gate error decreases as γ2/∆2 and is indepen-
g E2
generic states, the fidelity can even be markedly higher, dent of C. We have assumed Ω∼∆E2/8 resulting in a gate
time ∼ 400/γ. Solid/dashed lines are analytical results and
and improving with increasing particle number N [29]
symbols are numerical simulations (see [29]). For both plots,
In the special case of only two qubits, the Toffoli gate
we have assumed κ=100γ.
is referred to as a CZ-gate, and in this case, we can even
improve the gate to have an arbitrarily small error by
combining it with single qubit rotations. For the gen-
So far, we have assumed a model where there is no
eral Toffoli gate discussed above, we needed ∆ = 0 to
e decayfrom|E(cid:105)→|g(cid:105). Inrealatoms,therewill,however,
ensure the correct phase evolution, but making the sin-
alwaysbesomedecay|E(cid:105)→|g(cid:105)withadecayrateγ >0.
gle qubit transformations |0(cid:105) → e−i∆0t/2|0(cid:105) and |1(cid:105) → g
The result of such an undetectable decay is that both
e−i(∆1−∆0)t/2|0(cid:105), at the end of a driving pulse of length
the CZ-gate and the Toffoli gate will have an error ∼
t =|π/(∆ −2∆ +∆ )|,ensurestherightphaseevo- √
CZ 2 1 0 γ /(γ C). To make this error small, it is thus essential
lution of the CZ-gate without any constraints on ∆ . g
e tosuppressthebranchingratioγ /γ. Belowweshowhow
Hence, it is possible to tune ∆ to eliminate the detri- g
e to suppress γ by driving the |g(cid:105) → |E(cid:105) transition with
mental effect of having a qubit dependent decay rate. g
√ a two photon process. As a result, we realize a CZ gate
Choosing ∆ = γ 4C+1 and ∆ = 1Cγ2/∆ ensures
E 2 e 2 E with an error arbitrary close to zero and a Toffoli gate
Γ = Γ = Γ , and thus removes all dissipative errors
0 1 2 with an error scaling as 1/C even for a realistic atomic
from the heralded gate. The conditional error is then
system.
limited only by non-adiabatic effects, that can in prin-
Specifically we think of a level structure for the aux-
ciple be made arbitrarily small by reducing the driving
√
iliary atom, shown in Fig. 3, where we still assume
strength. The success probability is 1−P ∼ 6/ C in
s
|E(cid:105) ↔ |f(cid:105) to be a closed transition. For simplicity, we
the limit C (cid:29) 1 (see Fig. 2a). We thus have a heralded
have also assumed |E (cid:105) ↔ |g(cid:105) to be a closed transition.
twoqubitgatewitharbitrarilysmallerrorwithasuccess 2
Such a level structure could, e.g. be realized in 87Rb as
probabilitythatcanapproach1(itispossibletodecrease
shown in Fig. 3. We assume that a microwave field cou-
the scaling factor of the probability from ∼6 to ∼3.4 at
ples the two excited states such that we can have a two
theexpenseofanerrorscalingas1/C bytuning∆ ,∆ ).
E e
photontransitionfrom|g(cid:105)→|E(cid:105)andthatΩissmall,al-
We now consider the gate time. The gate time of the
√
lowingforaperturbativetreatmentofthecoupling. Thus
Toffoli gate is t ∼ 4π Cγ/Ω2 and for the CZ-gate we
√T
wecanmapthesystemtoasimplethree-levelatomwith
havet ∼15π Cγ/(2Ω2)forC (cid:29)1. Sincet >t we
CZ CZ T levels |g(cid:105),|E(cid:105) and |f(cid:105) and a decay rate γ˜ and drive Ω˜
focus on t . The gate time is set by the strength (Ω) g
CZ
between|g(cid:105)and|E(cid:105),determinedbythetwophotondriv-
of the driving pulse, which is limited by non-adiabatic
ing process as shown in Fig. 3. The dynamics are thus
errors. This is investigated in the supplemental mate-
similar to what we have already described for the sim-
rial where we also verify our analytical results numeri-
ple three level atom except that we have the extra decay
cally [29]. Assuming realisitc parameters of κ = 100γ √
√
γ˜ that introduces an error in the gates ∼ (γ˜ /γ)/ C,
[23, 31], we find that a driving of Ω = Cγ/4 keeps g g
as previously described. In the limit C (cid:29) 1, we find
the non-adiabatic error of the gate below 4 · 10−5 for
C ≤ 1000. The gate times decreases as 1/√C as shown γ˜g/γ ∼ γ4gγΩ∆2M2W. Thus by increasing ∆E2, we can in prin-
E2
in Fig. 2a. For a cooperativity of 100 the gate time is ciple make these errors arbitrarily small. The error of
≈1 µs for typical atomic decay rates. the CZ-gate for different ∆ is shown in Fig. 2b, as-
E2
4
-ΔE! ! -ΔE ! menetnatnsgl(elmineknst),swwahpicphinagre[36s]u.bBseyquoerngatlnyizcinognntehceterdepuesaitnegr
!!!!!!!! in a tree structure, the probabilistic nature of the gate
Δ! E2 ! ΩMW ! ! ! can be efficiently circumvented. The success rate of dis-
g
! f! tributingentanglementacrossthetotaldistanceL,scales
! Ω~ gf! as ∼ (L/L0)1−log2(3/p), where p < 1 is the success prob-
Ω
ability of the swap, L is the total distribution distance
! ! and L0 is the length between the links [17] (note that in
(a) ! (b) ! thelimitp→1,theaboveexpressionunderestimatesthe
rate, e.g., forp=1theactualrateis∼3timesfasterfor
128links). Thisisasubstantialimprovementoverdirect
FIG. 3. (Color online) (a) Level structure of the auxiliary
transmission where the success rate scales exponentially
atom and the transitions driven by a weak laser (Ω), a mi-
with L. For a realization with nuclear spin memories
crowave field (Ω ) and the cavity (g ). We assume that
MW f
|E(cid:105)↔|f(cid:105)isaclosedtransitionand,forsimplicity,wealsoas- where the swap can be performed deterministically the
sumethat|E2(cid:105)↔|g(cid:105)isaclosedtransitionbutthisisnotane- ratecanscaleevenbetteras∼log2(L/L0)−1. Inorderto
cessity. Here|r(e),r(e)(cid:105)withr=1,2,3referstohowtheatom maintain the favorable scaling without resorting to time
may be realized in the (52P ) states |F(e) = r,m(e) = r(cid:105) consuming purification, the total number of links, N
3/2 max
52S of Rb87. (b) Effective three-level atom realized by shouldbekeptbelowN ∼−ln(F )/((cid:15) +(cid:15) ),where
1/2 max final 0 g
mapping the two-photon drive to give an effective decay rate F is the required fidelity of the final distributed pair
γ˜ and an effective drive Ω˜. final
g and (cid:15) ,(cid:15) (cid:28) 1 are the errors of the initial entanglement
0 g
generation and the entanglement swapping respectively.
Thus, it is essential that the errors are kept small, which
suming an initial state of (|0(cid:105) + |1(cid:105))⊗2. Note that in can be obtained with the heralded gate.
order to prevent an increasing scattering probability of
Inconclusion,wehaveintroducedaheraldedtwo-qubit
level |E2(cid:105), we need to have Ω ∝ C1/4 resulting in a
MW quantumgatewithaconditionalfidelityarbitrarilyclose
gate error that is independent of the cooperativity. [29].
tounityandanN-qubitToffoligatewithanerrorscaling
The success probability and time of the gates are the
as1/C. Thegateshaveabuilt-inerrordetectionprocess,
same as before with Ω → Ω˜ ∼ Ω2M∆WE2Ω. With similar whichremovesthenecessityofextractingtheerrorbythe
considerations about the validity of our perturbation as
morecomplicatedprocessofentanglementpurificationor
before, we find that for realistic parameters, we can use
quantum error correction. Our gate is designed for the
Ω = ∆ /8,Ω ∼ 4γC1/4 resulting in a gate time of
E2 MW specific case of optical cavities, and allows exploiting re-
∼10µsfortypicalatomicdecayratesandC (cid:46)1000[29].
alisticsystemsforquantumcommunication,eventhough
Asanexampleimplementation, weconsiderultra-cold theerrorratewouldinhibitthiswithdeterministicgates.
87Rb atoms coupled to nanophotonic cavities [23, 31]. Similar advantages can be realised in other systems such
There are some additional errors originating from the asthosebasedoncircuitQED,wherecertainerrorscould
extra states in the 87Rb atoms in this case. In Ref. [29], beheraldedandthusalleviatethedauntingrequirements
we treat these errors and find that with a detuning of of fault tolerant computation.
∆ =100γ and a cooperativity of C ≈100, a heralded
E2 We thank Jeff Thompson for helpful discussions and
CZ gate with ∼ 67% success probability and a heralded
gratefully acknowledge the support from the Lundbeck
error of ≈ 10−3 can be realized in ≈ 10 µs time. This
Foundation, NSF, CUA, DARPA, AFOSR MURI, and
justifiesneglectingatomicdecoherencewhichistypically
ARL. The research leading to these results has received
much slower. Alternatively the gate can be implemented
funding from the European Research Council under
withatom-likesolid-statequbitssuchasNVandSiVcen-
the European Union’s Seventh Framework Programme
ters in diamond [32]. These systems can exhibit closed
(FP/2007-2013) / ERC Grant Agreement n. 306576 and
transitionsandlong-livedelectronicspinstateswhichare
through SIQS (grant no. 600645) and QIOS (grant no.
theessentialrequirementforthegate[33],whilehighco-
306576).
operativities are possible in diamond nanocavities [34].
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6
SUPPLEMENTAL MATERIAL
This supplemental material to the article ”Heralded quantum gates with integrated error detection in optical
cavities” describes the details of the perturbation theory and the derivation of the effective Hamiltonian Hˆ and
eff
effective Lindblad operators. We describe the situation both with and without a two-photon drive. Furthermore,
we present the results of a numerical simulation of the full dynamics of the gates to verify the results found with
perturbation theory and address the question of how strong a drive we can allow for. In the end, this determines
the gate time as described in the article. Finally we discuss of the additional errors described in the final part of the
article.
PERTURBATION THEORY
We will now give the details of the perturbation theory and the derivation of the effective operators together with
the success probabilitites, gate times and gate errors (see Tab. S1). Our perturbation theory is based on the effective
operator formalism described in Ref. [30].
Gate Origin of error Error Probability Time
γ =0 0 √
CZ-gate γg >0 ∼ γ√g ∼1− √6C ∼ 15π2Ω2Cγ
g γ C
Γ (cid:54)=Γ (cid:46)0.3 √
Toffoli γi >0j ∼Cγ√g ∼1− √3C ∼ 4πΩ2Cγ
g γ C
TABLE S1. The errors, success probabilities and gate times of the N-qubit Toffoli gate and the CZ-gate considered in the
article. Notethatthebranchingfractionγ /γcanbemadearbitrarilysmallusingafardetunedtwo-photondrivingasexplained
g
inbelow. Γ istherateofdetectableerrorsforthequbitstatewithiqubitsinstate|1(cid:105). ThesuccessprobabilityoftheCZ-gate
i
can be increased at the expense of an error scaling of 1/C as explained in the article.
First, we treat the simplest situation where the auxiliary atom is directly driven to an excited state |E(cid:105) by a weak
classical drive Ω as shown in Fig. S4 (reproduced from Fig. 1 in the article). Note that we allow for some decay from
|E(cid:105)→|g(cid:105)withdecayrateγ asopposedtothesituationinthearticle. Wewilllaterconsiderthesituationwherethis
g
decay rate is suppressed using a two-photon drive. The level structure of the qubit atoms are also shown in Fig. S4.
! !
! !

!! ! ! ! !! ! ! !
g
g

g
Ω f Ω gf
! !
! ! ! !
(b) ! ! (a) (b) ! ! (a)
FIG. S4. (a) Level structure of the qubit atoms. Only state |1(cid:105) couples to the cavity and we assume that the excited level
decaystosomelevel|o˜(cid:105),possibleidenticalto|f(cid:105)or|0(cid:105). (b)Levelstructureoftheauxiliaryatomandthetransitionsdrivenby
the weak laser (Ω) and the cavity (g ). We allow for some decay from |E(cid:105)→|g(cid:105) with decay rate γ .
f g
The Hamilton describing the system in a proper rotating frame is given by Eqs. (1)-(3) in the article and is
7
reproduced here
Hˆ =Hˆ +Vˆ +Vˆ†, (S3)
e
Hˆ =∆ |E(cid:105)(cid:104)E|+g (aˆ|E(cid:105)(cid:104)f|+H.c)
e E f
(cid:88)
+ ∆ |e(cid:105) (cid:104)e|+g(aˆ|e(cid:105) (cid:104)1|+H.c), (S4)
e k k
k
Ω
Vˆ = |E(cid:105)(cid:104)g|, (S5)
2
where we have assumed for simplicty that all couplings (g,Ω) are real and k labels the qubit atoms ((cid:126)=1). We have
defined ∆ =ω −ω −ω , and ∆ =ω −ω −ω +ω −ω where ω is the laser frequency and otherwise ω is
E E g L e e g L f 1 L x
the frequency associated with level x. Note that we assume the cavity frequency to be ω =ω +ω −ω such that
c L g f
we are on resonance with the |g(cid:105)→|E(cid:105)→|f(cid:105) two-photon transition.
√
The dissipation in the system is assumed to be described by Lindblad operators such that Lˆ = κaˆ describes the
√ √ 0
cavity decay with decay rate κ, Lˆ = γ |g(cid:105)(cid:104)E|, and Lˆ = γ |f(cid:105)(cid:104)E| describes the decay of the auxiliary atom and
g g f f
√
Lˆ = γ|o˜(cid:105) (cid:104)e| describes the decay of the qubit atoms (k = 1,2...N). As described in the article |o˜(cid:105) may or may
k i
not coincide with |0(cid:105) or |1(cid:105). Assuming that Ω is weak (Ω2/∆ (cid:28) ∆ and Ω (cid:28) g), we can treat the driving as a
E E
perturbationtothesystem. AsshowninRef.[30],thedynamicsofthesystemisthengovernedbyaneffectivemaster
equation of the form
ρ˙ =i(cid:104)ρ,Hˆ (cid:105)+(cid:88)Lˆeffρ(Lˆeff)†− 1(cid:16)(Lˆeff)†Lˆeffρ+ρ(Lˆeff)†Lˆeff(cid:17), (S6)
eff x x 2 x x x x
x
whereρisthedensitymatrixofthesystem, Hˆ isaneffectiveHamiltonian, andLeff areeffectiveLindbladoperators
eff x
with x=0,g,f,k. The effective operators are found from:
1 (cid:16) (cid:17)
Hˆ =− Vˆ† Hˆ−1 +(Hˆ−1)† Vˆ (S7)
eff 2 NH NH
Lˆeff =Lˆ Hˆ−1Vˆ, (S8)
x x NH
where
Hˆ =Hˆ − i (cid:88)Lˆ†Lˆ , (S9)
NH e 2 x x
x
is the no-jump Hamiltonian. The Hilbert space of the effective operators can be described in the basis of {|g(cid:105),|f(cid:105)}
of the auxiliary atom and the states {|0(cid:105),|1(cid:105),|o˜(cid:105)} of the qubit atoms. To ease the notation, we define the projection
operators Pˆ which projects on to the states with n qubits in state |1(cid:105). From Eq. (S7) and (S8) we then find:
n
Hˆ = (cid:88)N −Ω2Re(cid:40) i∆˜e/2+nC (cid:41)|g(cid:105)(cid:104)g|⊗Pˆ
eff 4γ ∆˜ (i∆˜ /2+C )+∆˜ nC n
n=0 e E f E
N
= (cid:88)∆ |g(cid:105)(cid:104)g|⊗Pˆ (S10)
n n
n=0
Lˆeff = (cid:88)N √1 (cid:112)Cf∆˜eΩ |f(cid:105)(cid:104)g|⊗Pˆ
0 2 γ∆˜ (i∆˜ /2+C )+n∆˜ C n
n=0 e E f E
N
= (cid:88)reff |f(cid:105)(cid:104)g|⊗Pˆ (S11)
0,n n
n=0
√
Lˆeff = (cid:88)N 1 (i∆˜e/2+nC)Ω γg|g(cid:105)(cid:104)g|⊗Pˆ
g 2∆˜ (i∆˜ /2+C )+n∆˜ C γ n
n=0 e E f E
N
= (cid:88)reff |g(cid:105)(cid:104)g|⊗Pˆ (S12)
g,n n
n=0
√
Lˆeff = (cid:88)N 1 (i∆˜e/2+nC)Ω γf|f(cid:105)(cid:104)g|⊗Pˆ
f 2∆˜ (i∆˜ /2+C )+n∆˜ C γ n
n=0 e E f E
8
N
= (cid:88)reff |f(cid:105)(cid:104)g|⊗Pˆ (S13)
f,n n
n=0
√
Lˆeff = (cid:88)N − √1 (cid:112)Cf CΩ |f(cid:105)(cid:104)g|⊗|o˜(cid:105) (cid:104)1|⊗Pˆ
k 2 γ∆˜ (i∆˜ /2+C )+n∆˜ C k n
n=1 e E f E
N
= (cid:88)reff|f(cid:105)(cid:104)g|⊗|o˜(cid:105) (cid:104)1|⊗Pˆ , (S14)
n k n
n=1
where we have defined the cooperativities C =g2 /γκ for the qubit (auxiliary) atoms and the complex detunings
(f) (f)
∆˜ γ =∆ −iγ /2 and ∆˜ γ =∆ −iγ/2. Note that we have defined the parameters reff ,reff ,reff and reff in Eqs.
E E f e e 0,n g,n f,n n
(S11)-(S14) to characterize the decays described by the Lindblad operators. Note that reff = 0. In our calculations
0
we parametrize the difference between the auxiliary atom and the qubit atoms by C = αC and γ = βγ to easier
f f
treat the limit of C (cid:29)1 that we are interested in.
Success probability and fidelity
Eqs. (S11)-(S14) show that the effect of all Lindblad operators, except Lˆeff, is that the state of the auxiliary atom
g
is left in state |f(cid:105). All these errors are thus detectable by measuring the state of the auxiliary atom at the end of the
gate. For the heralded gates where we condition on measuring the auxiliary atom in state |g(cid:105) at the end of the gates,
these detectable decays therefore do not effect the fidelity of the gates but only the success probability. The rate Γ
n
of the detectable decays for a state with n qubits in state |1(cid:105) is Γn =(cid:12)(cid:12)r0eff,n(cid:12)(cid:12)2+(cid:12)(cid:12)(cid:12)rfeff,n(cid:12)(cid:12)(cid:12)2+(cid:12)(cid:12)rneff(cid:12)(cid:12)2 and assuming an initial
qubit state described by density matrix ρ the success probability of the gates is
qubit
N
P = (cid:88)Tr(cid:110)e−Γntgateρ Pˆ (cid:111), (S15)
success qubit n
n=0
where t is the gate time and Tr denotes the trace.
gate
Havingremovedthedetrimentaleffectofthedetectableerrorsbyheraldingonameasurementoftheauxiliaryatom
the fidelity of the gates will be determined by more subtle, undetectable errors (see below). We define the fidelity, F
of the gate as
1
F = (cid:104)ψ|(cid:104)g|ρ˜ |g(cid:105)|ψ(cid:105), (S16)
P qubit
success
where we have assumed that the ideal qubit state after the gate is a pure state |ψ(cid:105) and ρ˜ is the actual density
qubit
matrix of the qubits and the auxiliary atom after the gate operation.
N-qubit Toffoli gate
As shown in the article, the effective Hamiltonian in Eq. (S10) is sufficient to make a Toffoli gate by putting the
qubit atoms on resonance (∆ =0). We will now treat the worst case and average fidelities of the general Toffoli gate
e
referred to in the article. The undetectable errors limiting the fidelities are the following.
√
• As described in the article the energy shifts of the coupled qubit states are all ∆ ∼ Ω2/(4γ C) in the
n>0
limit C (cid:29)1. However, to higher order in C, we find corrections on the order O(Ω2/C3/2) to the energy shifts,
√
which depend on the number of qubits that couples. The gate time of the Toffoli gate is t ∼4π Cγ/Ω2 and
T
consequently,thehigherordercorrectionsgiveunevenphaseshiftsontheorderofO(C−1)forthecoupledqubit
states at the end of the gate. This leads to a phase error in the fidelity of O(C−2).
• The difference between the rates of detectable errors (Γ ) for different qubit states changes the relative weight
n
of the qubit states during the gate. This error wil be O(C−1) as shown below.
• For γ >0 the undetectable decay from |E(cid:105)→|g(cid:105) in the auxiliary atom will destroy the coherence between the
g
qubit states. We find that this error will be ∼ γ√g . For now, we will assume that γg =0 and thus ignore this
γ C
9
0 0
10 10
Upper
10−1 N=5
N=10
−2
10 N=15
ss Upper
e
Error10−3 Psucc N=5
− N=10
−4 1
10
N=15
−1
10
−5
10 (a) (b) (b)
−6
1010 0 101 102 103 10 0 101 102 103
Cooperativity Cooperativity
FIG. S5. (Color online) (a) Gate error of the Toffoli for different initial states as a function of cooperativity. We have plotted
thegenericerrorforN =5,10,and15andtheupperboundoftheerror. NotethatthegenericerrordecreasesasN increases.
We have fixed ∆ such that Γ = Γ and have assumed that α = β = 1. (b) The failure probabilities 1−P and
0 1 success,up
1−P as a function of cooperativity. 1−P is plotted for N =5,10,15. We have used the same assumptions
success,gen success,gen
as in (a). In general, the failure probability only have a weak dependence on N. Note that the line for 1−P ,N =5
success,gen
coincides with 1−P .
success,up
error since we will show that we can suppress the branching fraction γ /γ arbitrary close to zero by having a
g
two photon driving.
Assuming that γ = 0, the dominating source of error limiting the performance of the Toffoli gate is thus the
g
difference between the rates of the detectable errors for the qubit states. We tune ∆ such that Γ = Γ and the
E 0 1
largest difference between the detectable errors is thus between the completely uncoupled state and the state with all
qubitatomsinstate|1(cid:105). Asaresult,wecanfindanupperboundonthefidelityoftheN qubitToffoligate,considering
an initial state |0(cid:105)⊗N +|1(cid:105)⊗N in the limit N → ∞ because this state experiences the largest difference between the
numberofcoupledanduncoupledqubits. Wefindthattheupperboundonthefidelityandthecorrespondingsuccess
probability is
π2α 1
F ∼1− (S17)
up 16(α+β)C
(α+2β)π 1
P ∼1− √ √ √ . (S18)
success,up
2 α α+β C
. Ingeneral,thefidelityofthegatewill,however,belargerthanwhatissuggestedabove. Consideringagenericinput
state (|0(cid:105)+|1(cid:105))⊗N with the same parameters as above, we find
απ2 1
F ∼1−k(N) (S19)
gen α+βC
(d(N)α+2β)π 1
P ∼1− √ √ √ , (S20)
success,gen
2 α α+β C
where k(N),d(N) are scaling factors which depend on the number of qubits N. We calculate k(N) and d(N)
numericallyforN =1−100usingtheperturbationtheoryandfindthatthattheybothdecreasewithN (seeFig.S5).
The upper bounded and generic fidelities and corresponding success probabilities are shown in Fig. S5 for different
number of qubits, N. As N increases we obtain higher generic fidelity, whereas the success probability is almost
independent of N.
CZ-gate
In the special case of only two qubits the Toffoli gate is referred to as a control-phase (CZ) gate. As shown in
the article, we can, in this case, completely remove the errors from the gate by choosing the detunings ∆ and ∆
E e
10
such that Γ =Γ =Γ and combining it with single qubit rotations we can ensure the right phase evolution. In the
0 1 2
general case where α,β (cid:54)=1, the detunings ∆ and ∆ are
e E
γ(cid:112) (cid:112)
∆ = β 4αC+β (S21)
E 2
αCγ2
∆ = . (S22)
e 2∆
E
The success probability of the gate is then
8β2+6βα+α2 1
P (cid:39)1−π √ √ , (S23)
success 8β3/2 α C
√ √
and we find that the gate time is t (cid:39) γπ α(α+2β)(α+4β) C in the limit C (cid:29)1.
CZ 2β3/2Ω2
Two-photon driving
We now describe the details of the implementation where the auxiliary atoms is driven by a two-photon process
as shown in Fig. S6 (reproduced from Fig. 4(a) in the article) in order to suppress the dominant undetectable error
caused by spontaneous decay of the auxilliary atom into the state |g(cid:105) (Lˆ ).
g
-ΔE -ΔE !
! !
!!!!!!!!
Δ Ω !
E2 MW
! !
! !
g
f!
!
~ g
! Ω f!

Ω

! !
(a) (b)
! !
FIG. S6. (a) Level structure of the auxiliary atom and the transitions driven by a weak laser (Ω), a microwave field (Ω )
MW
and the cavity (g ). We assume that |E(cid:105) ↔ |f(cid:105) is a closed transition and for simplicity we also assume that |E (cid:105) ↔ |g(cid:105) is a
f 2
closed transition but this is not a necessity. The figure also indicates how the levels could be realized in 87Rb. Here |r(e),r(e)(cid:105)
with r = 1,2,3 refers to state |F(e) = r,m(e) = r(cid:105) in 52S (52P ). (b) Effective three level atom realized by mapping the
F 1/2 3/2
two-photon drive to an effective decay rate γ˜ and an effective drive Ω˜
g
The Hamiltonian in a proper rotating frame is
Hˆ =Hˆ +Vˆ +Vˆ†, (S24)
e
Hˆ =∆ |E(cid:105)(cid:104)E|+∆ |E (cid:105)(cid:104)E |+g (aˆ|E(cid:105)(cid:104)f|+H.c)
e E E2 2 2 f
Ω
+ MW(|E(cid:105)(cid:104)E |+H.c.)
2 2
(cid:88)
+ ∆ |e(cid:105) (cid:104)e|+g(aˆ|e(cid:105) (cid:104)1|+H.c.), (S25)
e k k
k
Ω
Vˆ = |E (cid:105)(cid:104)g|, (S26)
2 2
where we have now defined ∆ = ω −ω −ω −ω , ∆ = ω −ω −ω and ∆ = ω −ω −ω −
E E g laser MW E2 E2 g laser e e g laser
ω +ω −ω . Here ω is the frequency of the laser drive (Ω), ω is the frequency of the microwave field
MW f 1 laser MW