Errors in trapped-ion quantum gates due to spontaneous photon scattering

0.39 MB·English

by  R. Ozeri

Errors in trapped-ion quantum gates due to spontaneous photon scattering
R. Ozeri, W. M. Itano, R. B. Blakestad, J. Britton, J. Chiaverini,∗ J. D. Jost,
C. Langer, D. Leibfried, R. Reichle,† S. Seidelin, J. H. Wesenberg, and D. J. Wineland
NIST Boulder, Time and Frequency division, Boulder, Colorado 80305
Weanalyzetheerrorintrapped-ion,hyperfinequbit,quantumgatesduetospontaneousscattering
ofphotonsfrom thegatelaserbeams. Weinvestigatesingle-qubitrotationsthatarebasedonstim-
ulated Raman transitions and two-qubit entangling phase-gates that are based on spin-dependent
optical dipole forces. This error is compared between different ion species currently being investi-
gated as possible quantum information carriers. For both gate types we show that with attainable
laser powers the scattering error can be reduced to below current estimates of the fault-tolerance
error threshold.
7 PACSnumbers: 03.67.Lx,03.65.Yz,03.67.Mn,32.80.-t
0
0
2 I. INTRODUCTION performed on trapped ions with laser light, using this
n level of error as a guideline [18].
a Quantumbits,orqubits,thatareencodedintointernal Inhis1975paper[19],Mollowshowedthattheeffectof
J statesoftrappedionsareaninterestingsystemforQuan- aquantumcoherentfieldonanatomisequivalenttothat
0 tumInformationProcessing(QIP)studies[1,2]. Internal ofaclassicalfieldplusaquantumvacuumfield. Theerror
3 states of trapped ions can be well isolated from the en- due to the interaction with light can be categorized into

vironment, and very long coherence times are possible two parts. The first is the error due to noise in classical
2
v [2, 3, 4, 5]. The internal states of several ion-qubits can laser parameters, such as intensity or phase [2, 20]. The
8 bedeterministicallyentangled,andquantumgatescanbe second part originates from the quantum nature of the
4 carriedoutbetweentwoion-qubits[6,7,8,9,10,11,12]. electromagnetic field and is due to vacuum fluctuations,
0 i.e. the spontaneous scattering of photons [21, 22].
Among different choices of internal states, qubits that
1
are encoded into pairs of ground state hyperfine or Zee-
1
man states benefit from negligible spontaneous decay
6
0 rates [2]. The small energy separation between the two A. Ion-qubit levels and transitions
/ states of such qubits (typically in the radio-frequency or
h
microwavedomain)allowsforphasecoherencebetweena
p MostionspeciesconsideredforQIPstudieshaveasin-
localoscillatorand a qubit superpositionstate overrela-
- glevalenceelectron,witha2S electronicgroundstate,
t tively long times [2, 4, 13, 14]. 1/2
n and 2P and 2P electronic excited states. Some of
a The quantum gates that are performed on hyperfine 1/2 3/2
the ions also have D levels with lower energy than those
u ion-qubits typically use laser beams. Since light couples
of the excited state P levels. Ions with a non zero nu-
q onlyveryweaklytotheelectronspin,spinmanipulations
: clearspinalsohavehyperfinestructureinalloftheselev-
v rely instead on the spin-orbit coupling of levels that are
els. Asmallmagneticfield istypicallyappliedto remove
i typically excited nonresonantly throughallowedelectric
X thedegeneracybetweendifferentZeemanlevels. Herewe
dipole transitions. Spin manipulations therefore require
consider qubits that are encoded into a pair of hyperfine
r a finite amplitude in the excited electronic state, and
a levelsofthe2S manifold. Figure1illustratesatypical
spontaneous scattering of photons from the laser beams 1/2
energy level structure.
during the gate is inevitable.
Toallowforastraightforwardcomparisonbetweendif-
Fault-tolerant quantum computation demands that
ferent ion species, we investigate qubits that are based
the error in a single gate is below a certain threshold.
on clock transitions, i.e. a transition between the F =
Current estimates of the fault-tolerance error threshold |
I 1/2,m =0 and F =I+1/2,m =0
rangebetween10−2to10−4 [15,16,17]. Theseestimates − F i≡|↑i | F i≡|↓i
hyperfine levels in the S manifold of ions with a half
rely on specific noise models and error-correction proto- 1/2
odd-integernuclearspinI,atasmallmagneticfield. For
cols and should be considered as guidelines only. How-
this transition, the total Raman and Rayleigh photon
ever, the general view is that for fault tolerance to be
scattering rates, as well as the Rabi frequency, are inde-
practical, the error probability in quantum gates should
pendent of I, and the comparison between different ion
be at least as small as 10−4. It is, therefore, worth ex-
species depends only on other atomic constants. Super-
ploring the limitations to the fidelity of quantum gates
positionsencodedintothesestatesarealsomoreresilient
against magnetic field noise [3, 5, 11]. Even though our
quantitative results apply only to this configuration, for
other choices of hyperfine or Zeeman qubit states (in-
∗Present address: Los Alamos National Laboratory, Los Alamos,
NM87545, USA. cluding those with I = 0) the results will not change
†Presentaddress: UniversityofUlm,89069Ulm,Germany. significantly.
2
2P
the electric dipole operator. The right and left circular
3/2
w polarization components of the b and r beams are b+/−
f andr , respectively. The P andP excited levels
+/− 1/2 3/2
D
are separated by an angular frequency ω .
2P f
1/2 The total spontaneous photon scattering rate from
these beams is given by [23]
γ 1 2
Γ = g2(b2 +b2)+g2(r2 +r2) + .
D total 3 b − + r − + ∆2 (∆ ω)2
5/2 (cid:20) − f (cid:21)
D (cid:2) (cid:3) (2)
3/2 HereγisthenaturallinewidthoftheP andP levels
w 1/2 3/2
r [27]. We now assume linearly polarized Raman beams
Ww
w with b = b = r = r = 1/√2 [23]. We further
3/2 b − + − − +
assume g =g g. The time for a π rotation is
b r
≡
π
|` \ τ = (3)
F=I - ½ π 2Ω
R
2S w
1/2 0 CombiningEq. (1),(2)and(3),theprobabilitytoscatter
F=I + ½ a photon during τπ is given by
|c \
πγ 2∆2+(∆ ω )2
f
P =( ) − . (4)
total
ω ∆(∆ ω)
f f
| − |
FIG. 1: Relevantenergy levels (not to scale) in an ion-qubit,
The dashed line in Fig. 2 shows P vs. ∆, where the
with nuclear spin I. The P and P excited levels are total
1/2 3/2
laser detuning is expressed in units of the excited state
separated by an angular frequency ω. The S electronic
f 1/2
groundstateconsistsoftwohyperfinelevelsF =I−1/2and fine-structure splitting, and the scattering probability is
F = I +1/2. The relative energies of these two levels de- given in units of γ/ωf. The total scattering probability
pendsonthesign ofthehyperfineconstantA andcanvary has a global minimum of
hf
betweenionspecies(inthisfigure,A isnegative). Thequbit
hf
isencodedinthepairofmF =0statesofthetwoF manifolds Pmin =2√2πγ/ωf, (5)
separated by an angular frequency ω0. Coherent manipula-
tions of the qubit levels are performed with a pair of laser whenthelaserdetuningisbetweenthetwofine-structure
beamsthataredetunedby∆fromthetransition totheP1/2 manifolds (∆ = (√2 1)ωf). The asymptotic value of
level,representedbythetwostraightarrows. Theangularfre- P for large positiv−e or negative detuning
total
quency difference between the two beams equals the angular
frequency separation between the qubit levels ωb−ωr = ω0. P =3πγ/ω, (6)
∞ f
SomeionspecieshaveDlevelswithenergiesbelowthePman-
ifold. Wavy arrows illustrate examples of Raman scattering is only slightly larger than the global minimum.
events.
Previous studies estimated decoherence by assuming
that any photon scattering will immediately decohere a
hyperfine superposition [23]. Under this assumption the
Gates are assumed to be driven by pairs of Raman
lowest possible gate error equals P and ions with a
beams detuned by ∆ from the transition between the min
smallγ/ω ratiobenefitfromalowergateerrorminimum.
S andthe P levels(See Fig. 1). We further assume f
1/2 1/2
Two kinds of off-resonance photon scattering occur in
thattheRamanbeamsarelinearlypolarizedandRaman
the presence of multiple ground states. Inelastic Raman
transitions are driven by both σ photon pairs and σ
+ −
scattering which transfers population between ground
photon pairs. The two beams in a Raman pair are des-
states, and Rayleigh elastic scattering which does not
ignated as red Raman (r) and blue Raman (b) by their
change ground state populations. Since no energy or
respective frequencies. In the following we also assume
angular momentum are exchanged between the photon
that ∆ is much larger than the hyperfine and Zeeman
and the ion internal degrees of freedom, no information
splitting betweenlevelsinthe groundandexcitedstates.
aboutthe qubitstate is carriedawayby aRayleighscat-
TheRabifrequencybetweenthetwoclockstatesis[23]
tered photon. Rayleigh scattering, therefore, does not
gbgr ωf necessarily lead to decoherence [23, 24, 25]. This was
Ω = (b r b r ) , (1)
R − − + +
3 − ∆(∆ ω) experimentally shown in [25] where, when of equal rate
f
−
from both qubit levels,off-resonanceRayleighscattering
where gb/r = Eb/r P3/2,F = I + 3/2,mF = Fdˆ of photons did not affect the coherence of a hyperfine
σˆ+ S1/2,F = I +1|/h2,mF = F /2~, Eb/r is the p|eak· superposition. Decoherence in the presence of light was
| i|
electricfieldoftheborr beamatthepositionofthe ion, showntobedominatedbyRamanscattering. Theguide-
respectively,anddˆ σˆ istherightcircularcomponentof line used in [23] is therefore overly pessimistic.
+
·
3
of the ion-qubit internal state. Following a spontaneous
Raman scattering event the ion-qubit is projected into
one of the ground states in the S manifold. For ions
1/2
withlowlyingDlevels,Ramanscatteringeventscanalso
transfer the ion from the qubit levels into one of the D
levels.
A. Single-qubit gate
Forasinglequbitgatewechoosetolookatthefidelity
of a π rotation around the x-axis of the Bloch sphere,
represented by the Pauli operator σˆ . This gate is as-
x
sumed to be driven by a co-propagating Raman beam
pair, where the frequency difference between the beams
equals the frequency separation between the two qubit
states ω . It is straightforwardto generalize this case to
0
other rotations.
As a measure of the error in the rotation, we use the
fidelityofthefinalstate(characterizedbydensitymatrix
FIG. 2: The solid line is the probability (PSE = PRaman)to
ρˆ ) produced by the erroneous gate as defined by
scatter a Raman photon (in units of γ/ωf) during a π ro- final
tation vs. the laser detuning in units of the excited state
F = Ψρˆ Ψ , (7)
fine-structure splitting. The dashed line is the probability h | final| i
for any type of scattering event (PSE = Ptotal) during the where Ψ is the ideal final state. Given an initial state
pulsevs. detuning. TheRamanscatteringprobabilitydecays | i
Ψ , Ψ can be written as
init
quadratically with ∆ for |∆|≫ωf. | i | i
Ψ =σˆ Ψ . (8)
x init
| i | i
Inthe presenceofoff-resonanceRamanscattering,the
In this paper we re-examine the errors due to sponta-
density matrix that describesthe state of the qubit after
neous photon scattering on single-qubit gates (rotations
the gate has the form
of the equivalent spin 1/2 vector on the Bloch sphere)
and two-qubit (entanglement) gates, where the qubits
ρˆ =(1 P )Ψ Ψ +ρˆ . (9)
final Raman ǫ
are based on ground state hyperfine levels and manipu- − | ih |
lated with stimulated Raman transitions. We compare The erroneous part of the density matrix ρˆǫ =
between different ion species and examine different Ra- iwi|iihi|,iscomposedofprojectorsintodifferentlevels
man laser parameters. In section II we analyze the con- |Pii of lower energy. Here PRaman = iwi is the proba-
tribution to the gate error due to spontaneous Raman bilityforaspontaneousRamanscatteringeventtooccur
P
scattering. Following a Raman scattering event the ion- during the gate.
qubit is projected into one of its ground states and spin Note that some Raman scattering events keep the ion
coherence is lost. This error was also addressed in [26]. within the qubit manifold. Using Eq. (7) the contribu-
InsectionIII we examinethe errordue toRayleighscat- tion of ρˆǫ to the fidelity is positive and not strictly zero.
tering. Rayleigh scattering error results primarily from For simplicity, we neglect this contribution and put a
the photon’s recoil momentum kick. We show that both lower bound on the gate fidelity
typesofgateerrorscanbereducedtosmallvalues,while
F 1 P . (10)
keeping the gate speed constant, with the use of higher ≥ − Raman
laser intensity. Inwhatfollowsweassumethisexpressiontobeanequal-
ity. The error in the gate due to spontaneous Raman
photon scattering is hence given by the Raman scatter-
II. RAMAN SCATTERING ERROR ing probability
ǫ 1 F =P . (11)
Raman
In a Raman photon scattering event energy and an- ≡ −
gular momentum are exchanged between the scattered We first examine the error due to Raman scattering
photon and the ion’s internal degrees of freedom. The back into the 2S manifold ǫ . The Raman scattering
1/2 S
polarizationandfrequencyofthe scatteredphoton(with rate back into the S manifold is calculated to be [25]
1/2
respect to those of the laser) become entangled with the
ions’internalstate. Therefore,aftertracingoutthe pho- Γ = 2γ g2(b2 +b2)+g2(r2 +r2) ωf 2.
tondegreesoffreedom,theions’spincoherenceislost. In Raman 9 b − + r − + ∆(∆ ω)
(cid:20) − f (cid:21)
other words, Raman scattering serves as a measurement (cid:2) (cid:3) (12)
4
For the same laser parameters as above, the probability
to scatter a Raman photon during the gate is

2πγ ω
f
P = =ǫ . (13)
Raman 3 ∆(∆ ωf) S 199Hg+
| − | -2 67 +
10 Zn
111 +
The solid line in Fig. 2 shows PRaman vs. ∆. The 25Cd+
Raman scattering probability decays quadratically with ) Mg
W 9 +
Be
∆ for |∆| ≫ ωf. Qualitatively, this is because Raman ( 171Yb+
scattering involves a rotation of the electron spin. Elec- 43 +
Ca
tron spin rotations are achieved through the spin-orbit 10-3 87Sr+
137 +
coupling in the excited state. This coupling has oppo- Ba
site signcontributions fromthe two fine-structure levels.
Thereforeaswedetunefarcomparedtothefinestructure
splitting, those two contributions nearly cancel [24].
-4
Using Eq. (1) we can write 10
-4 -3 -2
10 10 10
2πγ ΩR Gate Error
P = | |. (14)
Raman g2
The ratio γ/g2 can be expressed in terms of atomic con-
FIG.3: LaserpowerineachoftheRamanbeamsvs. theerror
stants, and the peak electric field amplitude E at the
inasingleiongate(πrotation)duetoRamanscatteringback
position of the ion [27] into the S manifold (obtained using Eq. (18)). Different
1/2
lines correspond to different ion species (see legend). Here
γ 4~ω33/2 we assume Gaussian beams with w0 = 20 µm, and a Rabi
= . (15)
g2 3πǫ0c3E2 frequency ΩR/2π = 0.25 MHz (τπ =1 µsec).
Here ω is the frequency of the transition between the
3/2
S and the P levels (D2 line), ǫ is the vacuum per-
1/2 3/2 0 case, assuming that w0 is constant, the required power
mittivity,andcisthe speedoflight. AssumingGaussian
would scale as the optical transition frequency cubed.
laser beams, at the center of the beam
Either way, ion species with optical transitions of longer
wavelengtharebetter suited in the sense that less power
4
E2 = P . (16) is required for the same gate speed and error require-
πw2cǫ
0 0 ments. In addition, high laser power is typically more
readily available at longer wavelengths. Finally, we note
Here is thepowerineachofthe Ramanbeamsandw
0
P that the error is independent of the fine-structure split-
is the beam waist at the position of the ion. The prob-
tingaslongaswehavesufficientpowertodrivethetran-
ability to scatter a single Raman photon can be written
sition. The transition wavelengths of different ions are
as
listed in Table I.
P = 2π|ΩR|~ω33/2w02 =ǫ . (17) Figure 3 shows the laser power needed per Raman
Raman 3c2 S beam for a given error due to spontaneous Raman scat-
P
tering into the S manifold. Here we assume Ω /2π =
1/2 R
This result is essentially the same as Eq. (5) of [26].
0.25MHz(τ =1µsec),andw =20µm. Differentlines
π 0
We can now rearrange this expression to put an upper
correspondtothedifferentionspecieslistedonthefigure
bound on the required power for a desired gate speed
legend. Table II lists the error in a single ion-qubit gate
and error
due to Raman scattering back into the S manifold for
1/2
2π 2πw the same parameters as in Fig. 3, and assuming 10 mW
= ( 0)2~ω Ω , (18)
P 3ǫ λ 3/2| R| in each of the Raman beams. The power 0 needed in
S 3/2 eachof the gate beams for ǫ =10−4 is alsoPlisted in the
S
where λ =c/ω . table. As can be seen, ions with shorter transitionwave-
3/2 3/2
Assume that the ratio of the beam waist to the tran- length require a larger detuning and accordingly higher
sition wavelengthis constant for different ion species. In laser power to maintain a low gate error. For most ions,
this case, the power needed to obtain a given Rabi fre- a few milliwatts of laser power is enough to reduce the
quency and to keep the error below a given value would gate error to below 10−4.
scale linearly with the optical transition frequency. A Equation(13)canbesolvedtogivetherequireddetun-
morerealisticassumptionmightbethattheRamanbeam ing values for a given ǫ . The number of such detuning
S
waistisnotdiffractionlimitedandisdeterminedbyother valuescomesfromthenumberofcrossingsofahorizontal
experimental considerations, such as the inter-ion dis- line, set at the desired error level, with the solid curve
tance in the trap or beam pointing fluctuations. In this in Fig. 2. When ǫ is higher than the minimum P
S Raman
5
Ion I γ/2π (MHz) ω0/2π (GHz) ωf/2π (THz) λ1/2(nm) λ3/2(nm) f−1
9Be+ 3/2 19.6 1.25 0.198 313.1 313.0 N.A.
25Mg+ 5/2 41.3 1.79 2.75 280.3 279.6 N.A.
43Ca+ 7/2 22.5 3.23 6.68 396.8 393.4 17
67Zn+ 5/2 62.2 7.2 27.8 206.2 202.5 N.A.
87Sr+ 9/2 21.5 5.00 24.0 421.6 407.8 14
111Cd+ 1/2 50.5 14.53 74.4 226.5 214.4 N.A.
137Ba+ 3/2 20.1 8.04 50.7 493.4 455.4 3
171Yb+ 1/2 19.7 12.64 99.8 369.4 328.9 290
199Hg+ 1/2 54.7 40.51 273.4 194.2 165.0 700
TABLEI:Alistofatomicconstantsofseveraloftheionsconsideredforquantuminformationprocessing. HereIisthenuclear
spin,γ isthenaturallinewidth of theP1/2 level[28,29,30,31,32,33,34,35], ω0 is thefrequencyseparation betweenthetwo
qubit states set by the hyperfine splitting of the S level [36, 37, 38, 39, 40, 41, 42, 43, 44], ω is the fine-structure splitting
1/2 f
[45],λ andλ arethewavelengthsof thetransitionsbetweentheS andtheP andP levels[45],respectively. The
1/2 3/2 1/2 1/2 3/2
branchingratio of decay from the Plevels to theD and the S levels is f [46, 47, 48, 49, 50,51, 52].
inside the fine-structure manifold, i.e. ǫS >8πγ/3ωf (see Ion ǫS/10−4 P0(mW) ∆0/2π(GHz) ǫD/ǫS ǫD∞/10−4
Fig. 2),thenfourdifferentdetuningvaluesyieldthesame 9Be+ 0.34 3.4 -203 N.A. N.A.
ǫ , two outside and two inside the fine-structure mani- 25Mg+ 0.47 4.7 −691 N.A. N.A.
S
fold. Whenǫ islowerthanthisvalue,onlytwodetuning 43Ca+ 0.17 1.7 −442 0.10 0.019
S
values,bothofwhichareoutsidethe fine-structureman- 67Zn+ 1.23 12.3 −1247 N.A. N.A.
ifold, yield ǫ . Those detuning values ∆ that are below 87Sr+ 0.15 1.5 −442 0.11 0.006
S 0
the S P transition (∆ < 0) and correspond to 111Cd+ 1.05 10.5 −1043 N.A. N.A.
1/2 1/2
ǫ = 10−→4 are listed in Table II for different ions. For 137Ba+ 0.11 1.1 −418 0.51 0.012
S
most ions ∆ is in the few hundred gigahertz range, and 171Yb+ 0.2 2 −411 0.005 0.00006
0
its magnitude is much smaller than ω. 199Hg+ 2.3 23 −1141 0.002 0.00003
f
We now consider the error for various ions caused by
Raman scattering into low lying D levels, ǫ . As tran-
D TABLEII:A list of errors in asingle-qubit gate (π rotation)
sitions between levels in the S and the D manifolds do
due to spontaneous photon scattering. The error due to Ra-
not necessarily involve electron spin rotations, the error
man scattering back into the S1/2 manifold ǫS is calculated
suppression discussed preceding Eq. (13) will not occur.
with the same parameters as Fig. 3: Gaussian beams with
Instead, Raman scattering rate into the D levels will be w0 =20µm,asingleionRabifrequencyΩR/2π =0.25 MHz
given by the total scattering rate times a fixed branch- (τπ =1 µsec), and 10 mW in each of the Raman beams. P0
ing ratio f. When driven resonantly the P and P isthepower(inmilliwatts) neededineachofthebeams,and
1/2 3/2
levels decay to the D manifold with different (but often ∆0/2π is the detuning (in gigahertz) for ǫS =10−4. The ra-
similar) branching ratios. Here we assume that for a de- tio between errors due to Raman scattering to the D and S
tuninglargecomparedtothefine-structuresplitting,the manifolds, ǫD/ǫS, is given when ǫS = 10−4. The asymptotic
branchingratioisessentiallyindependentofthelaserde- valueof ǫD in the|∆|≫ωf limit is ǫD∞.
tuning and is given by the average of the two resonant
branching ratios [46, 47]. Table I lists f, for various ion
species,obtainedfrom[48, 49,50, 51, 52]. The errordue ω and (perhaps with the exception of 137Ba+) ǫ is the
f S
to Raman scattering into D levels is given by more dominant source of error. The ratios ǫ /ǫ when
D S
ǫ =10−4(i.e. ∆=∆ )aregiveninTableIIfordifferent
ǫ =fP . (19) S 0
D total ions.
Using Eq. (13) we can write the ratio of the errors due Duetotheasymptoticvalueofthetotalscatteringrate
to Raman scattering into the different manifolds in the ∆ ω limit (Eq. (6)), ǫ has an asymptotic
f D
| | ≫
valuewhichgivesalowerboundtotheRamanscattering
ǫD 3f 2∆2+(∆ ωf)2 error
= − . (20)
ǫ 2 ω2
S (cid:18) f (cid:19)
3πγf
For ∆ < ω, Raman scattering error is dominated by ǫ = . (21)
| | f D∞ ω
scatteringbackintothe S levels. Forionswith1/f f
1/2
≫
1 the two errors become comparable at a detuning ∆
≃
√2ω/3√f. Whenthedetuningbecomeslargecompared Table II lists ǫ for various ions. For all ion species
f D∞
tothefine-structuresplitting,scatteringintolow-lyingD considered this value is below the assumed estimates for
levels dominates. For most ions considered here, ∆ < the fault tolerance threshold.
0
| |
6
B. Two-qubit gate B
Auniversalquantumgatesetis complete with the ad-
ditionoftwo-qubitentanglinggates. During the lastfew
years there have been several proposals and realizations
of two ion-qubit gates [1, 8, 9, 10, 11, 12, 53, 54, 55].
Here we focus on gatesthat use spin-dependent forcesin
order to imprint a geometric phase on certain collective
spin states [8, 9, 11, 12, 53, 54, 55]. We examine only
Trap axis
gates that are implemented with a continuous non reso-
nant pulse rather than those using multiple short pulses
[56]. Again, to compare different ion species we examine
ion-qubits that are encoded into hyperfine clock states.
Whenthe laserdetuning is largecomparedto the hyper-
fine splitting, the differential light force between clock
levels is negligible [4, 57]. However, a phase gate can be
LLaasseerr bbeeaammss
applied between spin states in the rotated basis (super-
positions of clock states that lie on the equatorial plane
oftheBlochsphere)[53,54]. Inthisschemetheionstra-
FIG. 4: Schematic of Raman laser beam geometry assumed
verse a trajectory in phase space that is conditioned on
for the two-qubit phase gate. The gate is driven by two Ra-
their mutual spin state (in the rotated basis [11]). The
manfields,eachgeneratedbyaRamanbeampair. Eachpair
phase the ions acquire is proportional to the total area
consists of two perpendicular beams of different frequencies
encircled in phase space. This geometric phase gate was that intersect at theposition of theions such that the differ-
demonstrated in [8] and was realized on clock states in ence in their wave vector lies parallel to the trap axis. One
[11]. beam of each pair is parallel to the magnetic field which sets
This form of phase gate is implemented with two dif- the quantization direction. The beams’ polarizations in each
ferent Raman fields that are slightly off-resonance with pair are assumed to be linear, perpendicular to each other
and to the magnetic field. Wavy arrows illustrate examples
upperandlowermotionalsidebandsofthespin-fliptran-
of photon scattering directions.
sition. For simplicity, we assume here that the gate is
driven by two independent pairs of Raman beams, i.e.
a total of four beams. Most experimental implementa-
As in the single-qubit case, Raman scattering will
tions of this phase gate thus far have used a three-beam
project one of the ion qubits into one of its states be-
geometry [8, 11]. It is straightforward to generalize the
low the P manifold. In the appropriate basis the ideal
treatment below to the three-beam case.
gate operation is represented by [9, 11]
Typicalconditionsforthegatearesuchthattheangu-
larfrequencydifferencebetweenthebeamsis(ω +ω
0 trap
δ) in one Raman pair and (ω ω +δ) in the othe−r. 1 0 0 0
0 trap
Here ωtrap is the angular frequ−ency of the normal mode Uˆ =0 eiφ 0 0, (22)
that the gate excites and δ is the Raman field detuning 0 0 eiφ 0
fromthatmotionalsideband. Undertheseconditionsthe 0 0 0 1
 
ions will traverse K full circles in phase space for a gate  
duration of τ = 2πK/δ. Typically δ is chosen to be where typically φ = π/2. The output state is Ψ =
much smallergatthean ωtrap to avoid cou|pl|ing to the pure Uˆ|Ψiniti. Thedensitymatrixfollowingtheerroneou|sgiate
spin flip (“carrier”) transition or the motional “specta- will be of the form
tor” mode, and K is usually chosen to be 1 to minimize
ρˆ =(1 2P )Ψ Ψ +ρˆ , (23)
the gate time. final Raman-gate ǫ
− | ih |
Figure 4 depicts the assumed geometry of the laser
where P is the probability that one of the ions
Raman-gate
beams. The two beams comprising each Raman pair in-
scattered a Raman photon during the gate [58, 59]. For
tersect at right angles at the position of the ions, such
a gate consisting of K circles, the Raman detuning δ is
thatthe difference intheir wavevectorsis parallelto the
chosen such that the gate time is given by [53]
trapaxis. Withthischoice,theRamanfieldscouple,toa
veryhighdegree,only to the motionalongthe trapaxis. π √K
Thebeampolarizationsareassumedtobelinear,perpen- τgate = √K =τπ . (24)
2Ω η η
R
dicularto eachother andto the magnetic fieldaxis. The | |
beams’ relative frequencies can be arranged such that Here η = ∆kz is the Lamb-Dicke parameter, where
0
the final state is insensitive to the optical phase at the ∆k = √2k is the wave vector difference between the
L
ions’ position [57]. Generalizing this treatment to other two beams that drive the gate (for the particular geom-
Raman beam geometries is straightforward. etry of Fig. 4), and k is the laser beam wave vector
L
7
magnitude. The root mean-square of the spatial spread Ion ǫS/10−4 P0(mW) ∆0/2π(THz) ǫD/ǫS ǫD∞/10−4 η
of the ground state wave function of one ion for the nor- 9Be+ 3.6 36 -1.20 N.A. N.A. 0.194
mal mode that the gate excites is 25Mg+ 11.1 111 -7.28 N.A. N.A. 0.130
43Ca+ 13.6 136 -10.42 1.01 1.06 0.071
z = ~/4Mω , (25) 67Zn+ 41.1 411 -24.96 N.A. N.A. 0.11
0 trap
87Sr+ 26.5 265 -20.34 0.52 0.50 0.048
q
whereM is the massofanindividualion. The single ion 111Cd+ 64.3 643 -35.44 N.A. N.A. 0.081
carrier Rabi frequency Ω is given in Eq. (1). 137Ba+ 37.4 374 -30.67 1.65 1.46 0.034
R
Usingsimilarconsiderationstothoseusedinthesingle 171Yb+ 57.5 575 -32.89 0.01 0.007 0.038
ion gate and, as before, neglecting any (positive) contri- 199Hg+ 149.7 1497 -49.52 0.003 0.001 0.078
bution of ρˆ to the fidelity, we can set an upper bound
ǫ
for the power needed for a certain error in the gate due
TABLE III: A list of different errors in a two-qubit phase
to Raman photon scattering into S levels
1/2 gate dueto spontaneous photonscattering. Theerror dueto
= 2π(2πw0)2~ω Ω 4√K. (26) RlaatemdaansssucmatitnegrinGgaubsasciakninbteoamthsewSit1h/2wm0=an2i0foµldm,,ǫaS,giastecatlicmue-
P 3ǫS λ3/2 3/2| R| η τgate=10 µs, ωtrap/2π =5 MHz, a single circle in phase space
(K=1), and 10 mW in each of the four Raman beams. P0
This required gate power is 4√K/η times larger than is the power in milliwatts needed in each of the beams, and
that needed for the same error in the single ion-qubit π ∆0/2π is the detuning in gigahertz for ǫS =10−4. The ratio
rotationgiveninEq. (18). Thefactorof√K/η isdue to betweenerrorsduetoRamanscatteringtotheDandSman-
thelongertwoion-qubitgatedurationcomparedtosingle ifolds, ǫD/ǫS,isgivenwhen ǫS =10−4. Theasymptoticvalue
of ǫD in the |∆|≫ωf limit is ǫD∞. The Lamb-Dicke param-
qubitrotations,andthefactorof4isduetothepresence
eter η for the above trap frequency is also listed for different
of two ions and the pair of required Raman fields.
ions.
Whencomparingdifferentionspecieswecanfixdiffer-
ent parameters, depending on experimental constraints
or requirements. For example, here we choose as fixed
in the table. Here, heavier ions need a larger detuning
parameters the beam waist w (for the reasons given
0
andcorrespondinglyhigherlaserpowertomaintainalow
above), the gate time τ (assuming a certain compu-
gate gate error. For most ion species, hundreds of milliwatts
tation speed is desired), and the mode frequency ω
trap of laser power per beam and a detuning comparable or
(which sets the time scale for various gates). With these
even larger than ω are needed to reduce the gate error
choices,heavierionspaythepriceofsmallerη andthere- f
to the 10−4 level.
fore higher power requirements per given gate time and
error. A different approach would be to choose η fixed, Asintheone-iongate,Ramanscatteringintolowlying
in which case heavier ions will need a lower ω . For Dlevelswill addto the gate error. This errorisgivenby
trap
a fixed gate time a lower ω leads to stronger, off-
trap
resonant, coupling of the Raman fields to the carrier or 4√K
the other motional “spectator” mode, and, therefore, to ǫ =2fP = fP . (29)
D total-gate total
η
a larger error due to this coupling [60, 61]. The Lamb-
Dicke parameters for the different ions for ω /2π = 5
trap
MHz are listed in Table III. Here Ptotal-gate is the probability that one of the ions
With the choice where w0, τgate, and ωtrap are fixed, scattered a photon during the two-qubit gate, and Ptotal
we can write is the one-qubit gate scattering probability given in Eq.
(4). SinceboththeRamanandthetotalscatteringprob-
8π2 K
= w2ω Mω . (27) abilities increase by the same factor as compared to the
P 3ǫS τgate 0 3/2 trap one-qubit gate, the ratio of the two errors ǫD/ǫS will re-
mainthesameasgivenbyEq. (20). TableIIIlistsǫ /ǫ
D S
Or equivalently, for the different ions when ǫ = 10−4. Notice that for ǫ
S S
= 10−4 some ions require ∆ & √2ω /3√f. For those
ǫ = 8π2 K w2ω Mω . (28) ions ǫ is no longer negligi|ble0|comparefd to ǫ .
S 3 τ 0 3/2 trap D S
gate
P Scattering into a low lying D level will, again, set a
Figure5showsthepowerneededvs. errorinatwo-qubit lower bound on the total error. Combining Equations
gateduetoRamanscatteringbackintotheS manifold (29) and (6) we find this lower bound to be
1/2
for w = 20 µm, τ = 10 µs, ω /2π = 5 MHz and
0 gate trap
K = 1. Table III lists ǫ for the various ion species and
S 3πγf 4√K
the same laser parameters as Fig. 5, assuming a power ǫ = . (30)
D∞
ω η
of 10 mW is used in each of the four Raman beams. f
Alternatively,the power andthe detuning ∆ needed
0 0
in each of the gate beamPs for ǫ = 10−4 are also listed Table III lists ǫ for the different ion species.
S D∞
8
the gate operation. Therefore, Rayleigh scattering has a
negligible effect on single-qubit gates.

In the two-qubit phase gate, a mode of motion is ex-
cited that is entangled with the two-ion collective spin
199 +
Hg state. Inthiscase,recoilfromphotonscatteringperturbs
111 +
1 171Cd+ the ion’s motion through phase space and contributes to
Yb
67 + the gate error. The ion-qubit trajectory is distorted in
Zn
W) 137Ba+ twoways. Thelargerdistortionarisesfromthedirectre-
( 4837Sr++ coil momentum displacement. A second, much smaller,
Ca distortionarisesfromthecontributionoftherecoiltothe
25 +
Mg appearanceofnonlinearitiesinthe gateoperationdue to
9 +
0.1 Be
deviations from the Lamb-Dicke regime (see subsection
IIIB).
In subsection IIIA we calculate the error in a two-
qubitgate due to directrecoilphase-spacedisplacement.
-4 -3 -2 InsubsectionIIIBweelaborateonthe twoothersources
10 10 10
of error mentioned above, namely errors due to uneven
Gate Error
Rayleigh scattering rates and errors due to deviations
from the Lamb-Dicke regime.
FIG.5: LaserpowerineachoftheRamanbeamsvs. theerror
in a two ion entangling gate due to Raman scattering back
into the S manifold (obtained using Eq. (27)). Different A. Rayleigh scattering recoil error
1/2
linescorrespondtodifferentionspecies(seelegend). Herewe
assume Gaussian beams with w0 = 20 µm, a gate time τgate In elastic Rayleigh scattering, energy and momentum
=10µs,ωtrap/2π =5MHz,andasinglecircleinphasespace are not exchanged between the ions’ and the photons’
(K = 1).
internal degrees of freedom. However, momentum and
energyareexchangedbetweenthephotons’andtheions’
externaldegreesoffreedom. Thescatteredphotondirec-
III. RAYLEIGH SCATTERING ERROR
tion will be different from that of the laser beam, caus-
ing the ion to recoil. This recoil acts on the ion-qubit
Since Rayleigh photon scattering is elastic, no en- asaphase-spacemomentumdisplacement,distortingthe
ergy or angular momentum is transferred between the ion’strajectorythroughphasespaceandcausinganerror
photons’ and the ions’ internal degrees of freedom. in the entangling-gate phase. Note that the momentum
Therefore,thesedegreesoffreedomremainuncorrelated. imparted to the ion (and therefore the deviation from
Rayleigh scattering does not necessarily lead to direct the desired gate phase) and the momentum that is car-
spin decoherence. ried by the scattered photon (namely its scattering di-
In situations where Rayleigh scattering rates from the rection) are correlated. From this point of view the gate
two ion-qubit states are different, Rayleigh scattering of infidelity again arises due to the entanglement between
photonswilleventuallymeasurethe qubitstateandlead the scattered photons’ and ions’ (this time external) de-
to decoherence. In fact, the most common ion-qubit de- grees of freedom (for more details on this point of view
tection method relies on state selective Rayleigh scat- see Appendix A).
tering of photons on a cycling transition. In most Ra-
In the Lamb-Dicke regime (for a thermal state
man gates, however,the laser is detuned from resonance η√2n¯+1 1, where n¯ is the average mode popula-
by much more than the qubit levels’ energy separation, ≪
tion),thegateoperationcanbeapproximatedasaseries
∆ ω , and the laser polarization is typically linear to
0 of finite displacements in phase space [9]
≫
suppressdifferentialStarkshifts. Underthese conditions
Rayleigh scattering rates from the two qubit levels are
almost identical and the error due to the rate difference N
isnegligible. Fortheclocktransitionqubitstatesconsid- Uˆ = Dˆ(∆α ). (31)
k
ered here, Rayleigh scattering rates are almost identical,
k=1
Y
regardless of the laser polarization (for more details on
this error see subsection IIIB). Displacements through phase space, ∆α , are condi-
k
The main effect of Rayleigh scattering is the momen- tionedonthejointspinstateofthetwoions,anddepend
tum recoil it imparts to the ion-qubit. For a single- onthegateparameters. Forcertaingateparameters,the
qubit gate, the Raman beams are usually arranged displacement is zero for the two parallel spin states
|↑↑i
in a co-propagating geometry. In this configuration, and , and non zero with opposite sign for the
| ↓↓i | ↑↓i
since the Lamb-Dicke parameter is very small (η and states, where and hereafter refer to the ro-
≃ |↓↑i ↑ ↓
(ω /ω )k z ), the effect of ion motion is negligible on tated basis rather than the clock levels. Using the com-
0 3/2 L 0
9
mutation relations between phase space displacements (a) |wx\ (b)
P’ P’
Dˆ(α)Dˆ(β)=Dˆ(α+β)eiIm(αβ∗), (32)
we can write the gate operation as a single displacement
times an overall phase [9]
X’ X’
N
Uˆ =Dˆ( ∆αk)eiIm(PNj=2∆αjPjl=−11∆αl), (33)
k=1
X
where N is the total number of infinitesimal dis-
placements. When the net displacement is zero, i.e.
N ∆α = 0, the two-ion motion returns to its ini- |xw\
k=1 k
tial state and spin and motion are disentangled at the
P
end of the gate. The gate operation on the affected spin
states can be written as
FIG. 6: Schematics of the trajectories traversed by the ions
in phase-space during the gate. The acquired phase is pro-
Uˆ =Iˆeiφ. (34)
portional to the encircled area. (a) Phase-space trajectory
of an ideal gate. (b) Phase-space trajectory of an erroneous
The phase acquired, gatewhereaphotonwasscatteredduringthegatedrive. The
short straight arrows represent the recoil displacement. The
N j−1 grey-shadedareaisproportionaltothephaseerror. Thearea
φ=Im( ∆αj ∆αl), (35) that is added to the | ↑↓i trajectory is subtracted from the
j=2 l=1 |↓↑i trajectory.
X X
isthesameforthe and statesandproportional
|↑↓i |↓↑i
totheencircledphase-spacearea. Figure6ashowsanex- can write the position operator for for ion i along this
ample for the trajectory traversedin phase space during axis as
an ideal gate in a reference frame rotating at the mode
zˆ =Z +z (aˆ +aˆ†)+z (aˆ +aˆ†), (39)
frequency ω . i i 0 0 0 1 1 1
trap
Inthepresenceoflaserlightthereisafiniteprobability where Z is the equilibrium position of the ion and z ,
i 0/1
PRayleigh-gate thataRayleighphotonwillbe scatteredby aˆ† and aˆ are the root mean-square of the ground
one of the ions during the gate [58]. 0/1 0/1
state spatial spread, and the creation and annihilation
The probability for a Rayleigh scattering event to oc-
operators,respectively,ofthe twomotionalmodes 0 and
cur during a single-qubit gate is given by the difference
1. We assume the gate is performed by exciting mode 0.
between Eq. (4) and Eq. (13)
Recoil into mode 1 does not distort the gate dynamics
directly. Thispartofthephotonrecoilwilladdminutely
πγ3∆2 2∆ω +ω2/3
P = − f f . (36) to the gate error through its contribution to the gate
Rayleigh
ω ∆(∆ ω)
f | − f | nonlinearity discussed in section IIIB. We neglect this
contribution for the moment and write the recoil opera-
Since in the limit of ∆ ω all scattering events are
| | ≫ f tion as
Rayleigh scattering, P and P have the same
Rayleigh total
asymptotic value P∞. Uˆrecoil =eiqzz0(aˆ0+aˆ†0) eβaˆ0−β⋆aˆ†0 Dˆ(β), (40)
Using Eq. (36) and the factor for the extra required ≡ ≡
power in the two-qubit gate where qz is the projection of q along the trap axis, and
β =iq z .
z 0
4√Kπγ 3∆2 2∆ω +ω2/3 The recoil action on the two-ion crystal is, therefore,
PRayleigh-gate = − f f . (37) a momentum displacement β in phase space. The mag-
ηω ∆(∆ ω)
f (cid:18) | − f | (cid:19) nitude of β depends on the wave-vector difference be-
Theeffectontheionmotionwouldbethatofmomentum tween the scattered photon and the beam from which
recoil it was scattered, and the projection of this momentum
difference along the trap axis. Since the same recoil dis-
Uˆ =eiq·ˆri. (38) placement is applied to all spin states, motion will still
recoil
be disentangled from spin at the end of the gate. Er-
Hereq=k k isthewavevectordifferencebetween rors are therefore due to the change in area encircled in
L Scat
−
the scattered photon and the laser beam from which it phase space. Figure 6b illustrates a gate trajectory that
was scattered, and ˆr is the position operator of the ion is distorted due to photon recoil.
i
that scattered the photon (i = 1,2). We will neglect The erroneous gate can be again written as a sum of
recoil into directions other than along the trap axis. We displacements, those due to the gate drive and β, which
10
occurs at some random time during the gate. For a par- and using Eq. (7), we can write for the gate fidelity
ticular two-ion spin state,
N M F = µ2+ei∆φ κ2+e−i∆φ γ 2+ δ 2 2, (50)
Uˆ = Dˆ(∆α )Dˆ(β) Dˆ(∆α ). (41) | | | | | | | |
ǫ k l (cid:12) (cid:12)
k=YM+1 Yl=1 The most(cid:12)relevant fidelity for fault toleranc(cid:12)e considera-
tionsis thataveragedoverallpossibleinitialstates. The
Byuseofthecommutationrelation(32),thegateiswrit-
fidelity we calculate here is that due to the worst-case
ten as
input state, that is the input state that minimizes Eq.
Uˆ =Dˆ(β)ei(φ+∆φ), (42) (50). The worst-case fidelity is clearly smaller than the
ǫ
average fidelity and therefore gives a conservative esti-
where the phase error ∆φ is determined to be mate to the gate error [62]. The worst-case fidelity for
this kind of error was calculated in [63]. For ∆φ/φ < 1
M N it is
∆φ=Im β ∆α∗+ ∆α β∗ . (43)
l k
" #
Xl=1 k=XM+1 F =cos2∆φ. (51)
Writing displacements as a function of the gate time
This minimal fidelity is the result of an input state with
∂α(t) κ2 = γ2 =1/2.
∆α =∆α(t ) dt, (44)
k k | | | |
→ ∂t
Photon scattering occurs in only a small fraction
we can write the phase error as a function of the gate PRayleigh-gate of the gates. Averaging over all gates per-
displacement before and after the scattering time t formed, we get
scat
∆φ=Im β tscat ∂α∗(t)dt+β∗ tgate ∂α(t)dt . Fgate =1 PRayleigh-gate(1 cos2∆φ ) 1 ǫR, (52)
− −h i ≡ −
∂t ∂t
(cid:20) Z0 Ztscat (cid:21)
(45)
wherethebracketscorrespondtoanaverageoverallpos-
Since α(t )=α(0)=0 we get
gate sible ∆φ’s, i.e. due to different values of β and different
∆φ=Im[βα∗(t ) β∗α(t )]. (46) scattering times.
scat scat
− To perform the above average we need to explicitly
Since the accumulated displacement at the moment the write the different displacements. In the rotating frame
photon was scattered α(t ) is of equal magnitude but the cumulative gate displacement after a time t can be
scat
opposite sign for the and states, we get written as [9, 53]
|↑↓i |↓↑i
∆φ = ∆φ ∆φ. (47)
↑↓ − ↓↑ ≡ α(t)= i [e−iδt 1]eiΦL, (53)
Theerroneousgatecanbethereforerepresentedbythe 2√K −
operator
where Φ is the gate phase, determined by the optical
L
1 0 0 0 phase difference betweenthe gate beams at the ion’s po-
Uˆǫ =00 ei(φ+0∆φ) ei(φ−0∆φ) 00. (48) sbietiownr.itItnenthaesrotatingframe,the recoildisplacementcan
0 0 0 1
 
  β = β eiωtrapt. (54)
The final state fidelity following the erroneous gate de- | |
pends on Ψ . For a general initial state
init
| i Substituting equations (54) and (53) into Eq. (46), we
Ψ =µ +κ +γ +δ , (49) can write the averagedterm in the fidelity
init
| i |↑↑i |↑↓i |↓↑i |↓↓i
δ |β|max 2δπ β
cos2∆φ = S(β )cos2 | | [cos((ω +δ)t Φ ) cos(ω t Φ )] dtdβ . (55)
trap L trap L
h i 2π | | √K − − − | |
Z|β|=0 Zt=0 (cid:20) (cid:21)
Here we assume that the probability of scattering at dif- ferent time intervals during the gate is uniform. The