Randomized benchmarking of single qubit gates in a 2D array of neutral atom qubits

0.47 MB·English

by  T. Xia

Randomized benchmarking of single qubit gates in a 2D array of neutral atom qubits
T. Xia, M. Lichtman, K. Maller, A. W. Carr, M. J. Piotrowicz, L. Isenhower, and M. Saffman
Department of Physics, University of Wisconsin, 1150 University Avenue, Madison, WI 53706
We characterize single qubit Clifford gate operations with randomized benchmarking in a 2D
array of neutral atom qubits, and demonstrate global and site selected gates with high fidelity.
An average fidelity of F2 = 0.9983(14) is measured for global microwave driven gates applied to
a 49 qubit array. Single site gates are implemented with a focused laser beam to Stark shift the
microwavesintoresonanceataselectedsite. AtStarkselectedsinglesitesweobserveF2 =0.9923(7)
and an average spin flip crosstalk error at other sites of 0.002(9).
5 PACSnumbers: 03.67.-a,31.10.Jk,42.50.Dv,03,67.Lx.
1
0
2 Qubits encoded in hyperfine states of neutral atoms sites. Arbitrary rotations on the Bloch sphere are im-

b are one of several promising approaches for scalable im- plemented using variable length and phase microwave
e plementations of quantum information processing[1]. In pulses. In contrast to previous experiments which used
F
this letter we demonstrate and characterize single qubit either adiabatic pulses that do not provide full control

8 gate operations in a 2D array of up to 49 neutral atom on the Bloch sphere[7], or spatially periodic Stark shift-
1 qubits encoded in long lived hyperfine states. Using a ing techniques which do not address single sites[8, 9], we
microwavefieldweimplement arbitraryBlochspherero- demonstrate full control at single sites of a 2D qubit ar-
]
h tations on either the entire 2D qubit array in parallel, ray. Using RB techniques we characterize the fidelity of
p or on single sites that are selected by an auxiliary Stark Clifford group gates, as well as the crosstalk during site
- shifting laser beam[2]. Single atom qubits are stochasti- selected gates[10].
t
n cally loaded into the array, with an average of 29 sites
a RB was introduced in [3] as an efficient approach for
filledforthedatareportedhere. Thisisthelargestnum-
u characterization of quantum gate fidelities. It has sev-
berofindividuallycontrollablequbitsforwhichquantum
q eral advantages compared to full tomography including
[ gate operations have been characterizedto date. The fi-
resource requirements that scale linearly with the num-
delity of the global operations, site selected operations,
3 ber of qubits and the capability of distinguishing gate
and crosstalk during site selected operations are quanti-
v errorsfrom state preparationand measurement(SPAM)
1 fied using randomized benchmarking (RB)[3].
errors. RBhasbeenusedforcharacterizationofone-and
4
Control of individual qubits in a spatially extended two-qubitgates,aswellasquantumprocesses,onavari-
0
2 arrayis animportantcapability inongoingefforts to de- etyofqubitplatformsincludingions[3,11],nuclearmag-
0 velop scalable quantum processors. Atomic qubits en- netic resonance[12], superconductors [13], neutral atoms
. coded in hyperfine ground states can be controlled with [9, 14, 15], and quantum dots[16]. We encode qubits
1
0 optical or microwave frequency fields. Optical fields in the Cs clock states with 0 f = 3,mf = 0 ,
| i ≡ | i
5 can be tightly focused to address individual qubits as 1 f = 4,m = 0 . Our implementation of RB
f
1 has been demonstrated in several experiments[4]. When u|sies≡th|e complete set oif 24 Clifford gates 1 . These
: C
v the spatial separation of qubits is comparable to, or are generated from the set I,Rj( π/2),Rj(π) where
Xi less than, the opticalwavelength,addressingby focusing Rj(θ) = e−ıθσj/2 with σj {Pauli ±matrices abo}ut axes
alone is not sufficient to suppress crosstalk to neighbor- j = x,y,z. We use constant amplitude pulses of mi-
r
a ing sites. Addressing with subwavelength resolution can crowaveradiationresonantwiththe ωq =2π 9.19GHz
×
be achieved using quantum interference techniques[5] or 0 1 clocktransitionforR rotations. Phaseshifting
x
| i↔| i
by using an additional external field gradient to select a themicrowavesprovidesR rotations. R operationsare
y z
desired site. This latter method has been implemented implemented by composing x and y axis rotations. The
with magnetic field gradients[6], or with auxiliary Stark microwavepulses usedfor eachCliffordgateare listedin
shifting optical beams in conjunction with microwave the supplementary material.
fields[2, 7–9]. The use of microwave fields for qubit con-
Themainelementsoftheatomicexperimentareasde-
trolis particularlyconvenientsinceboth the globalrota-
scribed in [17]. In brief, a two-dimensional array of blue
tions which are the starting point for many quantum al-
detuned optical traps is defined using 780 nm light pro-
gorithms,aswellassinglequbitcontrolneededforgates,
jectedintoapyrexultrahighvacuumcell. The7 7=49
can be implemented with the same control hardware. ×
site array has a 3.8 µm site to site spacing and trap
In this letter we demonstrate global and site selected depths of 400 µK for Cs atoms. The Cs atoms are
∼
single qubit gates using microwave drive fields with a collectedina 2Dcoolingregion,transferredto the pyrex
tightly focused Stark shift beam. We derive optimal val- cell with a push beam, and then trapped and cooled in
ues for the Stark shift which minimize crosstalk to other a 3D magneto-optical trap (MOT). The array is then
2
a) counts loovaedrilnapg 06.70% % 44 b) 1.0a) b0).02223 0.02224 0.00325
33 <N>=28.9
>0.8
ces y |0 30 31 32
occurren 0 2p6h7oto3e4l4ectron4s60 probabilit00..46 .09237 38 .01939
0.2 0.058 0.014 0.014
3.8 µm 0.0
0 50 100 150 200 250 300
number of sites loaded time (µs)
FIG. 1. (color online) Qubit array measurements. a) False- FIG. 2. (color online) a) Microwave Rabi oscillations at a
colorfluorescenceimageofsingleatomqubitsfilteredwithan singlesiteduringglobaladdressing. Eachpointistheaverage
independent component analysis. Each pixel views a region of50measurementsandthefittedRabifrequencyisΩ=2π
×
0.62 0.62 µm The image is a composite averaged over 500 4.74 kHz. b) Oscillations on site 31 using Stark addressing
expo×sures. b) Histogram of the number of occupied sites for with the average probability of measuring a neighboring site
500 array loading measurements. The inset shows the single in the wrong state due to crosstalk during a 3π oscillation
atom photoelectron counts at a single site for 2000 measure- indicated numerically. The Rabi frequency for this data was
ments. TheaverageoverlapoftheGaussianfitsatall49sites Ω = 2π 8.5 kHz and the sites are numbered from 0 in the
×
was 0.0004. upperleft corner to 48 in thelower right corner.
turned on, the MOT quadrupole field is turned off, and ues are 17 s for the 1/e atom lifetime, 0.59 s for T1 and
the captured atoms are cooled to 5-10 µK using polar- 14 ms for T∗. The T∗ value, which is measured using
2 2
ization gradient cooling. About 5 ms of near resonant microwave Ramsey spectroscopy, is dominated by mag-
lightat852nmisusedtoinvokephoto-assistedcollisions netic noise and finite temperature motional effects[19].
which remove atoms in multiply occupied sites. After The9.19GHzmicrowavesourceislockedtoaGPSdisci-
this preparationstep multiple atoms are notobservedat plinedcrystaloscillator. Thefrequencyisslightlyshifted
any site. The presence of an atom is detected by fluo- from the free space Cs clock transition due to the mag-
rescence imaging as shown in Fig.1. We observe single netic bias field andthe 500Hz lightshift atthe center
∼
atomloadingratesapproaching70%atafewsites,which of each trapping site. A maximum T∗ of 50 ms has
2
∼
is suggestive of repulsive light assisted collisions[18]. On been observed at a few sites which we attribute to vari-
averageweload60%ofthearraysiteswithasingleatom ations in the cooling efficiency and atom temperature.
in each experimental run. We anticipate thatthe T∗ value canbe substantiallyim-
2
An initial fluorescence image reveals which sites are proved in future experiments using trap compensation
loadedwithqubitsforeachexperimentalrun. Theatoms techniques[20].
are then optically pumped into 1 using π polarized894 We proceed with RB experiments to measure the fi-
| i
nm light resonant with 6s ,f = 4 6p ,f = 4 delity of single qubit gate operationswith microwavera-
1/2 1/2
| i → | i
and π polarized 852 nm repump light resonant with diation from a horn external to the vacuum cell driving
6s ,f = 3 6p ,f = 4 . The quantization axis all qubits in parallel. Random Clifford gate sequences of
1/2 3/2
| i → | i
is perpendicular to the plane of the array and is defined length ℓ are generated with each gate chosen uniformly
by a 0.15 mT magnetic bias field. After the quantum from 1. The average pulse area per gate was 7π/4 (see
C
gate operations described below a state sensitive mea- supplemental material). We start with all qubits in 1 .
| i
surementisperformed. Tomeasuretheprobabilityof 0 At the end of each sequence we add a final gate which,
| i
we push out atoms in f = 4 with unbalanced resonant intheabsenceoferrors,shouldtransferthequbitsto 0 .
| i
light pressure (6s ,f = 4 6p ,f = 5 ) and then In the presence of depolarization errors the probability
1/2 3/2
| i →| i
measure the presence of an atom by integrating the flu- of measuring 0 is
| i
orescence from MOT light with detuning −7γ6p3/2 and 1 1
resonant saturation parameter 3.3 for 20 ms. This re- P|0i = 2 + 2(1−dif)(1−d)ℓ. (1)
sults in highfidelity discriminationofthe qubitstates as
is seen in Fig. 1b). To measure the probability of 1 Heredif isthedepolarizationprobabilityassociatedwith
| i
we apply a R (π) microwave pulse before the push out state preparation,the final transfer gate, and state mea-
x
and fluorescence measurement. All sites in the arrayare surement, while d is the averagedepolarizationof a Clif-
measured in parallel using camera detection of the flu- ford gate. Using the standard definition[21] of the fi-
orescence. The dominant error in state measurement is delity of two density matrices ρ,ρ′ given by F(ρ,ρ′) =
the small probability of transferringanatom from f =4 Tr[p√ρρ′√ρ] one can readily show that the square of
to f =3 during the pushing out step. the average gate fidelity is F2 = 1 d/2. The quantity
Inpreparationforquantumgateexperimentsthequbit F2 is equal to the average fidelity o−f a Clifford gate F
a
array characteristics are measured. Array averaged val- introduced in [3].
3
1.0 TABLE I. Results of RB fidelity measurements for global
(first five rows) and single site addressing. The last three
y |>00.8 rhaoEnwdxtsiEannrxetthhEenxnativthetrheaeagraervaceyrroaasgvseetracalrkgoesfdsotrcarltkohseesrtrnaolekraroeexnscttluhndeeiiengnghtbitrohereansriertaaeyrs-,,
bilit0.6 ubits15 est nheighib−or sites.
roba0.4 mber of q150 hhddiif4i747sitseitses 00.0.009352±±00..0006267
p0.2 nu F2 0.9983 0.0014
0 47 sites ±
0.994 0.996 0.998 1.00
0.0 1 23 nFu2mb45er of gate67s 89 100 (cid:10)FFmm22ianx(cid:11) 00..99993999±±00..00000073
F2 0.9923 0.0007
FIG. 3. (color online) Probability of measuring the correct single site ±
output state at site 27 of the array for 7 RB sequences. dif xt 0.037 0.027
h i ±
Each sequence was truncated at 10 different lengths ℓ = Ext 0.002 0.009
1,12,23,34,45,56,67,78,89,100 . Eachdatapointisanav- h i ±
{ } Ext nn 0.014 0.02
erage of 50 measurements. The inset shows a histogram of h i ±
gate fidelities for 47 of the 49 array sites. Two sites were hExti−nn 0.0005±0.001
dropped dueto poor loading statistics.
by a factor scaling as Ω2/δ2. We then selected a de-
siredsite using a tightlyfocused 459nm beamwith1/e2
We applied 7 randomized Clifford gate sequences for intensity radii of w = 3.2, w = 2.7 µm detuned by
x y
all the trapping sites. Representative data from a single ∆S =2π 20 GHz from the 6s1/2,f =4 7p1/2,f =
site are shownin Fig. 3 together with a histogramof F2 4 transit×ion. The beam size| was choseni↔as|a compro-
i
acrossthearrayobtainedbyextractingdfromfitstoEq. mise between tight focusing which gives small crosstalk
(1) at each site. The results are summarized in Table I. to neighboring sites, and loose focusing which reduces
The highest fidelity seen at any site was 0.9999(3) with sensitivityto beammisalignmentonthe targetsite. The
an arrayaverageof 0.9983(14). These values are compa- intensityofthe 459nmbeamwasadjustedsuchthatthe
rable to the highest fidelity neutral atom gates reported induced differential Stark shift of states 0 , 1 was set
| i | i
to date[14], who reported a global average RB fidelity equal to δ to bring a selected site into resonance[2]. The
of 0.99986(1). An indication of where these experimen- 459 nm Stark beam was σ+ polarized and propagated
tal results stand in relation to theoretical thresholds for normalto the plane of the array. Using a pair of orthog-
fault tolerant quantum computing can be found by con- onal acousto-optic modulators the Stark beam could be
sulting Table 8 in Ref. [22]. scanned to a desired qubit site with a switching speed
In order to understand the sources of the observeder- under 0.5 µs.
rors we simulated the RB experiment allowing for de- The choice of detuning δ for single site gates is a
tuning from ω by up to 100 Hz, which corresponds to tradeoffbetweenlessthanperfectsuppressionofthemi-
q
the average differential Stark shift of the trapped qubit crowave field at small δ and excessive photon scattering
states across the array. We also included pulse length from the Stark beam at large δ. For a given value of δ
timingerrorsofupto200nscorrespondingto0.2%drifts thephotonscatteringcanbereducedbyworkingatlarge
of the microwave power. Accounting for these imperfec- opticaldetuning∆ ,butnotcompletelyeliminatedsince
S
tionspredictsgateerrorsseveraltimessmallerthanthose forlarge∆ ,δ ω /∆2 andtendstozero. Anoptimized
S ∼ q S
observed. Including a density matrix coherence decay working point which reduces the need for large δ can be
factor[19]α(t,T2∗)=1/2+(1/2)/[1+0.95(Tt2∗)2]3/2 wees- found by choosing a detuning for which the off-resonant
timate F2 =1 d /2=1 [1 α( t ,T∗)]/2. Putting coupling to non-selected sites gives a pulse area which is
t =h θi /Ω−h=i(7π/4)−/(2π− h4.i7C41 kH2z) = 185 µs amultiple of4π andthereforedoesnotdisturbthe qubit
ahnidC1T2∗ =h 2i.C71ms we recover the×observed (cid:10)F2(cid:11)47 sites = states. For a pulse area of θR on the targeted qubit the
0.9983fromTableI.ThemedianT∗observedinthearray condition for minimal disturbance of non-targeted sites
2
is overtwice longer at 7.0 ms. We conclude that the fac- is δ/Ω = (cid:0)n216π2/θ2 1(cid:1)1/2 with n an integer. Thus
R−
torslimitingthegatefidelityfoundfromRBexperiments the leakageerrorshouldhaveafirstlocalminimum for a
are a combination of finite T∗ which could be improved π pulse at δ/Ω √15.
2
≃
using echo techniques or trap compensation[20], and er- Thisestimatecanbeverifiedbyacalculationwhichav-
rors in the pulse length and detuning. eragesoverallpossiblestates ofthe non-targetedqubits.
To performgates on individual qubit sites we detuned Let the initial state be θ,φ = cos(θ)0 +eiφsin(θ)1 .
| i 2 | i 2 | i
themicrowavefrequencyω byδ =ω ω 2π 33kHz. This state receives a unitary transformation θ,φ
q
− ≃ × | i →
This detuning suppresses the microwave qubit rotation U (θ ,δ)θ,φ with U (θ ,δ) the operator for a θ ro-
j R j R R
| i
4
0.1
a) Rz(π/2) 1.0
0.08
Rz(π)
>0.8
0
0.06 |
y
t0.6
0.04 bili
a
or b0.4
r 0.02 Rx(π/2) Rx(π) o
r r
e p
k 0.2
al 0
st 0.1 b)
s
o 0.0
cr 0.08 1 15 number of 2g9ates 43 50
0.06 FIG. 5. (color online) RB data at Stark beam addressed site
31 for 10 RB sequences. Each data point is an average of 50
0.04 measurements.
0.02
0 12(1−dif)(1−dxt)ℓ after each Clifford sequence was ap-
0 2 4 6 8 10 plied. DroppingtheStarkaddressedsite andsiteswhose
δ/Ω
R loadingwaspooryieldsanaverage dxt . Thearrayaver-
h i
FIG. 4. (color online) Crosstalk error for R (π/2)(blue solid agedbackgrounderroronnon-addressedsitesperClifford
x
line), Rx(π)(yellow dashed line), Rz(π/2)(blue dotted line), was Ext = dxt /2 = 0.002(9). Due to the finite size
andRz(π)(yellowdashed-dottedline) rotations. Thevertical of thhe Stiark bheami there was intensity overlap to near-
dashed lines mark values of √16n2 1. Panel a) is the error
− est neighbor sites that was as high as 5%, resulting in
from Eq. (2) averaged over input states and panel b) is the
increased crosstalk compared to further away sites. The
error for theinitial state 1 .
| i crosstalkvaluesforthe neighboringsitesaregiveninTa-
ble I. The average crosstalk error is comparable to the
tation about axis j detuned by δ. The fidelity of the average error of global gates. However, this result was
transformed state with respect to the original state, av- obtainedforaninitialstateinthenontargetedsitesof 0
| i
eraged over the Bloch sphere, is and therefore essentially correspondsto a spin flip error.
Itistobeexpectedthatmeasurementswitharbitraryini-
F(θR,δ)= R0πdθ sin(θ)R02πdφ|hθ,φ|Uj(θR,δ)|θ,φi|2. tainadl sbt)a)t.esUwltoimuladteyliye,ldahsliigghhetrlyersrmoraslle(rcoSmtaprakrebFeaigms,. a4nad)
4π
(2) larger values of δ/Ω than have been demonstrated here
ThecrosstalkerrordefinedasExt =1 F(θR,δ)isshown should be used for effective crosstalk suppression.
−
in Fig. 4 for several elements of 1. We see that the In summary we have demonstrated high fidelity sin-
C
simple estimate of δ/Ω √15 for a R (π) rotation is gle qubit gate operations in a 2D array of neutral atom
x
≃
verified by the full calculation. Since the error is min- qubits. Using microwavepulses we perform either paral-
imized at different detunings for different rotations the lel gates on all qubits, or gates on single qubits selected
detuning should be dynamically adjusted in tact with by a Stark shifting beam. The results reported,together
the gate being performed. In the demonstration de- withthedemonstrationoftwo-qubitentanglementinthe
scribed below we have simply used a fixed detuning of arrayusingRydbergblockadegates,whichwewillreport
δ/Ω= 2π×33 kHz =3.88 √15. on elsewhere[23], are a step towards scalable quantum
2π×8.5 kHz ≃
To characterize site selected gates we applied 10 ran- computing with neutral atom qubits.
domized Clifford sequences as shown in Fig. 5. Each This work was supported by the IARPA MQCO pro-
sequence was truncated at 8 different lengths 1, 8, 15, gram through ARO contract W911NF-10-1-0347.
{
22, 29, 36, 43, 50 . Averaging over the 10 sequences
yields an average g}ate fidelity F2 =0.9923, giving a per
gate error which is about 4.5 times larger than for the
array averagedglobal gates. We attribute this to fluctu-
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1
SUPPLEMENTARY MATERIAL FOR RANDOMIZED BENCHMARKING OF SINGLE QUBIT GATES
IN A 2D ARRAY OF NEUTRAL ATOM QUBITS
In this supplementary material we specify the microwave pulses used for each of the 24 gates in the single qubit
Cliffordgroup 1. ThegroupelementsandourimplementationarelistedinTableI.Notethatrotationsof π/2were
C −
implemented as +3π/2 rotations. We therefore used microwave pulses which were longer than necessary giving an
averagepulse area per Clifford gate of θ =7π/4. If all 3π/2 operations had been replaced by π/2 rotations this
h iC1 −
would have reduced the average pulse area to 55π/48. As discussed in the main text the finite T∗ time accounts for
2
about half of the observedgate error. This could be improvedon simply by using the shorter π/2 rotations instead
−
of 3π/2 rotations.
2
index x axis y axis z axis U R3 R2 R1 θtotal
1 0
1 I I I - - - 0
0 1!
1 0
2 I I π/2 e−ıπ/4 Rx(π/2) Ry(π/2) Rx(3π/2) 5π/2
0 i!
1 0
3 I I π i - R (π) R (π) 2π
x y
− 0 1!
−
1 0
4 I I π/2 eıπ/4 R (π/2) R (3π/2) R (3π/2) 7π/2
x y x
− 0 i!
−
0 1
5 I π I 1 - - R (π) π
y
− 1 0!
−
0 1
6 I π π/2 eıπ/4 R (π/2) R (π/2) R (π/2) 3π/2
x y x
− i 0!
0 1
7 π I I i - - R (π) π
x
− 1 0!
0 1
8 π I π/2 e−ıπ/4 Rx(π/2) Ry(3π/2) Rx(π/2) 5π/2
i 0!
−
1 1
9 π π/2 I √−2i 1 1! - Rx(π) Ry(π/2) 3π/2
−
1 1
10 I π/2 I 1 - - R (3π/2) 3π/2
− √2 1 1! y
−
11 π/2 I π/2 e−ıπ/4 1 1 - R (3π/2) R (π/2) 2π
√2 i i! y x
−
12 π/2 π π/2 eıπ/4 1 1 - R (3π/2) R (3π/2) 3π
− √2 i i! y x
−
1 1
13 π −π/2 I √i2 1 −1! - Rx(π) Ry(3π/2) 5π/2
− −
14 −π/2 I π/2 e−√ıπ2/4 1i −i1! - Ry(π/2) Rx(3π/2) 2π
1 1
15 I π/2 I √12 1 −1 ! - - Ry(π/2) π/2
16 −π/2 π π/2 eı√π2/4 1i −1i! - Ry(π/2) Rx(π/2) π
− −
17 π/2 π/2 I e−ıπ/4 1 i - R (3π/2) R (3π/2) 3π
− − √2 1 i! x y
−
18 π/2 π/2 I eıπ/4 1 i - R (3π/2) R (π/2) 2π
− √2 1 i! x y
−
1 i
19 π/2 π I i - R (3π/2) R (π) 5π/2
− √2 i 1! x y
− −
1 i
20 π/2 I I 1 - - R (3π/2) 3π/2
− √2 i 1! x
21 π/2 −π/2 I eı√π2/4 11 −ii! - Rx(π/2) Ry(3π/2) 2π
− −
1 i
22 π/2 I I √12 i −1! - - Rx(π/2) π/2
−
1 i
23 π/2 π I √−2i i −1! - Rx(π/2) Ry(π) 3π/2
−
24 π/2 π/2 I e−√ıπ2/4 11 −ii! - Rx(π/2) Ry(π/2) π
TABLEI.ElementsoftheClifford group 1 forasinglequbit. Thenotationθ incolumn2,3or4isshorthandforRj(θ)about
C
the corresponding axis. The resulting operator U is due to the sequence U U U , reading right to left. The operators have
x y z
been factored so thatthefirst nonzero element in thetop rowis unity. Themicrowave implementation R3R2R1,reading right
toleft,usedtherotationsshowninthelastthreecolumns. ThesegivethesameCliffordoperationsasU,uptoirrelevantglobal
phase factors.