Optimal quantum control of multi-mode couplings between trapped ion qubits for scalable entanglement

1.4 MB·

by  T. Choi

Optimal quantum control of multi-mode couplings
between trapped ion qubits for scalable entanglement
T. Choi,1 S. Debnath,1 T. A. Manning,1 C. Figgatt,1 Z.-X. Gong,1,2 L.-M. Duan,2 and C. Monroe1
1Joint Quantum Institute, University of Maryland Department of Physics and
National Institute of Standards and Technology, College Park, MD 20742
2Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
(Dated: January 9, 2014)
We demonstrate high fidelity entangling quantum gates within a chain of five trapped ion qubits
by optimally shaping optical fields that couple to multiple collective modes of motion. We individ-
ually address qubits with segmented optical pulses to construct multipartite entangled states in a
programmable way. This approach enables both high fidelity and fast quantum gates that can be
scaled to larger qubit registers for quantum computation and simulation.
4
1 Trapped atomic ion crystals are the leading archi- represented by the 2S hyperfine “clock” states within
1/2
0 tecture for quantum information processing, with their 171Yb+, denoted by |0(cid:105) and |1(cid:105) and having a splitting of
2
unsurpassed level of qubit coherence and near perfect ω /2π =12.642821GHz[14]. Weinitializeeachqubitby
0
n initialization and detection efficiency [1, 2]. Moreover, optically pumping to state |0(cid:105) using laser light resonant
a
trappedionqubitscanbecontrollablyentangledthrough with the 2S ↔ 2P transition near 369.5 nm.
J 1/2 1/2
theirCoulomb-coupledmotionbyapplyingexternalfields We then coherently manipulate the qubits with a
8
that provide a qubit state-dependent force [3–6]. How- mode-lockedlaserat355nmwhosefrequencycombbeat
] ever, scaling to large numbers of ions N within a single notes drive stimulated Raman transitions between the
h
crystal is complicated by the many collective modes of qubit states and produce qubit state-dependent forces
p
motion,whichcancausegateerrorsfrommodecrosstalk. [15, 16]. The Raman laser is split into two beams, one
-
t Such errors can be mitigated by coupling to a single mo- illuminating the entire chain and the other focused to
n
a tional mode, at a cost of significantly slowing the gate a waist of ∼ 3.5 µm for addressing any subset of ad-
u operation. The gate time τg must generally be much jacent ion pairs in the chain, with a wavevector differ-
q longer than the inverse of the frequency splitting of the ence ∆k aligned along the x-direction of transverse mo-
[
motional modes, which for axial motion in a linear chain tion. We finally measure the state of each qubit by ap-
1 implies τg (cid:29) 1/ωz > N0.86/ωx, where ωz and ωx are plying resonant laser light near 369.5 nm that results in
v thecenter-of-massaxialandtransversemodefrequencies state-dependent fluorescence [14] that is imaged onto a
5 [7]. For gates using transverse motion in a linear chain multi-channel photo-multiplier tube (PMT) for individ-
7
5 [8], we find τg (cid:29) ωx/ωz2 > N1.72/ωx. In either case, the ual qubit state detection. We repeat each experiment at
1 slowdown with qubit number N can severely limit the least 300 times and extract state populations by fitting
. practical size of trapped ion qubit crystals. to previously measured fluorescence histograms [17].
1
0 In this letter, we circumvent this scaling problem by When a constant state-dependent force is applied to
4 applyingqubitstate-dependentopticalforcesthatsimul- the ion qubits, the multiple incommensurate modes gen-
1 taneously couple to multiple modes of motion. We ad- erally remain entangled with the qubits following the in-
:
v dresssubsets ofionsimmersedin afive-ionlinearcrystal teraction, thereby degrading the quantum gate fidelity.
i and engineer laser pulse shapes to entangle pairs of ions However, more complex optical pulses can be created
X
with high fidelity while suppressing mode crosstalk and that satisfy a set of constraints for disentangling every
r
a maintaining short gate times [8, 9]. The pre-calculated mode of motion following the gate. This optimal control
pulseshapesoptimizetargetgatefidelity,achievingunity problem involves engineering a sufficiently complex laser
for sufficiently complex pulses. In the experiment, we pulse that maximizes or even achieves unit fidelity.
concatenatetheseshapedgatestoentanglemultiplepairs The qubit state-dependent optical force is applied by
of qubits and directly measure multi-qubit entanglement generating bichromatic beat notes near the upper and
in the crystal. Extensions of this approach can be scaled lower motional sideband frequencies at ω0 ± µ, where
to larger ion chains and also incorporate higher levels of the detuning µ is in the neighborhood of the motional
pulse shaping to reduce sensitivity to particular experi- mode frequencies. Using the rotating wave approxima-
mental errors and drifts [10–12]. tion in the Lamb-Dicke limit, the evolution operator of
the dipole interaction Hamiltonian becomes [9, 18, 19]
In the experiment, five 171Yb+ ions are confined in
a three-layer linear rf trap [13] with transverse center-
 
of-mass (CM) frequency ranging from ω /2π = 2.5 −
x
4.5MHzandaxialCMfrequencyωz/2π =310−550kHz, Uˆ(τ)=exp(cid:88)φˆi(τ)σˆx(i)+i(cid:88)χi,j(τ)σˆx(i)σˆx(j). (1)
with a minimal ion separation of ∼5 µm. Each qubit is i i,j
2
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|ts0ot(cid:105)att±he-e|d1pe(cid:105)phseatnsaedteeosnftftohdleliospbwliacthchereommternaattjiecocftboterhaieetsnioo±tneαsii,[ms2u(0cτ]h.)Titnhhaipsthitashsaee nilgnatnElgnatnE00..24 t4il.t3 4D D e et TTEE ucc55hhxxtuooeepp--nssooeennnrrrreeissyyiinmmigg ttn((aammcfeegignnonnv4ee entttt  .-s((nnppscf3titteaouuv/ 6gppn2enllssmt-suu seeptellea ssu ng(n(((eeltMmstetM h pepx((eu)euteHnpHhllosxts. e)ezrppez y)ou).))))lrs ye)) 4C.M38 PP --4202-4CM-2 XX0 123452snrttssssshhdtd seeeeeeggggggPm..... e15234n --t44202-4 -2 X0 tilt2 4
space of the mth motional mode according to [9]
(cid:90) τ FIG. 1. Improvement of entangled state creation using pulse
α (τ)=iη Ω (t)sin(µt)eiωmtdt. (2) shaping on N = 2 trapped ion qubits. (a) Comparison of
i,m i,m i
0 Bell state entanglement fidelity for a constant pulse (black)
Here, η = b ·∆k(cid:112)(cid:126)/2Mω is the Lamb-Dicke pa- versus a five-segment pulse (red) over a range of detuning
i,m i,m m µ,showingsignificantimprovementwiththesegmentedgate.
rameter, b is the normal mode transformation matrix
i,m (b)Thesegmentedpulsepattern, parameterizedbytheRabi
forioniandmodem[21],ωm isthefrequencyofthemth frequency Ωi(t) with the particular detuning µ near the 2nd
motionalmode,andM isthemassofasingle171Yb+ion. (“tilt”) motional mode (arrow in (a)) and measured state fi-
The second term of Eq. (1) describes the entangling in- delity≥94(2)%. (c)Phasespacetrajectories(arbitraryunits)
teraction between qubits i and j, with [9] subject to pulse sequence in (b) for both CM and tilt modes
of the two ions. The five-segment pulse pattern brings the
(cid:88) (cid:90) τ(cid:90) t(cid:48) two trajectories back to their origins, simultaneously disen-
χ (τ)=2 η η Ω (t)Ω (t(cid:48))
i,j i,m j,m i j tangling both modes of motion from the qubits.
(3)
m 0 0
×sin(µt)sin(µt(cid:48))sin[ω (t(cid:48)−t)]dtdt(cid:48).
m
In Eqs. (2-3), the time-dependent Rabi frequency Ω (t) Figure 1a shows theoretical and measured fidelity of
i
ontheithionisusedasacontrolparameterforoptimiza- the Bell state Uˆ(τg)|00(cid:105) = |00(cid:105)+i|11(cid:105) for both a sim-
tionofthegateandisassumedtoberealwithoutlossof ple constant pulse and a five segment pulse on a two-ion
generality. (We could alternatively vary the detuning µ chain, as a function of detuning µ for a fixed gate time
over time for control [22].) τg = 104 µs. For two ions, the five segments provide
InordertoperformanentanglingXX gateontwoions full control (2N +1 = 5), meaning that a pulse shape
a and b in a chain of N ions, we apply identical state- canbecalculatedateachdetuningthatshouldyieldunit
dependent forces to just these target ions and realize fidelity. As seen in Fig. 1a, a constant pulse can be op-
Uˆ(τ ) = exp[iπσˆ(a)σˆ(b)/4]. This requires χ (τ ) = π/4 timized to achieve high fidelity, but only at detunings µ
g x x a,b g
along with the 2N conditions α (τ ) = 0 so that the whose frequency difference from the two modes is com-
a,m g
phase space trajectories of all N motional modes return mensurate [19], which in this case has many solutions
to their origin and disentangle the qubits from their mo- spaced by 1/τg. The observed fidelity of the constant
tion [4–6]. These constraints can be satisfied by evenly pulse follows theory, with uniformly lower fidelities con-
partitioning the pulse shape Ω (t) = Ω (t) into 2N +1 sistent with known errors in the system. On the other
a b
segments[8,9],reducingtheproblemtoasystemoflinear hand,highfidelitiesofthefivesegmentpulseareobserved
equations with a guaranteed solution. The detuning and over a wide range of detunings for the same gate time,
gatedurationbecomeindependentparameterssothatin with the details of a particular pulse sequence shown in
principle, the gate can be performed with unit fidelity at Fig. 1b-c. We measure the fidelity by first observing
any detuning µ (cid:54)= ω and any gate speed on any two the populations of the |00(cid:105) and |11(cid:105) states, then extract-
m
ions in a chain, given sufficient optical power. ing their coherence by repeating the experiment with
When the gate is faster than the trap frequencies an additional global π/2 analysis rotation R(π/2,φ) and
[23, 24], the motion of the target ions is excited and measuring the contrast in qubit parity as the phase φ is
stopped faster than the response time of the chain. In scanned [25].
this case, the motion is better described using the basis WhenthenumberofionsinachainincreasestoN >2,
set of “local modes” involving only the two target ions, itbecomesdifficulttofinddetuningsµofaconstantpulse
therebyreducingthecontrolproblemto2N+1=5equa- whose difference frequencies µ−ω from all modes are
m
tions regardless of the total number of ions in the chain nearly commensurate, without significantly slowing the
[9, 24]. In the experiment, the minimum achievable gate gate. Figure 2 shows the state fidelity for a constant
time of τ ∼20 µs is considerably longer than the trap pulseversusanine-segmentpulseforentanglingadjacent
g
period of 2π/ω < 1 µs, implying that all 2N +1 con- ion pairs 1&2 or 2&3 within a five ion chain, with gate
x
trolparametersarerequired. However,ajudiciouschoice time τ = 190 µs. We find significant improvement over
g
of detuning can often reduce the number of parameters a wide range of detunings when using more segments,
required to achieve near-unit gate fidelities [8, 9]. even though fewer than 2N +1 = 11 control parame-
3
(a) A further advantage of using multi-segment pulses is
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2.50 2.55 2.60 tuning. AsseeninFig. 3,aconstantpulseisexpectedto
(b) DeDteutnuinnigng  /2 ( M(MHHzz))
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2.50 2.55 2.60
DeDtuetnuinnign g /2 ( M(MHHz)z) qubits in a linear crystal, we produce tripartite entan-
(c)
PP --4202CM-2 X0 X2 123456789 ssssssssssnrtttttthhhhhhdtd eeeeeeeeeseggggggggg4Pg.........m 192345678e--n 4202t til-2t 0X2 P4 --02423r-2d 0X2 P4 --02424t-2h 0-000X...505 -0.25 0.0 4P0.5--02425t-2h 0-000X...505-0.5 2 0.0 4 0.5 gcanhlceeraidgoinhsssbt(oatshrteeeepsafiFbixryigse.dcoofln4atacsha)ee.tretbnhWearaeteimenatgsdaiitranwgbeooatrtXidioceXanrlsltygaoanstadhedusiddtirtenelaesalsltyfihnvceereaeiroaieontsnest
a GHZ-type state
FIG.2. EntanglementofqubitpairswithinachainofN =5
trappedions. (a)Comparisonoftheoreticalentangledstatefi- |000(cid:105)→|000(cid:105)+i|110(cid:105)+i|011(cid:105)−|101(cid:105). (4)
delityforaconstantpulse(black)versusanine-segmentpulse
The measured state populations are consistent with the
(red)whenthegateisperformedonionpair1&2. Theblack
above state, as shown in Fig. 4b.
arrow indicates the optimal detuning for the constant pulse.
Therightpanelsshowmeasuredparityoscillationforthegate In order to measure the coherences of the three-qubit
detuningindicatedbytheredarrowalongwiththesegmented subsystem, we apply analysis rotations R(π/2,φ) to any
pulse pattern used at this detuning. (b) Same as (a), except two of the three qubits, then measure their parity as
the gate is performed on ion pair 2&3, with the gate detun- before. (The individual rotations are accomplished by
ing indicated by the blue arrow. (c) Phase space trajectories
adiabatically weakening the axial trap confinement and
(arbitrary units) for the solution on ion pair 1&2 in at the
shuttling the ions so that the focused Raman laser beam
detuning indicated by the red arrow in (a).
addressesjustthetargetion.) Asthephaseφoftheanal-
ysisrotations isscanned, the parityshouldoscillate with
y period π or 2π when the third ion is post-selected to be
tile in state |0(cid:105) or |1(cid:105), respectively, as seen in Fig. 4c for one
d
if e of the pairs. By measuring the contrasts of the two par-
ta itycurvesforeachofthethreepossiblepairsconditioned
ts
lac optimized constant pulse upon the measured value of the third, we obtain the six
ite fixed constant pulse coherences of the final state. Combined with the state
ro optimized 9-segment pulse
ehT fixed 9-segment pulse populations (Fig. 4b), we calculate a quantum state fi-
delity of 79(4)% with respect to Eq. (4). This level of
-2 -1 0 1 2
fidelity is consistent with the compounded XX gate fi-
Detuning error Dm/2p (kHz)
delities (∼ 95% each) and the discrimination efficiency
(∼93%) for post-selection of the third qubit.
FIG. 3. Theoretical entangled state fidelity as a function of
To prove genuine tripartite entanglement within the
detuning error ∆µ. The black (red) line corresponds to a
constant(nine-segment)pulseshapeonionpair1&2inafive- five ion chain, we use single qubit rotations to trans-
ionchainatthedetuningsindicatedbytheblack(red)arrows form the state given by Eq. 4 into a GHZ “cat” state
in Fig. 2a. The multi-segment approach is less sensitive to |000(cid:105) + i|111(cid:105) [26]. As shown in the circuit of Fig.
detuning (or trap frequency) fluctuations. 4d, this is achieved by applying a Z-rotation opera-
tionR (−π/2)=R(−π/2,0)R(π/2,π/2)R(π/2,0)tothe
z
middle ion only followed by R(π/2,0) rotations to all
ters are utilized. Using nine-segment pulses, we achieve three ions. We finally measure the parity of all three
state fidelities over 95(2)% for ion pairs 1&2 and 2&3 at qubitswhilescanningthephasesofsubsequentR(π/2,φ)
the detunings indicated by the red and blue arrows in analysispulses,andtheoscillationwithperiod2π/3with
Fig. 2a-b. In this overconstrained case, the calculation a contrast of over 70% (Fig. 4e) verifies genuine tri-
becomes an optimization problem, where more weight is partite entanglement [25]. This is a conservative lower
given to the closing of more influential phase space tra- limittotheentanglementfidelity,givenknownerrorsand
jectories (Fig. 2c). crosstalk in the rotations and the measurement process.
4
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[1] R. Blatt and D. Wineland, Nature 453, 1008 (2008).
|0
4 [2] C. Monroe and J. Kim, Science 339, 1164 (2013).
|0
5 [3] J. Cirac and P. Joller, Phys. Rev. Lett 74, 4091 (1995).

(e) 1.0 [4] K. Molmer and A. Sorensen, Phys. Rev. Lett 82, 1835
(1999).
y ytira 0.5 [5] E.Solano,R.deMatosFilho,andN.Zagury,Phys.Rev.
tiraP Z 0.0 A 59, R2539 (1999).
PH [6] G.Milburn,S.Schneider,andD.James,Fortschritteder
G-0.5
Physik 48, 801 (2000).
-1.0 Esixnp(3e r i m)ent Simulated [7] J. Schiffer, Phys. Rev. Lett 70, 818 (1993).
0 π 2π
[8] S.-L. Zhu, C. Monroe, and L.-M. Duan, Phys. Rev. Lett
PhPahsea se ( ra d)
97, 050505 (2006).
[9] S.-L. Zhu, C. Monroe, and L.-M. Duan, Europhys. Lett
FIG. 4. Programmable quantum operations to create tripar- 73, 485 (2006).
tite entanglement. (a) Circuit for concatenated XX gates [10] G. Kirchmair et al., New. J. Phys 11, 023002 (2009).
betweenions1&2and2&3andπ/2analysisrotationsofions [11] D. Hayes et al., Phys. Rev. Lett 109, 020503 (2012).
1&2 with phase φ. (b) Measured population after XX gates [12] Y. Tomita, J. Merrill, and K. Brown, New. J. Phys 12,
on ions 1&2 and 2&3, where PN denotes the probability of 015002 (2010).
finding N ions in the |1(cid:105) state. (c) Parity oscillations of ions [13] W.Hensingeretal.,Appl.Phys.Lett88,034101(2006).
1&2withthephaseφoftheπ/2analysisrotations,afterpost- [14] S. Olmschenk et al., Phys. Rev. A 76, 052314 (2007).
selecting the state of the third ion, with periods π (left) and [15] D. Hayes et al., Phys. Rev. Lett 104, 140501 (2010).
2π (right) for the two states |0(cid:105)3 and |1(cid:105)3, respectively (see [16] W.Campbelletal.,Phys.Rev.Lett105,090502(2010).
Eq. (4)). (d)SchematicforcreatingaGHZ“cat”stateusing [17] M. Acton et al., Quantum Inf. Comp 6, 465 (2006).
twoXX gatesonions1&2and2&3asbefore,withadditional [18] S.-L. Zhu and Z. D. Wang, Phys. Rev. Lett. 91, 187902
individualqubitrotations,followedbyπ/2analysisrotations (2003).
of all three ions with phase φ. (e) Three-ion parity oscilla- [19] K. Kim et al., Phys. Rev. Lett 103, 120502 (2009).
tionwithphaseφoftheπ/2analysisrotations. Theredsolid [20] P. Lee et al., J. Opt. B 7, S371 (2005).
lineisfittothedatawithperiod2π/3,whilethebluedashed [21] D. F. V. James, Appl. Phys. B 66, 181 (1998).
line is the expected signal assuming a perfect cat state with [22] S. Korenblit et al., New. J. Phys 14, 095024 (2012).
known systematic measurement errors. [23] J.Garcia-Ripoll,P.Zoller,andJ.Cirac,Phys.Rev.Lett
91, 157901 (2003).
[24] L.-M. Duan, Phys. Rev. Lett. 93, 100502 (2004).
The simulated blue dashed curve in the same figure de- [25] C. Sackett et al., Nature 404, 256 (2000).
picts what we expect to measure given our known errors [26] W. Dur, J. Cirac, and R. Tarrach, Phys. Rev. Lett 83,
and assuming a perfect initial state. 3562 (1999).
We have shown how a single control parameter can [27] C.Shen,Z.-X.Gong,andL.-M.Duan,Phys.Rev.A88,
052325 (2013).
be used to mitigate multi-mode couplings between a col-
[28] T. Monz et al., Phys. Rev. Lett. 102, 040501 (2009).
lection of qubit, but this approach can be expanded to
[29] R. Feynman, Int. J. Theor. Phys. 21, 467 (1982).
include additional parameters, such as spectral and spa-
[30] H. Walther, B. T. H. Varcoe, B.-G. Englert, and T.
tialaddressingofeachqubit[22,27]. Thiscouldallowfor Becker, Reports on Progress in Physics 69, 1325 (2006).
the efficient implementation of more complicated quan- [31] M. H. Devoret and R. J. Schoelkopf, Science 339, 1169
tumcircuits,suchasToffoli[28]andothergatesinvolving (2013).