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Architecture for a large-scale ion-trap quantum computer

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Architecture for a large-scale ion-trapquantum computer
D. Kielpinski
*
, C. Monroe
†
& D. J. Wineland
‡
*
Research Laboratory of Electronics and Center for Ultracold Atoms, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
†
FOCUS Center and Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1120, USA
‡
Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80305, USA
...........................................................................................................................................................................................................................
Among the numerous types of architecture being explored for quantum computers are systems utilizing ion traps, in whichquantumbits(qubits)areformedfromtheelectronicstatesoftrappedionsandcoupledthroughtheCoulombinteraction.Althoughthe elementary requirements for quantum computation have been demonstrated in this system, there exist theoretical andtechnical obstacles to scaling up the approach to large numbers of qubits. Therefore, recent efforts have been concentrated onusing quantum communication to link a number of small ion-trap quantum systems. Developing the array-based approach, weshow how to achieve massively parallel gate operation in a large-scale quantum computer, based on techniques alreadydemonstrated for manipulating small quantum registers. The use of decoherence-free subspaces signiﬁcantly reducesdecoherence during ion transport, and removes the requirement of clock synchronization between the interaction regions.
A
quantum computer is a device that prepares andmanipulatesquantumstatesinacontrolledway,offeringsigniﬁcant advantages over classical computers in taskssuch as factoring large numbers
1
and searching largedatabases
2
. The power of quantum computing derivesfrom its scaling properties: as the size of these problems grows, theresourcesrequiredtosolvethemgrow inamanageableway.Henceauseful quantum computing technology must allow control of largequantum systems, composed of thousands or millions of qubits.The ﬁrst proposal for ion-trap quantum computation involvedconﬁningastringofionsinasingletrap,usingtheirelectronicstatesasqubitlogiclevels,andtransferringquantuminformationbetweenionsthroughtheirmutualCoulombinteraction
3
.Alltheelementary requirementsforquantumcomputation
4
—
includingefﬁcientquan-tum state preparation
5–7
, manipulation
7–10
and read-out
7,11,12
—
havebeendemonstrated inthissystem.Butmanipulatingalargenumberof ions in a single trap presents immense technical difﬁculties, andscaling arguments suggest that this scheme is limited to compu-tationsontensofions
13–15
.Onewaytoescapethislimitationinvolvesquantum communication between a number of small ion-trapquantum registers. Recent proposals along these lines that usephoton coupling
16–18
and spin-dependent Coulomb interactions
19
have not yet been tested in the laboratory. The scheme presentedhere, however, uses only quantum manipulation techniques thathave already been individually experimentally demonstrated.
The quantum CCD
To build up a large-scale quantum computer, we have proposed a‘quantum charge-coupled device’ (QCCD) architecture consistingof a large number of interconnected ion traps. By changing theoperating voltages of these traps, we can conﬁne a few ions in eachtrap or shuttle ions from trap to trap. In any particular trap, we canmanipulate a few ions using the methods already demonstrated,while the connections between traps allowcommunication betweensets of ions
13
. Because both the speed of quantum logic gates
20
andthe shuttling speed are limited by the trap strength, shuttling ionsbetween memory and interaction regions should consume anacceptably small fraction of a clock cycle.Figure 1 shows a diagram of the proposed device. Trapped ionsstoring quantum information are held in the memory region. Toperform a logic gate, we move the relevant ions into an interactionregion by applying appropriate voltages to the electrode segments.In the interaction region, the ions are held close together, enablingthe Coulomb coupling necessary for entangling gates
3,21
. Lasers arefocusedthroughtheinteraction regiontodrivegates.Wethenmovethe ions again to prepare for the next operation.We can realize the trapping and transport potentials needed forthe QCCD using a combination of radio-frequency (r.f.) andquasistatic electric ﬁelds. Figure 1 shows only the electrodes thatsupport the quasistatic ﬁelds. By varying the voltages on theseelectrodes, we conﬁne the ions in a particular region or transportthemalongthelocaltrapaxis,whichliesalongthethinarrowsinFig.1. Two more layers of electrodes lie above and below the staticelectrodes,asshowninFig.2.Applyingr.f.voltagetotheouterlayerscreates a quadrupole ﬁeld that conﬁnes the ions transverse to thelocaltrapaxisbymeansoftheponderomotiveforce
22
.Thisgeometry allowsstabletransportoftheionsaround‘T’and‘X’junctions,sowecan build complex, multiply connected trap structures.
Figure 1
Diagram of the quantum charge-coupled device (QCCD). Ions are stored inthe memory region and moved to the interaction region for logic operations. Thinarrows show transport and conﬁnement along the local trap axis.
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Electrodestructuresfor the QCCD are relativelyeasy to fabricate.A number of ion traps have been built by laser-machining slits inaluminawafers and evaporating gold electrodes onto the alumina
23
.These traps have geometries similar to that needed for the QCCD,andhavespacingsbetweenthestaticandr.f.electrodesoffractionsof amillimetre,allowingconﬁnementfrequenciesupto20MHzforr.f.ﬁeld frequencies of
,
250MHz. Similar electrode structures arecurrently being constructed from heavily doped silicon using stan-dard microfabrication techniques
24
. Here the silicon acts as theconductiveelectrodematerial,whileglassspacersanodicallybondedto the silicon insulate the r.f. electrodes from the static electrodes.A ﬁrst step towards a QCCD has been taken at the NationalInstitute of Standards and Technology (NIST) by constructing apair of interconnected ion traps; the individual traps are similar tothoseusedinpreviouswork
23
andareseparatedby1.2mm.Efﬁcientcoherent transport of a qubit between the two traps was demon-strated by performing a Ramsey-type experiment involving the twotraps, where no loss of contrast within the experimental error(
,
0.6%) was observed
25
. Transport times were as short as
,
50
m
s,withcorrespondingionvelocitiesgreater than10ms
2
1
(seealsoref.26). The transport did not cause any heating of the ion motion orshortening of ion lifetime in the trap.To maximize the clock speed of the QCCD, we need to transportions quickly. However, the entangling gate demonstrated in pre-vious work at NIST
9,21
has low error only for ions cooled near thequantum ground state. To recool the ions after transport and tocounteract the effects of heating
23
, we propose to use sympatheticcooling of the ions used for quantum logic by another ionspecies
13,27,28
. Conﬁning both species in the interaction region letsus use the cooling species as a heat sink, with the Coulombinteraction transferring energy between the two species, as exper-imentally demonstrated in refs 29–31.
Decoherence
Whereas the decoherence and gate errors in single-trap quantumregisters have already been characterized, additional decoherencecan occur during ion transport. For instance, the energy splitting of the qubit states of an ion depends on the magnetic ﬁeld at the ionthrough the Zeeman effect. During ion transport, the spatialvariations of the magnetic ﬁeld strength along the transport pathcause the qubit states to acquire a path-dependent relative phase
a
,so that, for example,
j #
l
þ j "
l
!
j #
l
þ
e
i
a
j "
l
over transport. If wedo not know
a
, we have lost the phase information, effectively dephasingthequantumstate.Butknowing
a
forallrelevantpathsistantamount to characterizing the magnetic ﬁeld on micrometrelength scales across the entire device, a very difﬁcult task.Retaining phase information during a computation also requiresaccurate positioning of the ions in the interaction regions. As thelogic gate parameters depend on the phase of the driving laser ﬁeldsattheionpositions,theionpositionsandlaserpath lengthsmustbecontrollable with accuracies much better than an optical wave-length. Although this does not place unreasonable constraints onthe accuracy of the voltage sources driving the QCCD electrodes,stray electric ﬁelds emanating from the electrodes or mechanicalvibration of the QCCD can readily move the ions a fraction of awavelength from their nominal positions, effectively adding arelative phase
a
of the type considered above.
Decoherence-free encoding
We can reduce these sources of decoherence by several orders of magnitudebyencodingeachqubitintoadecoherence-freesubspace(DFS) of two ions
32–34
. This DFS is spanned by the states
j
0
l
¼ j #"
l
,
j
1
l
¼ j "#
l
of the ions. We refer to the ions as ‘physical qubits’ andcall the effective two-level system formed by
j
0
l
,
j
1
l
a ‘logical qubit’.If the state
j "
l
of ion
j
acquires a phase
a
j
over and above thepredicted phase
a
0
for the transport path, we see
j
0
l
þ j
1
l
!
e
i
a
1
j #"
l
þ
e
i
a
2
j "#
l
¼ j
0
l
þ
e
i
Da
j
1
l
ð
1
Þ
where we write
Da
;
a
2
2
a
1
:
If
Da
¼
0, we see that the DFSlogical qubit is unaffected by the acquired phase. Here the phases
a
i
are themselves unknown, but the ions acquire the same unknownphase, a process called collective dephasing. The logical qubitdecoheres only insofar as the dephasing fails to be collective.As an example, assume that the physical qubit energies have alineardependenceonmagneticﬁeldandthattheﬁeldvarieslinearly overanextendedQCCDdeviceofsize10cm.Ifeachpairofphysicalqubitscomprisingalogicalqubitisseparatedonaverageby10
m
m,alogical qubit dephases more slowly than a physical qubit by a factorof 10
4
. Again, Stark shifts of the qubit levels can be induced by theelectric ﬁelds that push the ions from place to place, but theirdephasing effects on the logical qubits are also suppressed. Ingeneral, the effect of any external ﬁeld varying over a length scale
L
ext
and inducingenergyshifts of
p
th power inthe ﬁeld issuppressedby a factor (
L
ext
/
d
)
p
for DFS encoding. A DFS-encoded qubit istherefore robust against decoherence incurred during transport.We can also perform universal quantum logic in the DFS, as wenow show. If we hold two ions in an interaction region, we can usethe entangling gate of refs 9 and 21 to apply the operatorU
2
ð
v
;
f
1
;
f
2
Þ ¼
cos
v
½
I
1
^
I
2
þ
i
sin
v
½
X
f
1
^
X
f
2
¼
cos
v
I
DFS
þ
i
sin
v
X
DFS
Df
12
ð
2
Þ
X
f
;
Xcos
f
þ
Ysin
f
ð
3
Þ
where X, Y are the Pauli operators and the superscript ‘DFS’indicates that the operator acts in the DFS logical basis. Thoughthe individual phases
f
1
,
f
2
depend sensitively on the path-lengthdifferences of the driving lasers over the macroscopic distance fromthe lasers to the ions
13
, the DFS gate phase
Df
12
¼
f
1
2
f
2
dependsonlyonthemicroscopicpath length ofthedriving laser betweenthetwo ions, which can be readily controlled by small adjustments of the trap voltages
9,12,24
. The DFS encoding makes the computationinsensitive to spatial phase ﬂuctuations; as we will see, it protectsagainst temporal phase ﬂuctuations aswell. We can set
v
over awiderange of values
21,24
, so the two-ion entangling gate lets us performarbitrary rotations of a single logical qubit. Using the same entan-gling gate on four ions, we can obtain the operatorU
4
¼
1
ﬃﬃﬃ
2
p ½
I
1
^
I
2
^
I
3
^
I
4
2
i
X
f
1
^
X
f
2
^
X
f
3
^
X
f
4
ð
4
Þ¼
1
ﬃﬃﬃ
2
p ½
I
DFS
^
I
DFS
2
i
X
DFS
Df
12
^
X
DFS
Df
34
ð
5
Þ
Figure 2
Conﬁguration of radio-frequency (r.f.) and static (d.c.) electrodes for theQCCD. Dotted regions indicate insulating spacers. Applying high r.f. voltage to the twoouter electrode layers while keeping the inner layer at r.f. ground creates the r.f.quadrupole ﬁeld shown by the arrows. This ﬁeld provides trapping potentials forconﬁnement transverse to the local trap axis, which points perpendicular to the pageand is located at the central black dot. View in Fig. 1 is from the top of this ﬁgure.
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2002 Nature PublishingGroup
whereions1and2encodeonelogicalqubitandions3and4encodeanother, and we write
Df
34
¼
f
3
2
f
4
:
This operator is equivalentto an XOR in the logical basis up to rotations of single logicalqubits
35
, so the operators of equations (2) and (5) sufﬁce foruniversal quantum logic.To use the DFS encoding in a large-scale quantum computation,we initialize the ions in pairs to the state
j # "
l
. Each pair of ionsremains in the DFS through the quantum computation, so thelogical qubits resist transport decoherence and all other types of collective dephasing. Read-out of the DFS qubit is straightforward,as we need only discriminate between
j # "
l
and
j " #
l
. All theoperators needed for universal quantum logic in the DFS havealready been experimentally implemented
9,24
, so we should be ableto use the DFS encoding in a large-scale quantum computer.Notably, all logic gate operations can be accomplished by uniformly illuminating the ions in the interaction region, removing the needfor tightly focused laser beams.
Logic gate synchronization
The DFS encoding also removes the requirement of clock synchro-nization between logic gates, a major but little-recognized obstacleto large-scale parallel quantum computation. As the energy levelsof our physical qubits are non-degenerate, we must keep track of theresulting phase accumulation to preserve the quantum informationin the physical qubit basis
34
. Parallel operations taking place inmany interaction regions thus require clocks that remain synchro-nized over the whole computation time
36
. Synchronization canbecome very difﬁcult for many qubits: for a transition frequency
q
0
between
j #
l
and
j "
l
, the two components of the state
j #
l
N
þ j "
l
N
acquire a signiﬁcant relative phase in a time
,
1/(
N
q
0
).Tomaintainphasestabilityofthecomputation,wethereforerequire a frequency reference with fractional frequency stability much better than
&
1/(
N
q
0
t
) at an averaging time
t
equal to theduration of the quantum computation.Tobeconcrete,weconsidertrapped
40
Ca
þ
ionsasqubits,withtheground S
1/2
state and metastable D
5/2
states as logic levels. Thissystem is being investigated for quantum computation bya numberof groups
7,15,37
. Here the transition frequency is 412THz, compar-abletothe533THzoperating frequencyofthecurrently moststablelaser oscillator
38
, which has a fractional frequency instability of 3
£
10
2
16
as 1s averaging time. If the computation takes about 1s,equal to the lifetime of the metastable D
5/2
state, we see that currenttechnology barely provides the appropriate phase stability for evenone ion. Of course, this argument can be regarded as too simplisticbecause the requirements on phase stability can be reduced by invoking error correction
39
; however, this comes at the cost of increased overhead
18
. For qubit levels with a transition frequency in the microwave regime, local oscillators of the required phasestabilityexist,buttheopticalpathlengthsbetweenthedriving lasersand each of the interaction regions must be stable at the nanometrelevel over the course of the computation
13
, a daunting task for a10cm QCCD device.On the other hand, as the logic levels of a DFS-encoded qubit aredegenerate, we do not need phase synchronization at all to performa logic operation within the DFS. Operations in the DFS are alsoindependentoftheopticalpathlengthsofthedrivinglasers,becausethe phases
Df
12
,
Df
34
in equations (2) and (5) depend only on thedistance between the two ions comprising a logical qubit. Theuniversal gate-set constructed above allows us to perform highly parallel computations in the DFS without synchronization betweengatesseparatedintimeorspace.Similarconsiderationswouldapply to other quantum computing architectures.
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Acknowledgements
We acknowledge the experimental contributions of the NIST Ion Storage group, and alsoJ. Beall for assistance with microfabrication. We thank D. Leibfried and M.A. Rowe forcomments on the manuscript. D.K. and D.J.W. were supported by the US NationalSecurity Agency (NSA), Advanced Research and Development Activity (ARDA) and theOfﬁce of Naval Research. C.M. was supported by the US NSA, ARDA and the NationalScience Foundation ITR programme.
Correspondence and requests for materials should be addressed to D.K.(e-mail: utonium@mit.edu).
progress
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2002 Nature PublishingGroup

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