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Architecture for a large-scale ion-trap

quantum 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 which

quantum bits (qubits) are formed from the electronic states of trapped ions and coupled through the Coulomb interaction. Although

the elementary requirements for quantum computation have been demonstrated in this system, there exist theoretical and

technical obstacles to scaling up the approach to large numbers of qubits. Therefore, recent efforts have been concentrated on

using quantum communication to link a number of small ion-trap quantum systems. Developing the array-based approach, we

show how to achieve massively parallel gate operation in a large-scale quantum computer, based on techniques already

demonstrated for manipulating small quantum registers. The use of decoherence-free subspaces signiﬁcantly reduces

decoherence during ion transport, and removes the requirement of clock synchronization between the interaction regions.

Aquantum computer is a device that prepares and

manipulates quantum states in a controlled way, offering

signiﬁcant advantages over classical computers in tasks

such as factoring large numbers

1

and searching large

databases

2

. The power of quantum computing derives

from its scaling properties: as the size of these problems grows, the

resources required to solve them grow in a manageable way. Hence a

useful quantum computing technology must allow control of large

quantum systems, composed of thousands or millions of qubits.

The ﬁrst proposal for ion-trap quantum computation involved

conﬁning a string of ions in a single trap, using their electronic states

as qubit logic levels, and transferring quantum information between

ions through their mutual Coulomb interaction

3

. All the elementary

requirements for quantum computation

4

—

including efﬁcient quan-

tum state preparation

5–7

, manipulation

7–10

and read-out

7,11,12

—

have

been demonstrated in this system. But manipulating a large number

of ions in a single trap presents immense technical difﬁculties, and

scaling arguments suggest that this scheme is limited to compu-

tations on tens of ions

13–15

. One way to escape this limitation involves

quantum communication between a number of small ion-trap

quantum registers. Recent proposals along these lines that use

photon coupling

16–18

and spin-dependent Coulomb interactions

19

have not yet been tested in the laboratory. The scheme presented

here, however, uses only quantum manipulation techniques that

have 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 consisting

of a large number of interconnected ion traps. By changing the

operating voltages of these traps, we can conﬁne a few ions in each

trap or shuttle ions from trap to trap. In any particular trap, we can

manipulate a few ions using the methods already demonstrated,

while the connections between traps allow communication between

sets of ions

13

. Because both the speed of quantum logic gates

20

and

the shuttling speed are limited by the trap strength, shuttling ions

between memory and interaction regions should consume an

acceptably small fraction of a clock cycle.

Figure 1 shows a diagram of the proposed device. Trapped ions

storing quantum information are held in the memory region. To

perform a logic gate, we move the relevant ions into an interaction

region by applying appropriate voltages to the electrode segments.

In the interaction region, the ions are held close together, enabling

the Coulomb coupling necessary for entangling gates

3,21

. Lasers are

focused through the interaction region to drive gates. We then move

the ions again to prepare for the next operation.

We can realize the trapping and transport potentials needed for

the QCCD using a combination of radio-frequency (r.f.) and

quasistatic electric ﬁelds. Figure 1 shows only the electrodes that

support the quasistatic ﬁelds. By varying the voltages on these

electrodes, we conﬁne the ions in a particular region or transport

them along the local trap axis, which lies along the thin arrows in Fig.

1. Two more layers of electrodes lie above and below the static

electrodes, as shown in Fig. 2. Applying r.f. voltage to the outer layers

creates a quadrupole ﬁeld that conﬁnes the ions transverse to the

local trap axis by means of the ponderomotive force

22

. This geometry

allows stable transport of the ions around ‘T ’ and ‘X’ junctions, so we

can build complex, multiply connected trap structures.

Figure 1 Diagram of the quantum charge-coupled device (QCCD). Ions are stored in

the memory region and moved to the interaction region for logic operations. Thin

arrows show transport and conﬁnement along the local trap axis.

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Electrode structures for the QCCD are relatively easy to fabricate.

A number of ion traps have been built by laser-machining slits in

alumina wafers and evaporating gold electrodes onto the alumina

23

.

These traps have geometries similar to that needed for the QCCD,

and have spacings between the static and r.f. electrodes of fractions of

a millimetre, allowing conﬁnement frequencies up to 20 MHz for r.f.

ﬁeld frequencies of ,250 MHz. Similar electrode structures are

currently being constructed from heavily doped silicon using stan-

dard microfabrication techniques

24

. Here the silicon acts as the

conductive electrode material, while glass spacers anodically bonded

to the silicon insulate the r.f. electrodes from the static electrodes.

A ﬁrst step towards a QCCD has been taken at the National

Institute of Standards and Technology (NIST) by constructing a

pair of interconnected ion traps; the individual traps are similar to

those used in previous work

23

and are separated by 1.2 mm. Efﬁcient

coherent transport of a qubit between the two traps was demon-

strated by performing a Ramsey-type experiment involving the two

traps, where no loss of contrast within the experimental error

(,0.6%) was observed

25

. Transport times were as short as ,50

m

s,

with corresponding ion velocities greater than 10 m s

21

(see also ref.

26). The transport did not cause any heating of the ion motion or

shortening of ion lifetime in the trap.

To maximize the clock speed of the QCCD, we need to transport

ions quickly. However, the entangling gate demonstrated in pre-

vious work at NIST

9,21

has low error only for ions cooled near the

quantum ground state. To recool the ions after transport and to

counteract the effects of heating

23

, we propose to use sympathetic

cooling of the ions used for quantum logic by another ion

species

13,27,28

. Conﬁning both species in the interaction region lets

us use the cooling species as a heat sink, with the Coulomb

interaction transferring energy between the two species, as exper-

imentally demonstrated in refs 29–31.

Decoherence

Whereas the decoherence and gate errors in single-trap quantum

registers have already been characterized, additional decoherence

can occur during ion transport. For instance, the energy splitting of

the qubit states of an ion depends on the magnetic ﬁeld at the ion

through the Zeeman effect. During ion transport, the spatial

variations of the magnetic ﬁeld strength along the transport path

cause the qubit states to acquire a path-dependent relative phase a,

so that, for example, j#lþj"l!j#lþeiaj"lover transport. If we

do not know a, we have lost the phase information, effectively

dephasing the quantum state. But knowing afor all relevant paths is

tantamount to characterizing the magnetic ﬁeld on micrometre

length scales across the entire device, a very difﬁcult task.

Retaining phase information during a computation also requires

accurate positioning of the ions in the interaction regions. As the

logic gate parameters depend on the phase of the driving laser ﬁelds

at the ion positions, the ion positions and laser path lengths must be

controllable with accuracies much better than an optical wave-

length. Although this does not place unreasonable constraints on

the accuracy of the voltage sources driving the QCCD electrodes,

stray electric ﬁelds emanating from the electrodes or mechanical

vibration of the QCCD can readily move the ions a fraction of a

wavelength from their nominal positions, effectively adding a

relative phase aof the type considered above.

Decoherence-free encoding

We can reduce these sources of decoherence by several orders of

magnitude by encoding each qubit into a decoherence-free subspace

(DFS) of two ions

32–34

. This DFS is spanned by the states j0l¼j#"l,

j1l¼j"#lof the ions. We refer to the ions as ‘physical qubits’ and

call the effective two-level system formed by j0l,j1la ‘logical qubit’.

If the state j"lof ion jacquires a phase a

j

over and above the

predicted phase a

0

for the transport path, we see

j0lþj1l!eia1j#"lþeia2j"#l¼j0lþeiDaj1lð1Þ

where we write Da ;a22a1:If Da ¼0, we see that the DFS

logical qubit is unaffected by the acquired phase. Here the phases a

i

are themselves unknown, but the ions acquire the same unknown

phase, a process called collective dephasing. The logical qubit

decoheres only insofar as the dephasing fails to be collective.

As an example, assume that the physical qubit energies have a

linear dependence on magnetic ﬁeld and that the ﬁeld varies linearly

over an extended QCCD device of size 10 cm. If each pair of physical

qubits comprising a logical qubit is separated on average by 10

m

m, a

logical qubit dephases more slowly than a physical qubit by a factor

of 10

4

. Again, Stark shifts of the qubit levels can be induced by the

electric ﬁelds that push the ions from place to place, but their

dephasing effects on the logical qubits are also suppressed. In

general, the effect of any external ﬁeld varying over a length scale

L

ext

and inducing energy shifts of pth power in the ﬁeld is suppressed

by a factor (L

ext

/d)

p

for DFS encoding. A DFS-encoded qubit is

therefore robust against decoherence incurred during transport.

We can also perform universal quantum logic in the DFS, as we

now show. If we hold two ions in an interaction region, we can use

the entangling gate of refs 9 and 21 to apply the operator

U2ðv;f1;f2Þ¼cosv½I1^I2þisin v½Xf1^Xf2

¼cosvIDFS þisin vXDFS

Df12 ð2Þ

Xf;X cos fþY sin fð3Þ

where X, Y are the Pauli operators and the superscript ‘DFS’

indicates that the operator acts in the DFS logical basis. Though

the individual phases f

1

,f

2

depend sensitively on the path-length

differences of the driving lasers over the macroscopic distance from

the lasers to the ions

13

, the DFS gate phase Df12 ¼f12f2depends

only on the microscopic path length of the driving laser between the

two ions, which can be readily controlled by small adjustments of

the trap voltages

9,12,24

. The DFS encoding makes the computation

insensitive to spatial phase ﬂuctuations; as we will see, it protects

against temporal phase ﬂuctuations as well. We can setvover a wide

range of values

21,24

, so the two-ion entangling gate lets us perform

arbitrary rotations of a single logical qubit. Using the same entan-

gling gate on four ions, we can obtain the operator

U4¼1

ﬃﬃﬃ

2

p½I1^I2^I3^I42iXf1^Xf2^Xf3^Xf4ð4Þ

¼1

ﬃﬃﬃ

2

p½IDFS^IDFS 2iXDFS

Df12 ^XDFS

Df34 ð5Þ

Figure 2 Conﬁguration of radio-frequency (r.f.) and static (d.c.) electrodes for the

QCCD. Dotted regions indicate insulating spacers. Applying high r.f. voltage to the two

outer 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 for

conﬁnement transverse to the local trap axis, which points perpendicular to the page

and is located at the central black dot. View in Fig. 1 is from the top of this ﬁgure.

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where ions 1 and 2 encode one logical qubit and ions 3 and 4 encode

another, and we write Df34 ¼f32f4:This operator is equivalent

to an XOR in the logical basis up to rotations of single logical

qubits

35

, so the operators of equations (2) and (5) sufﬁce for

universal 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 ions

remains in the DFS through the quantum computation, so the

logical 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# "land j" #l. All the

operators needed for universal quantum logic in the DFS have

already been experimentally implemented

9,24

, so we should be able

to 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 need

for 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 obstacle

to large-scale parallel quantum computation. As the energy levels of

our physical qubits are non-degenerate, we must keep track of the

resulting phase accumulation to preserve the quantum information

in the physical qubit basis

34

. Parallel operations taking place in

many interaction regions thus require clocks that remain synchro-

nized over the whole computation time

36

. Synchronization can

become very difﬁcult for many qubits: for a transition frequency

q

0

between j#land j"l, the two components of the state

j#lNþj"lNacquire a signiﬁcant relative phase in a time ,1/

(Nq

0

). To maintain phase stability of the computation, we therefore

require a frequency reference with fractional frequency stability

much better than &1/(Nq

0

t) at an averaging time tequal to the

duration of the quantum computation.

To be concrete, we consider trapped

40

Ca

þ

ions as qubits, with the

ground S

1/2

state and metastable D

5/2

states as logic levels. This

system is being investigated for quantum computation by a number

of groups

7,15,37

. Here the transition frequency is 412 THz, compar-

able to the 533 THz operating frequency of the currently most stable

laser oscillator

38

, which has a fractional frequency instability of

3£10

216

as 1 s averaging time. If the computation takes about 1 s,

equal to the lifetime of the metastable D

5/2

state, we see that current

technology barely provides the appropriate phase stability for even

one ion. Of course, this argument can be regarded as too simplistic

because 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 phase

stability exist, but the optical path lengths between the driving lasers

and each of the interaction regions must be stable at the nanometre

level over the course of the computation

13

, a daunting task for a

10 cm QCCD device.

On the other hand, as the logic levels of a DFS-encoded qubit are

degenerate, we do not need phase synchronization at all to perform

a logic operation within the DFS. Operations in the DFS are also

independent of the optical path lengths of the driving lasers, because

the phases Df

12

,Df

34

in equations (2) and (5) depend only on the

distance between the two ions comprising a logical qubit. The

universal gate-set constructed above allows us to perform highly

parallel computations in the DFS without synchronization between

gates separated in time or space. Similar considerations would apply

to other quantum computing architectures. A

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Acknowledgements

We acknowledge the experimental contributions of the NIST Ion Storage group, and also

J. Beall for assistance with microfabrication. We thank D. Leibfried and M.A. Rowe for

comments on the manuscript. D.K. and D.J.W. were supported by the US National

Security Agency (NSA), Advanced Research and Development Activity (ARDA) and the

Ofﬁce of Naval Research. C.M. was supported by the US NSA, ARDA and the National

Science Foundation ITR programme.

Correspondence and requests for materials should be addressed to D.K.

(e-mail: utonium@mit.edu).

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