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Architecture for a large-scale ion-trap quantum computer

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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.
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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^I2þisin 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
ffiffi
2
p½I1^I2^I3^I42iXf1^Xf2^Xf3^Xf4ð4Þ
¼1
ffiffi
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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NATURE| VOL 417 | 13 JUNE 2002 | www.nature.com/nature710 © 2002 Nature PublishingGroup
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.
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Quantum mechanics can speed up a range of search applications over unsorted data. For example, imagine a phone directory containing N names arranged in completely random order. To find someone`s phone number with a probability of 50{percent}, any classical algorithm (whether deterministic or probabilistic) will need to access the database a minimum of 0.5N times. Quantum mechanical systems can be in a superposition of states and simultaneously examine multiple names. By properly adjusting the phases of various operations, successful computations reinforce each other while others interfere randomly. As a result, the desired phone number can be obtained in only O({radical}(N)) accesses to the database. {copyright} {ital 1997} {ital The American Physical Society}
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