Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors
Подождите немного. Документ загружается.
9 Principles of Josephson-Junction-Based Quantum Computation 359
|¥
0
=
1
√
N
⎛
⎝
Ÿ
0
ıC
0
√
2ı
2
⎞
⎠
,
|¥
1
=
1
√
N
⎛
⎝
Ÿ
1
ıC
1
√
2ı
2
⎞
⎠
,
|¥
2
=
1
√
N
⎛
⎝
Ÿ
2
ıC
2
√
2ı
2
⎞
⎠
,
(9.100)
where
C
0
= P
−
−2
" +
/3 ,
C
1, 2
= C
±
=
−P
−
∓ i
√
3P
+
/2−2
" +
/3 ,
P
±
=
B
2
0
∓4"
2
+2" +
2
/
(
3B
0
)
,
B
0
=(3"
2
+
3
+3B
1
−8"
3
−6"
2
)
1/3
,
B
1
= i(12"
4
2
+3"
2
4
+12"
3
3
)
1/2
.
The above Eqs.(9.99),(9.100) allow for a direct com-
puting of the resonant contribution to the leakage at
the frequency of external field ! =
(
"
1
− "
0
)
/ from
which one obtains
L
(
t
)
t
=1−
Ÿ
4
0
+ Ÿ
4
1
+ Ÿ
4
2
+ Ÿ
2
0
Ÿ
2
1
+ ¡
4
N
4
, (9.101)
where ¡
4
=
Ÿ
0
Ÿ
1
(
Ÿ
0
C
1
+ Ÿ
1
C
0
)
and the nor-
malizing factor is
N
4
= Ÿ
4
0
+ Ÿ
4
1
+ Ÿ
4
2
+2Ÿ
2
0
Ÿ
2
1
+2Ÿ
2
1
Ÿ
2
2
+2Ÿ
2
0
Ÿ
2
2
.
(9.102)
Here Ÿ
0, 1, 2
= Ÿ
E
0, 1, 2
and the auxiliary function
Ÿ
is
Ÿ
=
− "
" − −
. (9.103)
The eigenenergies of the Hamiltonian (9.99) are ob-
tained as
E
0
= P
−
+
" +
3
,
E
1,2
= E
±
=−
P
−
2
+
" +
3
∓i
√
3
P
+
2
.
(9.104)
The corresponding analytical expression for the fi-
delity is much more cumbersome.
Typical calculation results of the fidelity and leak-
age probability versus time t are shown in Fig. 9.46.
Fig. 9.46. Fidelity of the SISIS two-qubit gate for the three
levels and different tunneling matrix elements =0.01,
0.6, and 0.9 for curves 1, 2, and 3, correspondingly. Dark
spots in the left inset show regions of maximum entangle-
ment. The right inset shows the related leakage probability
curves versus time
One can see that the fidelity of the SISIS gate shows
anomalies when the inter-qubit coupling is suffi-
ciently large, i.e.,when it exceeds the splitting energy,
J/" > 1. If the coupling is small, the qubit gate fi-
delity is close to 1.The leakage probabilityversus the
evolution time t oscillates with a large period which
depends on the qubit parameters and the inter-qubit
coupling, though it is small for t ≤ 10
3
.Optimal
working regions are associated with avoided level
crossing [108,109]where one achieves the maximum
entanglement between the quantum states of the two
SIS subjunctions. The entanglement is computed us-
ing Schmidt decomposition [110], which represents
any entangled state by a biorthogonal expression
with positive real coefficients
¦
(
1, 2
)
=
i=1
c
i
|˛
i
⊗
ˇ
i
:
, (9.105)
where |˛
i
and |ˇ
i
are orthonormal states of sub-
junctions 1 and 2 and c
i
are real and positive.The en-
tanglement of a partially entangled pure state [110]
is
E =−
i
c
2
i
log
2
c
2
i
, (9.106)
Since the inter-level spacing ", tunneling amplitude
, and the inter-qubit coupling J depend on the
bias current I
l
, the entanglement is computed self-
consistently. Typical results for entanglement are
360 S.E. Shafranjuk and J.B. Ketterson
shown in the contour plot (see the left inset in
Fig. 9.46), where we assumed the junction trans-
parency D =10
−4
, d =3,andl
i
/ = 5. Dark spots
in the {'
1
, '
2
}-plane correspond to the avoided level
crossing (working) regions with the maximum inter-
qubit entanglement.
The same QC coupling mechanism may be used
for a controlled readout of the two-qubit state. Since
the measurement is executed with weak supercur-
rents, there is no direct energy dissipation in the
multibarrier Josephson junction. Instead the dissi-
pation occurs rather indirectly via the tunneling
through the characteristic barriers, as was men-
tioned at the beginning of this section. Quantum
measurement of the qubit states is furnished in a
six-barrier multi-terminal SISI-SISIS-ISIS junction,
where the two additional SISI subjunctions act as
quantum meters. The SISI- and -ISIS sections mea-
sure the left and right side SIS subjunctions of the
SISIS junctionviathe QC coupling.Duringthe quan-
tum measurement event the quantum states of SISI-
and -ISIS sections are entangled with the adjacent
SIS subjunction for a short time t
m
t
dc
(t
dc
is the
decoherence time). To insure a sufficient sensitivity
of the measurement, one biases the measured SIS
subjunctionto get the upper level "
2
be located at the
top of the characteristic barrier U
1,2
. The readout
of the two-qubit states | ¥
1,2
:
is conducted by apply-
ing short pulses of ac supercurrent I
ac
1,2
(t)totheSIS
subjunctions. The "
2
level becomes populated with
probability p
2
(t). The system then tunnels out from
the populated state | 2with the tunneling rate "
2
.To
evaluate the energy dissipation W
2
, we establish the
time dependence of the upper level population p
2
(t).
This isdone by introducingofafourthfictional auxil-
iarystate| 3
n
(see,e.g.,[111]andreferencestherein).
The tunneling from the state | 2
1,2
to the auxiliary
state | 3
n
is described by the amplitude damping on
fictitious qubits [112] within the 4 × 4-state analy-
sis. The full space now includes 4 states for each of
the qubits affected by the interaction J,whichgives
in total the 8-dimensional Hilbert space of the real
two-qubit gate. Computation of the qubit character-
istics and simulation of the qubitwork both are con-
ducted numerically. Initially one finds the parame-
ters of the two-qubitHamiltonian (9.99)as described
in the former sections. Then the tunneling dynamics
is modeled by using the operator-sum representa-
tion, where an initial density matrix ˆ
i
is mapped to
the final density matrix ˆ
f
as
ˆ
f
=
m
ˆ
E
m
ˆ
i
ˆ
E
†
m
, (9.107)
where
ˆ
E
m
are the Kraus operators [112]. The nu-
mericanalysisisperformedbyslicingthetimede-
pendence
ˆ
E
m
(t) in many discrete steps as described
in [110]. The input parameters include the junction
parameters listed above, completed with the a r.f.
pulse amplitude and duration.We obtain the explicit
form of the density matrix ˆ
i
before the tunneling
event. Then ˆ
i
is mapped to ˆ
f
in accordance with
Eq. (9.107). Finally, we determine the time depen-
dence p
2
(t), which gives the dissipation W
2
for each
of the meters and for given parameters of the six-
barrier setup. In general, the dissipation introduced
by a meter depends on the measurement time t
m
and
sensitivity of the meter. The sensitivity actually de-
termines a minimal threshold magnitude of the en-
ergy dissipation W
2
, which must be sufficient for a
firm readout. For a two-qubit gate with parameters
used in Fig. 9.46 andJ =0.2,a most effective readout
is achieved when ln(t
dc
/t
m
)
∼
=
10 and I
ac
0
/I
c
∼
=
10
−2
,
where t
dc
is the decoherence time and I
ac
0
is the am-
plitude of the r.f. supercurrent pulse used to excite "
2
during the readout.
We conclude that the intrinsic coupling assures
a low decoherence and dephasing of the two-qubit
gate. If the measurement time t
m
is sufficiently short
while the tunneling current from the upper state
through the characteristic barrier is small, the de-
coherence and dephasing are caused by the leakage
only.The leakage comes from the induced changes in
the Hilbert space of the real gate as considered above.
This makes the intrinsic coupling mechanism to be
attractive for using in two-qubit gates.
The suggested model quantitatively describes
general properties of the two-qubit gate with an ex-
ternally controlled coupling.Since the coupling is in-
trinsic and coherent, it allows eliminating of addi-
tional circuit elements and wirings. The QC coupling
is tuned by the bias Josephson supercurrents which
typically are much smaller than currents circulating
9 Principles of Josephson-Junction-Based Quantum Computation 361
in flux qubits.No charge accumulates on the coupling
element in contrast to the case of the two-qubit gates
with a capacitive coupling. In this way noises and
external sources of decoherence in the system under
study can beminimized.Nevertheless one should pay
attention to remaining intrinsic sources of decoher-
ence, which persist during the two-qubit quantum
logic operations.
9.6 Conclusions
We have discussed how Josephson-junction devices
with suitable parameters can be manipulated in a
quantum-coherent fashion. They are clearly promis-
ing physical realizations (i.e. they can be the hard-
ware) of future devices to be used for quantum-
state engineering. We have discussed various modes
of operation associated with different designs (in
the charge- and the flux-dominated regimes), con-
straints on the parameters,various dephasing effects,
and the quantum-mechanical measurement process.
In order to demonstrate that the constraints on
the circuit parameters can be met by available tech-
nologies,wenow summarizethem andsuggest a suit-
able set.
(i) Necessary conditions for a Josephson charge
qubit are > E
C
E
J
, k
B
T. The supercon-
ducting energy gap must be large enough to
suppress quasiparticle tunneling. The temper-
ature also has to be low relative to the charg-
ing energy to assure initial thermalization, i.e.,
k
B
T E
C
, and to reduce dephasing effects. A
sufficient choice is k
B
T ∼ E
J
/2, since further
cooling does not reduce the dephasing (relax-
ation) rate in a qualitative way.
6
As an explicit
example we choose some reasonable parameters
and estimate the corresponding time scales: For
junctions with a capacitance C
J
=10
−15
F, corre-
sponding to (in temperatureunits) E
C
∼ 1K,and
a (smaller)gatecapacitance C
g
=0.5×10
−17
F(to
reduce the coupling to the environment), initial
thermalization is assured at a working tempera-
ture of 50 mK. The superconducting gap should
be slightly higher, > E
C
. We can therefore
conclude that aluminum is a suitable material.
If we further choose E
J
= 100 mK, the time scale
of one-qubit operations will be
1
op
= /E
J
∼
10
−10
s.
(ii) The impedance of the gate-voltage circuit is typ-
ically R ∼ 50§ that of the transmission line
that drives it. The associated voltage fluctuations
then limit the dephasing time to values of order
'
∼ 10
−4
s, thereby allowing coherent manipu-
lations of a single qubit.
(iii) To assuresufficiently fast two-bitoperations one
can choose C
L
∼ C
J
.Two-bitoperations are then
about 10
2
times slower than the one-bit opera-
tions with an associatedreduction the maximum
number.
(iv) The quantum measurement process introduces
additional constraints on the parameters, which
can be met in realistic devices as demonstrated
by the following example for typical parameters:
Forthejunctionand gatecapacitances we choose
C
T
=1.5 × 10
−17
F, C
SET
g
=0.5 × 10
−17
Frespec-
tively and, for the coupling capacitance between
the SET and qubit, C
int
=0.5 × 10
−17
F.
(v) Finally, a quantum measurement is needed to
read out the quantum information, either at the
final stage or during the computation, e.g., for
the purposes of error correction.
To conclude,fabrication and controlled coherent ma-
nipulations of Josephson-junction qubits are possi-
ble using present-day technologies. In these systems
fundamental properties of macroscopic quantum-
mechanical systems can be explored. Experiments
on elementary systems have now been performed
successfully. More elaborate designs as well as fur-
ther progress in materials and patterning technol-
ogy will provide longer coherence times and allow
sequences of coherent manipulations as well as scal-
ing to larger numbers of qubits. The application of
Josephson-junction systems as elements of a quan-
6
It will, however, affect relaxation far from the degeneracy point, i.e., for h = 0, or if we switch off the Hamiltonian,
H
ctrl
= 0. However, during manipulations E
J
is the typical energy difference and sets the time scale for both the
manipulation times and the dephasing.
362 S.E. Shafranjuk and J.B. Ketterson
tum computer, i.e., with a very large number of ma-
nipulations and large number of qubits,will remain
a challenge for some time. On the other hand, many
aspects of quantum information processing can ini-
tially be tested on simple circuits, as described here.
Additional fallout can be expected with further de-
velopments in quantum-state engineering.
We have also described how a dissipative quan-
tum system when coupled to a qubit can serve as a
quantum measuring device in an accessible range of
parameters. Explicitly we described a single-electron
transistor coupled capacitively to a charge qubit, and
the process of measurement has been discussed us-
ing the time evolution of the reduced density matrix
of the coupled system. It is found that the dephas-
ing time is shorter than the measurement time, and
the mixing time, i.e., the time scale over which the
transitions induced by the measurement occur, was
estimated. Similar scenarios were discussed for flux
qubits measured by a SQUID coupled to it induc-
tively,by switching the interqubit interactions on and
off.In terms ofthephysics,the two types of operation
allow the creation of arbitrary superpositions and
nontrivial coupled states, such as entangled states,
respectively.
Appendix. Elementary Quantum Logic
Operations
This chapter has focused primarily on individual
Josephson-based qubit strategies as well as a two-
qubit gate. In this Appendix we will briefly outline
some of the logical aspects associated with quan-
tum computing. Those wishing to pursue this as-
pect in depth are referred to the text by Nielsen and
Chuang [112].
We start by sketching the elementary quantum
logic operations. and the possibility of performing
them with macroscopic Josephson qubits. We then
focuson the feasibilityof implementing Shor’s quan-
tum factoring algorithm using such devices. This
will also gives us an opportunity to “design” a two-
dimensional Josephson qubit array, capable of per-
forming useful quantum computations. The discus-
sion closely follows that given in [113].
A.1 Boolean Logic
Boolean logic operates with two distinguishable
states – False or True, or simply 0 or 1 – and serves as
the basis for most contemporary computers.Now,ac-
cording to general quantum mechanical principles,if
there are two distinguishable quantum states, in ad-
dition to the system being in either the “0” or “1”
state, it can also exist in various coherent superpo-
sitions of these states. Because the smallest piece of
information in a two-state quantum system has this
new fundamental property we can call it a quan tum
bit or qubit.Insystemswithmorethanonequbit
one may introduce quantum logical gate operations,
which are the building blocks of a quantum com-
puter. Because of quantum mechanical superposi-
tions of qubits, a quantum computer has many more
degrees of freedom and with it new computational
capabilities not availablewith classical computers us-
ing conventional Boolean operations.
After the discovery of the quantum factorization
algorithm by Shor in 1994, it was proved that quan-
tum computers could perform certain important
tasks that cannot be completed by classical comput-
ers within a realistic period of time. The Shor algo-
rithm is exponentially faster than any known classi-
cal algorithm [114]. For this reason, quantum com-
putation theory stimulated efforts to realize practi-
calquantum computers.The Josephson-based qubits
areamongthemorepromisingdevicesproposedfor
performing quantum logic operations. In particu-
lar, they can be easily initialized, precisely manip-
ulated, and individually addressed by conventional
techniques. However, as noted of the beginning in
Sect. 9.1, macroscopic quantum systems tend to suf-
fer from the problem of decoherence,which degrades
and ultimately destroys quantum coherent superpo-
sitions by irreversible interactions with the environ-
ment.
As we have discussed in earlier sections,supercon-
ducting tunnel junction circuits have been designed,
fabricated, and characterized that have a high qual-
ity factor characterizing the quantum coherence (see,
e.g., [115–117]). Such experiments now demonstrate
that decoherence need not necessarily be an obsta-
cle in building a quantum computer from Josephson
circuit elements [117].
9 Principles of Josephson-Junction-Based Quantum Computation 363
If one defines the quantum computing process as
a set of quantum logic operations, such operations
can then be defined in terms of a unitary evolu-
tion of a quantum network that takes a specified ini-
tial state into some final state [4]. This network will
be a computing device consisting of an array quan-
tum logic gates, and each gate is a unitary operation
on one or more qubits. Quantum computations are
then accomplished by assembling quantum logic cir-
cuitsfrom many quantumlogic gates.Since a unitary
transformation is reversible, quantum gates need to
be reversible and cannot be assembled from their
classical counterparts.
We will start by describing the elementary quan-
tum logic operations, so-called “universal quantum
gates”. Following this some simulation work is pre-
sented in Sect.A.3.In Sect.A.4 we present the strategy
to perform Shor’s factoring algorithm using arrays of
superconducting Josephson qubits.
A.2 Required Properties
Figure 9.47 depicts a device consisting of a loop with
three junctions in series that encloses a magnetic
flux ¥ which is controlled by an external magnet.
When the enclosed magnetic flux is close to half a
superconducting flux quantum ¥
0
(= h/2e, where h
is Planck’s constant), the loop may have multiple sta-
ble persistent current states, and the system behaves
as a particle in a double-well potential. The two wells
are coupled via quantum tunneling through the bar-
rier between the wells, and the loop is a macroscopic
quantum two level system.
The system can be engineered to have two sta-
ble states designed as 0 and 1, involving oppositely
circulating persistent currents [118]. The device is
operated using resonant microwave modulation of
the enclosed magnetic flux applied through a super-
conducting control line; the control line is located on
top of the qubit,separated by a thin insulating layer.
Measurements can be made using superconducting
magnetometers (superconducting quantum interfer-
ence devices or SQUIDs). Two or more qubits can be
coupled by means of the flux that the circulatingper-
sistent current generates [119]. An alternate circuit
proposed in [120] consists of a Cooper pair bo x, a
Fig. 9.47. A three-junction Josephson qubit
superconducting island coupled to two small Joseph-
son junctions in a superconducting loop which in-
cludes a third, much larger, Josephson junction (see
Sect. 9.5.2). Arbitrary quantum states of the circuit
are manipulated by applying microwave pulses and
read out by a special readout circuit.
A.3 Universal Quantu m Gates
As in the case of classical computers, certain sets of
quantum gates are universal in the sense that any
quantum computation can be performed by appro-
priately connecting them. We now examine a mini-
mal set of universal quantum gates: i) the single qubit
rotation and ii) the quantum controlled-NOT gate, or
CNOT.An arbitrary singlequbitrotation can be writ-
ten as e
−i ˆt
=cost − i ˆ sin t for some Pauli matrix
combination ˆ = a ˆ
x
+b ˆ
y
+c ˆ
z
wherea
2
+b
2
+c
2
=1
andweusedthat,e.g.,
e
i
x
t
=
0 e
it
e
it
0
,e
i
z
t
=
e
it
0
0 e
−it
. (9.108)
By applying a magnetic pulse for certain duration,
a 3-Josephson junction qubit can be “rotated” (see
Sect. 9.4.3). Figure 9.48 shows, schematically, that the
unitary transformation U rotates the qubit a (which
is described by a 2×2 matrix).We use here the nota-
tions of [114]. The red arrow across the circle (which
denotes the rotation itself) indicates the “direction”
of rotation. Figure 9.49 depicts a controlled NOT
which is a two-qubit quantum logic gate that flips
the value of the second qubit if the value of the first
qubit is 1; i.e., |00→|00, |01→|10, |10→|11,
|11→|01. The system of two coupled qubits is
now described by the 4 × 4 matrix. The rotation af-
fects the lower right 2 × 2submatrix.Bycontrolling
364 S.E. Shafranjuk and J.B. Ketterson
Fig. 9.48. Single qubit rotation
Fig. 9.49. Controlled NOT (CNOT)
the coupling of two qubits, a so-called controlled not
gate can be performed as indicated in Fig. 9.49.
Given a pulse (i.e., by shifting the qubit phase
by ), a controlled NOT gate is constructed from
the controlled rotation gate. A controlled NOT can
be combined with single qubit rotations to perform
arbitrary quantum logic operations.
A.4 Quantum Sum and Carry
The addition of two quantum registers |a and |b
can be written as |a, b→|a, a + b. Since the input
(a, b) can be reconstructed from the output (a, a +b)
(i.e., there is no loss of information in the computa-
tion), the calculation can be carried out reversibly.
The sum operation can be implemented with two
CNOT gates (Fig. 9.50, where time goes from left to
right). However the carry operation has to be ob-
tained in a more complicated way [31] (Fig. 9.50).
A sequence of RF pulses with different frequencies
and durations are injected into the quantum network
during operation.During a pulse,the involved qubits
switch their states.
Fig. 9.50. Quantum sum
A.5 Shor’s Algorithm
Although there is still a long way to go before quan-
tum computers come into practice, it is believed that
they can solve problems that are intractable with
present classical computers, for example, the factor-
ization of large numbers into prime numbers. The
best algorithm available for factoring an integer n
using classical computation is exponential in time,
which rapidly becomes impractical with increasing
n. Shor discovered a quantum algorithm for fac-
toring, which is polynomial in n,alongwithsome
polynomial (in log n) processing time on a classi-
cal computer [4]. To factor an odd number n,Shor’s
algorithm first finds the least integer r such that,
x
r
≡ 1
(
modn
)
; i.e. the period of f
(
A
)
= x
A
mod n
for A from 0 to n −1,wherex is random with x < n
and gcd
(
x, n
)
= 1.Then it findsfactorsof n by calcu-
lating gcd(x
r/2
−1, n)andgcd(x
r/2
+1, n)ifr is even
and x
r/2
= ±1modn; otherwise, it repeats the algo-
rithm.Here,gcd(a, b) is the greatest common divisor
of a and b and it can be effectively computed with
Euclid’s algorithm on a classical computer. To per-
Fig. 9.51. The quantum carry operation as performed with RF pulses
9 Principles of Josephson-Junction-Based Quantum Computation 365
form Shor’s factoring algorithm, we start with two
quantum registers, one of which is prepared to be
superpositions |a and the other is in |0 state. We
compute x
a
(modn) in the second register and keep
the first register in the |a state, i.e., performing the
modular exponentiation
U
x,n
|a, 0→
a, x
a
(modn)
:
, (9.109)
whichcan be built fromareversiblenetworkof quan-
tum sum and carry operations [9].We next perform a
quantum Fourier transformation on the first register
to find the period r. The discrete Fourier transform
is a unitary transformation and can be implemented
by a network of quantum CNOT and qubit rotation
gates [4].Finally,we calculate the factors of nwith Eu-
clid’s algorithmby classical computing.We seethat in
the realizationof Shor’s factoringalgorithm,classical
computation is indispensable; i.e., the experimental
realization of a quantum network can be greatly sim-
plified by incorporating some classical components.
A.6 Conclusions
We have presented strategies for performing some
elementary quantum logic operations using macro-
scopic Josephson qubits,supported by some simula-
tions. The feasibility of building a quantum system
to perform Shor’s factoring algorithm was explored
by implementing the algorithm on a Josephson qubit
network.As is discussed by authors of [121]it should
eventually be possible to build a two-dimensional
Josephson qubit array,possibly accompanied by clas-
sical computing components, capable of perform-
ing useful quantum computations, e.g.Shor’s factor-
ing algorithm. But the practical realization depends
heavily on advances involvingthe individual devices.
Acknowledgements
We thank C. Bennemann for technical assistance
preparing the text.
References
1. D.Deutsch,Proc.R.Soc.LondonA400, 97 (1985)
2. D. DiVincenzo, Quantum computation, Science 270, 255 (1995); DiVincenzo, D., In: M esoscopic Electron Tr ansport,
ed. by L. Kouwenhoven, G. Sch¨on, and L. Sohn, NATO ASI Series E: Applied Sciences No. 345 (Kluwer, Dordrecht,
1997) p. 657; DiVincenzo, D., The physical implementation of quantum computation, Fortschr. Phys.48, 771 (2000)
3. Yu.Makhlin,G.Sch¨on,Rev.Mod.Phys.73, 357 (2001)
4. A. Ekert and R. Jozsa, Rev. Mod.Phys. 68(3),733 (1996)
5. P.W. Shor, in Proceedings of the 35th Annual Symposium on Foundation of Computer Science (IEEE Computer
Society, Los Alamitos,CA, 1994), p. 124
6. G.M. Palma, K.-A. Suominen, and A.K. Ekert, Proc. R. Soc. London A 452, 567 (1996); W. Zurek, Phys. Today 44(10),
36 (1991)
7. A. Barenco, Proc. R. Soc. London A 449, 679 (1995); D. Deutsch, A. Barenco, and A. Eckert, Proc. R. Soc. London A
449,669 (1995); H.Weinfurter,Europhys.Lett. 25, 559 (1994); S.Lloyd, Phys. Rev.Lett.75,346 (1995); A. Barenco,C.H.
Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J.A. Smolin, and H. Weinfurter, Phys. Rev. A 52,
3457 (1995)
8. J.I. Cirac and P.Zoller, Phys. Rev. Lett. 74, 4091 (1995)
9. Q.A.Turchette,C.J.Hood,W.Lange,H.Mabuchi,and H.J.Kimble,Phys.Rev.Lett.75, 710 (1997)
10. D. Cory, A. Fahmy, and T. Havel, Proc. Natl. Acad. Sci. USA 94, 1634 (1997); N.A. Gershenfeld and I.L. Chuang,
Science 275, 350 (1997); D.G. Cory, A.F. Fahmy, and T.F. Havel, In: Proceedings of the Fourth Workshop on Physics
and Computation, ed. by T. Toffoli,M. Biafore, and J.Leao (New England Complex Systems Institute, Boston,1996)
pp.87–91
11. A.Shnirman,G.Sch¨on, and Z. Hermon, Phys. Rev. Lett. 79, 2371 (1997)
12. D.A. Averin, Solid State Commun. 105, 659 (1998)
13. Y. Makhlin, G. Sch¨on, and A. Shnirman, Nature (London) 398, 305–307 (1999)
14. L.B.Ioffe,V.B.Geshkenbein,M.V.Feigelman,A.L.Faucher,and G.Blatter,Nature(London)398,679 (1999)
15. J.E.Mooij,T.P.Orlando,L.Tian,C.van derWal,L.Levitov,S.Lloyd,andJ.J.Mazo,Science285, 1036 (1999)
366 S.E. Shafranjuk and J.B. Ketterson
16. D. Loss and D.P. DiVincenzo,Phys. Rev.A 57, 120 (1998)
17. P. Zanardi and F. Rossi, Phys.Rev. Lett. 81, 4752 (1998)
18. G.K. Brennen, C.M. Caves, P.S. Jessen, and I.H. Deutsch, Phys. Rev. Lett. 82, 1060 (1999); D. Jaksch, H.-J. Briegel, J.I.
Cirac, C.W. Gardiner, and P. Zoller, Phys. Rev. Lett. 82, 1975 (1999)
19. B. Kane, Nature (London) 393,133 (1998); B.E.Kane et al., Phys. Rev. B 61, 2961 (2000)
20. M.Matters,W.J.Elion,andJ.E.Mooij,Phys.Rev.Lett.75, 721 (1995)
21. V. Bouchiat, D.Vion, P. Joyez, D. Esteve,and M. Devoret, Phys. Scr. T76, 165 (1998)
22. Y. Nakamura,Yu.A. Pashkin, and J.S. Tsai, Nature (London) 398, 786 (1999)
23. R.Fazio,G.M.Palma,andJ.Siewert,Phys.Rev.Lett.83, 5385 (1999); J. Siewert, R. Fazio, G.M. Palma, and E. Sciacca, J.
Low Temp. Phys. 118, 795 (2000); J. Siewert and R. Fazio, Phys.Rev. Lett. 87, 257905 (2001)
24. D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C.Urbina, D. Esteve, M.H. Devoret, Manipulating the Quantum
State of an Electrical Circuit, Science,Vol. 296, 886 (2002)
25. M. Tinkham An Introduction to the Theory of Superconductivity (Dover Publishers, NewYork,2004), Eq. (7.28)
26. I. Chiorescu,Y. Nakamura, C.J.P.M. Harmans, and J.E. Mooij, Science 299, 1869 (2003)
27. W.J.Elion,M.Matters,U.Geigenm¨uller, and J.E.Mooij,Nature (London) 371, 594 (1994)
28. J.E. Mooij, T.P. Orlando, L. Levitov, L. Tian, Caspar H. van der Wal, Seth Lloyd, Josephson persistent-current qubit,
Science, 285, 1036 (1999)
29. Y. Shimazu, Internal Report in Quantum Transportation Group of Delft University of Technology (04/05/2000)
30. I.L. Chuang, L.M.K.Vandersypen, X. Zhou, D.W. Leung, S. Lloyd, Experimental Realization of a Quantum Algorithm,
Nature 393 (1998)
31. V.Vedral,A.Barenco,A.Ekert,Phys.Rev.A,54(1), 147 (1996)
32. C. Monroe,D.M. Meekhof, B.E. King, W.M. Itano,and D.J. Wineland, Phys. Rev. Lett.75, 4714 (1995)
33. T.P.Orlando,J.E.Mooij,L.Tian,C.H.vanderWal,L.Levitov,S.Lloyd,andJ.J.Mazo,Phys.Rev.B60(15) 398 (1999)
34. N.P. Suh, The Principles of Design (Oxford University Press,Oxford,1990)
35. J.E. Mooij, Conference on Quantum Coherence and Decoherence, Santa Barbara, California, 1996 (Royal Soc. of
London, London, 1998)
36. M.F. Bocko,A.M. Herr, and M.F. Feldman, IEEE Trans. Appl. Supercond.7, 3638 (1997)
37. V. Bouchiat,D.Vion, P.Joyez,D.Esteve,andM.Devoret,Phys. Scr. T76,165 (1998);V.Bouchiat,Ph.D. thesis, Universite’
Paris (1997)
38. D.V. Averin and K.K. Likharev, In: Mesoscopic Phenomena in Solids, ed. by B.L. Altshuler, P.A. Lee, and R.A. Webb
(North-Holland,Amsterdam, 1991) Chap. 6, p. 173
39. P. Lafarge, M. Matters, and J.E. Mooij, Phys. Rev. B 54,7380 (1996)
40. A. van Oudenaarden, S.J.K.Vardy, and J.E. Mooij, Czech. J. Phys. 46, 707 (1996)
41. V. Bouchiat,D.Vion, P.Joyez,D.Esteve,andM.Devoret,Phys. Scr. T76,165 (1998);V.Bouchiat,Ph.D. thesis, Universite’
Paris (1997)
42. D.V. Averin and K.K. Likharev, In: Mesoscopic Phenomena in Solids, ed. by B.L. Altshuler, P.A. Lee, and R.A. Webb
(North-Holland,Amsterdam, 1991)
43. P. Lafarge, M. Matters, and J.E. Mooij, Phys. Rev. B 54,7380 (1996)
44. A. van Oudenaarden, S.J.K.Vardy, and J.E. Mooij, Czech. J. Phys. 46, 707 (1996)
45. L. J. Geerligs,V.F. Anderegg, and J.E. Mooij, Physica B 165, 973 (1990)
46. G. Zimmerli,T.M. Eiles, R.L. Kautz, and John M. Martinis,Appl. Phys. Lett. 61,
237 (1992)
47. S. Han, J. Lapointe, J.E. Lukens, In: Activated Barrier Crossing, ed. by G.R. Fleming and P. Hanggi (World Scientific,
Singapore,1993) pp. 241–267
48. R. Rouse, S. Han, and J.E. Lukens, Phys. Rev. Lett. 75, 1614 (1995)
49. A.O. Caldeira, and A.J.Leggett, Quantum tunneling in a dissipative system, Ann. Phys.(N.Y.) 149, 374 (1983)
50. R.F.Voss,and R.A.Webb, Macroscopic quantum tunneling in 1 mm Nb Josephson junctions,Phys. Rev. Lett. 47, 265
(1981)
51. J.M. Martinis, M.H. Devoret, and J. Clarke, Experimental tests for the quantum behavior of a macroscopic degree of
freedom: the phase difference across a Josephson junction, Phys. Rev. B 35, 4682 (1987)
52. J. Clarke, A.N. Cleland, M.H. Devoret, D. Esteve, and J.M. Martinis, Quantum mechanics of a macroscopic variable:
The phase difference of a Josephson junction, Science 239, 992 (1988)
9 Principles of Josephson-Junction-Based Quantum Computation 367
53. R. Rouse, S. Han, and J.E. Lukens, Observation of resonant tunneling between macroscopically distinct quantum
levels, Phys.Rev. Lett. 75, 1614 (1995)
54. J.R.Friedman,V.Patel,W.Chen,S.K. Tolpygo,and J.E.Lukens,Detection of a Schroedinger’s cat state in an r.f.-SQUID,
Nature (London) 406, 43 (2000)
55. C.H.van derWal,A.C.J.ter Haar,F.K.Wilhelm,R.N.Schouten,C.J.P.M.Harmans,T.P.Orlando,S.Lloyd,and J.E.Mooij,
Quantum superposition of macroscopic persistent-current states, Science 290, 773 (2000)
56. C.D. Tesche, Phys. Rev. Lett. 64, 2358 (1990); also,Johnson and M. Tinkham, ibid. 65, 1263 (1990)
57. P. Carruthers and M.M. Nieto, Rev. Mod. Phys. 40, 411 (1968)
58. C.D. Tesche, Physica B 165&166, 925 (1990); also Jpn. J.Appl. Phys.,Suppl. 26, 1409 (1987)
59. P. Joyez, A. Pilipe, D. Esteve, and M.H. Devoret, Phys. Rev. Lett. 72, 2458 (1994)
60. A.J. Leggett,1987, In: Chance and Matter,ed.byJ.Souletie,J.Vannimenus,andR.Stora(Elsevier,Amsterdam)p.395
61. Y. Makhlin, G. Sch¨on, and A. Shnirman, Nature (London) 398, 305–307 (1999)
62. Y. Nakamura, C.D. Chen, and J.S Tsai, Spectroscopy of energy-level splitting between two macroscopic quantum
states of charge coherently superposed by Josephson coupling.Phys. Rev. Lett. 79, 2328–2331 (1997)
63. M. Steffen, J.M. Martinis, and I.L. Chuang, Phys. Rev. B 68, 224518 (2003)
64. L. Emsley and G. Bodenhausen, Chem. Phys. Lett. 168, 297 (1990)
65. R.W.Simmonds,K.M.Lang,D.A.Hite,D.P.Pappas,andJ.M.Martinis,Phys.Rev.Letters,93, 077003 (2004)
66. C.T. Rogers and R.A Buhrman, Phys.Rev. Lett. 53, 1272(1984); C.T. Rogers and R.A. Buhrman, Phys Rev. Lett. 55, 859
(1985)
67. D.J.VanHarlingen, B.L.T. Plourde, T.L. Robertson, P.A Reichardt, and J. Clarke, Proceedings of the 3rd International
Workshop on Quantum Computing, Preprint (2004)
68. I. Chiorescu,Y. Nakamura, C.J.P.M. Harmans, and J.E. Mooij, Science 299, 1869 (2003)
69. D.Vion,A.Aassime,A.Cottet,P.Joyez,H.Pothier,C.Urbina,D.Esteve,andM.H.Devoret,Science296,886 (2002)
70. D. Loss and D.P. DiVincenzo,Phys. Rev.A 57, 120 (1998)
71. E.N. Bratus,V.S. Shumeiko, and G.A.B.Wendin, Phys. Rev. Lett. 74, 2110 (1995); D. Averin and A. Bardas, Phys. Rev.
Lett. 75, 1831 (1995); C.W.J. Beenakker,Phys. Rev. Lett. 67, 3836 (1991)
72. N. van der Post, E.T. Peters, I.K. Yanson, and J.M. van Ruitenbeek, Phys. Rev. Lett. 73, 2611 (1994); M.F. Goffman, R.
Cron,A.Levy Yeyati, P. Joyez, M. H. Devoret,D. Esteve,and C. Urbina,Phys. Rev. Lett.85, 170 (2000)
73. J.I. Cirac and P.Zoller, Phys.Rev. Lett. 74, 4091 (1995)
74. R.T.Wakai and D.J.Van Harlingen, Appl. Phys.Lett. 49, 593 (1986)
75. D.Vion,A.Aassime,A.Cottet,P.Joyez,H.Pothier,C.Urbina,D.Esteve,andM.H.Devoret,Science296,286-280 (2002)
76. D.V. Averin and C.Bruder, Phys Rev. Lett. 91, 057003 (2003)
77. D.C. White and H.H. Woodson, Electromechanical Energy Conversion (Wiley, New York, 1959) Chap. 1
78. Y. Nakamura,Yu.A. Pashkin, and J. S. Tsai, Nature (London) 398, 786 (1999)
79. A. Stern,Y.Aharonov and Y. Imry, Phys. Rev.A 41, 3436 (1990)
80. W.J.Elion,M.Matters,U.Geigenm¨uller, and J.E.Mooij,Nature (London) 371, 594 (1994)
81. M.H. Devoret, in Quantum Fluctuations,editedbyS.Reymaud,E.Giacobino,andJ.Zinn-Justin(Elsevier,NewYork,
1997),Chap. 10
82. T.P.Spiller,T.D.Clark,R.J.Prance,andA.Widom, In: Progress in Low Temper ature Physics,ed.by D.F. Brewer (Elsevier,
New York, 1992) Vol. XIII, Chap. 4
83. D.C. White and H.H. Woodson, Electromechanical Energy Conversion
(Wiley, New York, 1959) Chap. 1
8
4. S.E. Shafranjuk, Phys. Rev. B 74, 024521 (2006)
85. W.G.vanderWiel,S.DeFranceschi,J.M.Elzerman,T.Fujisawa,S.Tarucha,andL.P.Kouwenhoven,PreprintCondmat
0205350 (2002)
86. J.M.Martinis,S.Nam,J.Aumentado,andC.Urbina,Phys.Rev.Lett.89, 117901 (2002)
87. A. Blais, A. Maassen van den Brink, and A.M. Zagoskin, Phys. Rev. Lett. 90, 127901 (2003)
88. A.J.Berkley,H.Xu,R.C.Ramos,M.A.Gubrud,F.W.Strauch,P.R.Johnson,J.R.Anderson,A.J.Dragt,C.J.Lobb,andF.C.
Wellstood, Science 300, 1548 (2003)
89. R.C. Ramos, M.A. Gubrud, A.J. Berkley, J.R. Anderson, C.J. Lobb, and F.C. Wellstood, IEEE Trans. Appl. Supr. 11, 998
(2001); P.R. Johnson, F.W. Strauch,A.J. Dragt, R.C. Ramos, C.J. Lobb,J.R.Anderson, and F.C. Wellstood,Phys. Rev. B
67, 020509(R) (2003); F.W. Strauch, P.R. Johnson,A.J. Dragt, C.J. Lobb, J.R. Anderson, and F.C.Wellstood, Phys. Rev.
Lett. 91, 167005 (2003)
368 S.E. Shafranjuk and J.B. Ketterson
90. A. Shelankov, M. Ozana, Phys. Rev. B 61, 7077 (2000)
91. S.E. Shafranjuk and J.B. Ketterson, Phys. Rev. B 72, 212506 (2005)
92. I.P. Nevirkovets, S.E. Shafranjuk, and J.B. Ketterson, Phys. Rev. B 68,024514 (2003)
93. G.Burkard,D.Loss,andE.V.Sukhorukov,Phys.Rev.B61, R16303 (2000)
94. Y. Nakamura,Yu.A. Pashkin, and J.S. Tsai, Nature 398, 786 (1999)
95. J.R. Friedman et al., Nature 406,43 (2000)
96. C. H. van der Wal et al., Science 290, 773 (2000)
97. Y.Yu et al., Science 296, 889 (2002)
98. J.M.Martinis,S.Nam,J.Aumentado,andC.Urbina,Phys.Rev.Lett.89, 117901 (2002)
99. S.Russo,M.Kroug,T.M.Klapwijk,andA.F.Morpurgo,Phys.Rev.Lett.95, 027002 (2005)
100. A.V. Zaitsev, Zh. Eksp. Teor. Fiz.86, 1742 (1984) [Sov. Phys. JETP 59, 1015 (1984)]
101. S.E.Shafranjuk and J.B. Ketterson, Phys. Rev. B 72, 024509 (2005)
102. E. Merzbacher, Quantum Mechanics, 3rd ed. (Wiley, New York,1998)
103. A.V. Zaitsev, Zh. Eksp. Teor. Fiz.86, 1742 (1984) [Sov. Phys. JETP 59, 1015 (1984)]
104. S.E.Shafranjuk and J.B. Ketterson, Phys. Rev. B 72, 024509 (2005)
105. E. Prada and F.Sols, Eur. Phys.J. B 40, 379 (2004)
106. C.S. Owen and D.J. Scalapino, Phys.Rev. 164, 538 (1967)
107. G.A.Korn, T.M.Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968)
108. A.J. Berkley, H. Xu, R.C. Ramos, M.A. Gubrud, F.W. Strauch, P.R. Johnson, J.R. Anderson, A.J. Dragt, C.J. Lobb, and
F.C.Wellstood, Science 300, 1548 (2003)
109. R.C. Ramos, M.A. Gubrud, A.J. Berkley, J.R. Anderson, C.J. Lobb,and F.C. Wellstood, IEEE Trans. Appl. Supr. 11, 998
(2001); P. R. Johnson,F.W.Strauch,A. J.Dragt, R. C.Ramos, C. J. Lobb,J.R.Anderson, and F. C.Wellstood,Phys. Rev.
B 67, 020509(R) (2003); F.W. Strauch,P.R.Johnson, A.J. Dragt, C.J. Lobb, J.R.Anderson, and F.C.Wellstood,Phys. Rev.
Lett. 91, 167005 (2003)
110. A. Peres, Quantum Theory: Concepts and Methods (Kluwer, Dodrecht, 1993)
111. A. Blais,A. Maassen van den Brink,and A.M. Zagoskin,Phys. Rev. Lett. 90, 127901 (2003)
112. M. Nielsen, I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press,Cambridge
2000)
113. J. Han and P. Jonker, On Quantum Computing with Macroscopic Josephson Qubits, Proc. CAMP2000, Fifth IEEE
International Workshop on Computer Architectures for Machine Perception (Padova, Italy, Sep. 11–13, 2000),IEEE
Computer Society,Los Alamitos,California,USA,2000,69–78;A.Ekertand R.Jozsa,Rev.Mod.Phys.68(3), 733(1996)
114. P. Shor,Algorithms for quantum computation: Discrete log and factoring, Proc. 35th Annual Symp. on Foundations
of Computer Science, pp. 124–134 (1994)
115. C.H. van der Wal, A.C.J. ter Haar, F.K. Wilhelm, R.N. Schouten, C.J.P.M. Harmans, T.P. Orlando, S. Lloyd, J.E. Mooij,
Quantum superposition of macroscopic persistent-current states, Science 290, 773 (2000)
116. D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, M.H. Devoret, Manipulating the quantum
state of an electrical circuit, Science, 296, 886 (2002)
117. A.J.Leggett, Superconducting qubits — a major roadblock dissolved, Science 296, 861 (2002)
118. C.H. van der Wal, A.C.J. ter Haar, F.K. Wilhelm, R.N. Schouten, C.J.P.M. Harmans, T.P. Orlando, S. Lloyd, J.E. Mooij,
Quantum superposition of macroscopic persistent-current states, Science 290
, 773 (2000)
119.
J.E. Mooij, T.P. Orlando, L. Levitov, L. Tian, Caspar H. van der Wal, S. Lloyd, Josephson persistent-current qubit,
Science 285,1036 (1999)
120. D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, M.H. Devoret, Manipulating the quantum
state of an electrical circuit, Science 296, 886 (2002)
121. P.W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring. In: Proceeding of the 35th
Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press,Los Alamitos, CA, (1994)
pp.124–134