Holonomic quantum computation using rf superconducting quantum interference devices coupled through a microwave cavity

Title

Holonomic quantum computation using rf superconducting

quantum interference devices coupled through a microwave

cavity

Author(s)

Zhang, P; Wang, ZD; Sun, JD; Sun, CP

Citation

Physical Review A - Atomic, Molecular, And Optical Physics,

_{2005, v. 71 n. 4}

Issued Date

2005

URL

http://hdl.handle.net/10722/43158

Holonomic quantum computation using rf superconducting quantum interference devices

coupled through a microwave cavity

P. Zhang,1,2Z. D. Wang,1 J. D. Sun,1and C. P. Sun2

1_{Department of Physics, University of Hong Kong, Hong Kong, China} 2

Institute of Theoretical Physics, Chinese Academy of Science, Beijing, 100080, China

共Received 3 August 2004; revised manuscript received 29 November 2004; published 1 April 2005兲

We propose a different scheme to realize holonomic quantum computation with rf superconducting quantum interference device共SQUID兲 qubits in a microwave cavity. In this scheme associated with the non-Abelian holonomies, the single-qubit gates and a two-qubit controlled-PHASEgate as well as a controlled-NOTgate can be easily constructed by tuning adiabatically the Rabi frequencies of classical microwave pulses coupled to the SQUIDs. The fidelity of these gates is estimated to be possibly higher than 90% with the current technology. DOI: 10.1103/PhysRevA.71.042301 PACS number共s兲: 03.67.Lx, 03.65.Vf, 85.25.Cp

Since the proposal of holonomic quantum computation

关1兴, research on quantum gates based on Abelian or

non-Abelian geometric phases has attracted significant interest both experimentally and theoretically关2–10兴. It is believed that these quantum gates could be inherently robust against some local perturbations since the Abelian or non-Abelian geometric phases共holonomies兲 depend only on the geometry of the path executed. On the other hand, quantum informa-tion processing using Josephson-juncinforma-tion systems coupled through a microwave cavity has been paid particular atten-tion recently关11–19兴.

In this paper, we propose a different scheme to achieve holonomic quantum computation using superconducting quantum interference devices 共SQUIDs兲 in a cavity. Based on the non-Abelian holonomies, two noncommutating single-qubit gates and a two-qubit controlled-PHASEgate as well as a controlled-NOTgate are realized by tuning adiabati-cally the Rabi frequencies of classical microwave pulses coupled to the SQUIDs. The distinct advantages of the present scheme may be summarized as follows. 共i兲 The energy spectrum of each SQUID qubit may be adjusted by changing the bias field; 共ii兲 the strong-coupling limit g2_{Ⰷ共␥␬兲 may be easily realized, where g is the coupling} coefficients between the SQUID qubit and the cavity field,␬ the lifetime of the photon in the cavity, and␥the lifetime of the excited state of the SQUID qubit; 共iii兲 the decoherence caused by the external environment can be significantly sup-pressed; 共iv兲 the fidelity of these gates may be higher than 90% with the current technology.

We consider an rf SQUID with junction capacitance C and loop inductance L in a microwave cavity at a sufficiently low temperature such that the SQUID works in the quantum regime. The Hamiltonian of the rf SQUID can be written as

关15,20兴 Hs= Q2 2C+ 共⌽ − ⌽x兲2 2L − EJcos

冉

2␲ ⌽ ⌽0

冊

, 共1兲

where EJis the maximum Josephson coupling energy,⌽xthe external magnetic flux, and⌽0= h / 2e the flux quantum. The conjugate variables of this system are the total charge Q and the magnetic flux⌽ which satisfy

关⌽,Q兴 = iប. 共2兲

It is well known that in general the Hamiltonian of Eq.共1兲 is quite similar to that of a particle moving in a double well potential. By changing the device parameters C, L, and the control parameters EJ, ⌽x, one can control the structure of energy levels in the SQUID.

Let us address a共3+1兲-type system with three lowest lev-els 共兩a0典,兩a1典,兩g典兲 and an excited level 共兩e典兲 in the SQUID

共see Fig. 1兲. In the system, the 兩g典↔兩e典 transition with

energy-level difference␻egis coupled to a one-mode cavity field with frequency␻_cand the兩a_i典↔兩e典 transition with the energy-level difference␻ei共i=0, 1兲 is coupled to the classi-cal microwave pulse with the magnetic component as

Bi共r,t兲cos共␻0t兲 where ␻i is the energy difference between the states 兩ai典 and 兩e典 and Bi can be adiabatically changed. We may ensure that the “three-photon resonance” condition, i.e.,

␻eg−␻c=␻ei−␻i=⌬, 共3兲 is satisfied. In the interaction picture, the Hamiltonian of the system can be written as

FIG. 1. A schematic diagram of the energy level in the SQUID coupled to the single mode cavity field共with coupling constant g兲 and two microwave pulses共with coupling constants ⍀0and ⍀1兲. The three-photon resonance condition is satisfied and ⌬ is the detuning.

H =⌬兩e典具e兩 + ⍀₀共t兲兩e典具a₀兩 + ⍀₁共t兲兩e典具a₁兩 + ga兩e典具g兩 + H . c.

共4兲

where a共a†_{兲 is the photon annihilation 共creation兲 operator of} the cavity field. Here the coupling coefficients g can be writ-ten as

g = L−1

冑

␻c/共2␮0ប兲具g兩⌽兩e典

冕

S

Bc共r兲 · dS, 共5兲

where␮0is the vacuum permeability, S the surface bounded by the SQUID ring, and Bc共r兲 the magnetic component of the cavity field关15兴. The Rabi frequencies ⍀i共t兲 共i=0, 1兲 are proportional to the matrix elements 具ai兩⌽兩e典 关15兴. It is pointed out that the detuning ⌬ of the SQUID can be ad-justed by changing the bias field.

Regarding single-qubit operations, our scheme is similar to that proposed in Refs.关4,5兴. We choose the states 兩a0典 and

兩a1典 as the computational basis and 兩g典 as an ancillary state. When⍀0=⍀1= 0, the states兩ai典兩0典c共i=0, 1兲 span an eigens-pace of the Hamiltonian共4兲 with zero eigenvalue, where 兩0典c is the vacuum state of the cavity field. If the cavity is cooled to zero temperature and the quantum gate operations are switched off, i.e., the Rabi frequencies of all the classical microwave pulses are set to zero, the state ␣兩a0典+␤兩a1典 of the qubit is isolated from the state of the cavity and does not change with time. When the Rabi frequencies ⍀₀ and ⍀₁ change adiabatically along a close path C in the parameter space M and return the point corresponding to ⍀0=⍀1= 0

共we refer to this point as O兲, an initial state 兩⌿0典 = 共␣兩a0典 +␤兩a1典兲兩0典c

of the qubit-cavity composite system evolves according to the rule兩⌿0典→U共C兲兩⌿0典 关5,21兴. Here,

U共C兲 = P exp

冕

C

A 共6兲

is the non-Abelian holonomy associated with the path C and A =兺_␮A_␮d␭_␮ is the U共2兲-valued connection 共noe-form兲 ex-pressed as

A_␮ij=具Di共␭兲

冏

⳵ ⳵␭␮

冏

Dj共␭兲典, 共7兲

where兵␭_␮其 are the coordinates of the parameter space M and

兩Di共␭兲典 共i=0, 1兲 the basis of the eigenspace of the Hamil-tonian 共4兲 with zero eigenvalue 共hereafter we refer to the basis as dark states兲. Since the holonomy U共C兲 is a unitary transformation in the space spanned by the states兩a0典兩0典cand 兩a1典兩0典c, it can actually be considered as a unitary transfor-mation that only acts on the qubit state which is the super-position of兩a₀典 and 兩a₁典.

Without loss of generality, we assume that the coupling coefficient g is real and positive, and choose

⍀0= g tan共␰兲ei␾0,

共8兲

⍀1= g tan共␪兲sec共␰兲ei␾1,

where␪,␰苸关0,␲/ 2兲 and␾i苸关0,2␲兲 共i=0, 1兲. We take the angles␰,␪,␾0, and␾1 as the coordinates of the parameter space M. The dark states of this invariant subspace spanned by the states兵兩ai典兩0典c,兩g典兩1典c,兩e典兩0典c其 of the Hamiltonian 共4兲 can be written as the vector functions in M:

兩D0典 = cos共␰兲兩a0典兩0典c− sin共␰兲ei␾0兩g典兩1典c, 兩D1典 = − sin共␪兲sin共␰兲e

i共␾₁−␾_0兲兩a 0典兩0典c − sin共␪兲cos共␰兲ei␾1兩g典兩1典

c+ cos共␪兲兩a1典兩0典c. 共9兲 We wish to point out that our choices of the coordinates and dark states are quite different from those in Refs.关4,5兴. Here, the dark states兩D0典 and 兩D1典 are single valued at the point O共␪=␰= 0兲, e.g., 兩D_i共O兲典=兩a_i典兩0典_c.

It is well known that any single-qubit gate operation can be decomposed into the product of rotations about axes z and y: Rz共␸兲=exp共i␸␴z兲 and Ry共␸兲=exp共i␸␴y兲, where ␸ is the angle,␴z and␴yare Pauli matrices defined as

␴y= i共兩a1典具a0兩 − 兩a0典具a1兩兲,

共10兲

␴z=共兩a0典具a0兩 − 兩a1典具a1兩兲.

Therefore we need only to show the realization of Rz共␸兲 and Ry共␸兲. To realize the gate Ry共␸兲, we let the phases ␾0=␾1 = 0. The U共2兲 valued connections can be derived as

A_␰= − i sin共␪兲␴y, A␪= 0. 共11兲 After an adiabatic evolution along a closed path C in the parameter space M, the related unitary transformation 共ho-lonomy兲 is just the rotation about y axis U共C兲 = exp关i␸共C兲␴y兴, where the angle

␸共C兲 = −

冉

冕

C

sin共␪兲d␰

冊

共12兲

is dependent of the loop C. To achieve the gate Rz共␸兲, we can set␾0=␰= 0共i.e., ⍀0= 0兲 and change␪and␾1adiabatically. In this case, the nonzero connection is

A_␾ 1= i 1 2sin 2_{共␪兲共1 −␴} z兲. 共13兲

As a result, the holonomy associated with a close path C can be written as U共C兲 = e−i␹共C兲ei␹共C兲␴z_, 共14兲 where ␹共C兲 = −1 2

冕

sin 2_{共␪兲d␾} 1. 共15兲

It is easy to see that U共C兲 is just a rotation about the z axis up to a global phase.

We now illustrate how to realize the controlled-PHASE gate as well as the controlled-NOTgate via the non-Abelian holonomy in the present system of SQUID qubits, which is the main result of the present paper. At this stage, we

con-ZHANG et al. PHYSICAL REVIEW A 71, 042301共2005兲

sider the 共3+1兲-type energy-level structure of two rf SQUIDs in a microwave cavity. We assume that the兩e典↔兩g典 transition of each SQUID is coupled to the single mode cav-ity field. Also, we set the level spacing␻eibetween the states 共兩e典, 兩ai典, i=0, 1兲 to be different. The transitions 兩e典↔兩a0典 共or

兩a1典兲 in both SQUIDs are coupled to two distinguishable classical microwave pulses with different frequencies 关15兴 and the three-photon resonance condition of each SQUID is set to be satisfied. The Hamiltonian of such SQUIDs in a microwave cavity may be written as

H =

兺

l=1,2

关⍀0共l兲兩e典l具a0兩 + ⍀1共l兲兩e典l具a1兩 + H.c.兴

+

兺

l=1,2 关g共l兲_a兩e典 l具g兩 + H.c.兴 +

兺

l=1,2 ⌬共l兲_兩e典 l具e兩. 共16兲 Here ⌬共l兲_=␻ eg 共l兲_−␻ c 共l兲_=␻ ei 共l兲_−␻ i 共l兲 _共17兲

is the detuning of the lth SQUID and ⍀_i共l兲 is the Rabi fre-quency of the microwave pulse coupled to the transition

兩e典l↔兩ai典lof the lth SQUID.

We still choose the states兩a_i典_li = 0, 1 as the computational basis of the lth qubit. The two SQUIDs are coupled indi-rectly via the single mode cavity field. As in the previous discussions for the single-qubit gate case, it is seen that when all of the Rabi frequencies ⍀_i共l兲 of the classical microwave pulses are set to zero and the cavity is cooled to zero tem-perature, the two-qubit state, which can be written in terms of兩a_i典₁兩a_j典₂, is isolated from the state of the cavity and does not change with time. In the following, we show that the holonomy, associated with the adiabatic evolution of ⍀_i共l兲 along a closed path C starting from the point⍀_i共l兲= 0共we also refer to this point as O in the parameter space M, can be used to achieve a controlled-PHASEgate or a controlled-NOTgate of the two qubits.

We first choose

⍀0共l兲= g共1兲tan关␰共l兲兴e i␾₀共l兲_,

共18兲 ⍀1共l兲= g共2兲tan关␪共l兲兴sec关␰共l兲兴e

i␾₁共l兲 共l=1,2兲, and take␰共l兲_,␾ 0 共l兲_,␪共l兲_{, and ␾} 1 共l兲 _{as the independent} coordinates in the parameter space M. To realize the controlled-PHASEgate, we set␪共l兲=␾₁共l兲=␾₀共1兲= 0. This means that we set the Rabi frequencies ⍀₁共1兲and ⍀₁共2兲 to zero and only change⍀₀共1兲 and⍀₀共2兲adiabatically. It is found that the subspace spanned by the states

兩ai典1兩aj典2兩0典c,兩ai典1兩g典2兩1典c, 兩g典1兩g典2兩2典c,

兩g典1兩ai典2兩1典c,兩e典1兩ai典2兩0典c, 兩ai典1兩e典2兩0典c, 共19兲 兩e典1兩g典2兩1典c,兩g典1兩e典2兩1典c, 兩e典1兩e典2兩0典c共i, j = 0,1兲 is an invariance subspace共we call it as the subspace I兲 of the Hamiltonian 共16兲. After some tedious derivations, the dark states, i.e., the basis of the eigenspace of Hamiltonian 共16兲 with zero eigenvalue, can be obtained as

兩D00典 = ⌳00

−1_{关sin关␰共1兲兴sin关␰共2兲兴e}i␾₀共2兲_兩g典 1兩g典2兩2典c −

冑

2 cos关␰共1兲兴sin关␰共2兲兴ei␾0共2兲兩a

0典1兩g典2兩1典c +

冑

2 cos关␰共1兲兴cos关␰共2兲兴兩a₀典₁兩a₀典₂兩0典_c −

冑

2 sin关␰共1兲兴cos关␰共2兲兴兩g典1兩a0典2兩1典c兴, 兩D01典 = cos关␰共1兲兴兩a0典1兩a1典2兩0典c− sin关␰共1兲兴兩g典1兩a1典2兩1典c,

共20兲 兩D10典 = cos关␰共2兲兴兩a1典1兩a0典2兩0典c− sin关␰共2兲兴ei␾0

共2兲

兩a1典1兩g典2兩1典c, 兩D11典 = 兩a1典1兩a1典2兩0典c

where

⌳00=

冑

2 − sin2关␰共1兲兴sin2关␰共2兲兴. 共21兲 It is obvious that at the point O where␰共1兲=␰共2兲= 0, the above dark states are single valued:兩Dij共O兲典=兩ai典1兩aj典2兩0典c for i, j = 0, 1. The nonzero elements of U共4兲-valued connections are just A_␾ 0 共2兲 00,00 = i⌳₀₀−2兵2 − sin2关␰共1兲兴其sin2关␰共2兲兴, 共22兲 A_␾ 0 共2兲 10,10 = i sin2关␰共2兲兴.

When the system evolves adiabatically along a closed path C in the parameter space M and returns to the point O, the associated holonomy can be written as

U共C兲 = ei␩共C兲兩a0典2具a0兩_ei␾共C兲兩a0a0典具a0a0兩_. 共23兲 Here, the angles␩共C兲 and ␾共C兲 are defined as

␾共C兲 =

冕

⌳00 −2_{关2 − sin}2_关␰共1兲_{兴 − ⌳} 00 2 _兴sin2_关␰共2兲_兴d␾ 0 共2兲_, 共24兲 ␩共C兲 =

冕

sin2关␰共2兲兴d␾₀共2兲.

In the above expression of U共C兲, 兩a0典2具a0兩 is the projection operator of the second qubit to the state兩a0典2. It is easy to see that the holonomy U共C兲 is the production of a controlled-PHASEgate operation and a single-qubit rotation about the z axis operated on the second qubit. When␾共C兲⫽0, U共C兲 is a nontrivial two-qubit operation. If we choose the path C sat-isfying␩共C兲=0, we can obtain an explicit controlled-PHASE gate via the holonomy

U共C兲 = ei␾共C兲兩a0a0典具a0a0兩_. 共25兲 Moreover, we may also realize the controlled-NOTby set-ting ␪共1兲=␾₀共j兲=␾共j兲₁ = 0 and choose ␪共2兲,␰共1兲, and ␰共2兲 as the control parameters. In this case, the dark states can also be obtained with some straightforward derivations, but have quite complicated forms共not presented here兲. Our main re-sult is that, when

冕

C

sin关␪共2兲兴d␰共2兲−␪共C兲 = −␲/2, 共26兲

the holonomy associated with a closed path can be expressed as U共C兲 = e−i 4Ⲑ␲R_y共2兲关␪共C兲兴R_z共1兲

冉

␲ 4

冊

Rz 共2兲

冉

␲ 4

冊

UCNRz 共2兲

冉

₋␲ 4

冊

, 共27兲 where

U_CN=兩a₀典₁具a₀兩丢I共2兲+兩a₁典₁具a₁兩丢␴x共2兲 共28兲 is just the controlled-NOToperation. Here, I共2兲is the identity operator of the second qubit and the angle␪共C兲 is defined as

␪共C兲 = −

冕

C ␤−1_⌳ 01 −1_⌳ 00 −3_⌳ 2d␰共2兲−

冕

C ␤−1_⌳ 01 −1_⌳ 00 −3_⌳ 1d␰共1兲, 共29兲

where⌳₀₀is defined as before and the other coefficients are defined as

⌳01=

冑

2 − sin

2_{关␰共1兲兴cos}2_{关␰共2兲兴sin}2_{关␪共2兲兴,}

共30兲

␣= −共1/2兲⌳₀₀−1⌳₀₁−1sin2关␰共1兲兴sin关2␰共2兲兴sin关␪共2兲兴, and␤=

冑

1 +␣2_{. Therefore, up to a global phase factor, U共C兲} is just the product of control not operation and some single-qubit rotations.

In most cases, the Hamiltonian 共16兲 has a four-dimensional eigenspace with zero eigenvalue in the invariant subspace I. The basis of the eigenspace are just the dark states兩D_ij典. Nevertheless, it is pointed out that, in some very special cases, there may be accidental degeneracy in a sub-manifold共we call it the AD submanifold兲 of the parameter space M. In the AD submanifold, in addition to the dark states兩Dij典, the Hamiltonian has another two eigenstates with zero eigenvalue and thus the dimension of the Hamiltonian’s eigenspace with zero eigenvalue is 6 rather than 4. It is ap-parent that if the path of the adiabatic evolution of the Rabi frequencies cross the AD manifold, there might be a transi-tion from the four dark states兩Dij典 to the two external states. To avoid this kind of unwanted transition, we should control the evolution path of the Rabi frequencies in the parameter space M be far away enough from the AD submanifold. On the other hand, since the evolution of the Rabi frequencies in M is assumed to begin and end at the same point O where all the Rabi frequencies are set to zero, the accidental degen-eracy at O should be avoided. This can be implemented by adjusting the coupling strength g共l兲 via controlling the posi-tion of the SQUIDs in the cavity, or the detuning⌬共l兲of each SQUID via changing the bias fields. For instance, if the con-ditions

⌬共1兲_=⌬共2兲_{⫽ 0, g}共1兲_{= g}共2兲 _共31兲

or

⌬共1兲_=⌬共2兲_{= 0, g}共1兲_{⫽ g}共2兲 _共32兲

are satisfied, the accidental degeneracy at point O can be avoided.

Since the dark states have a nonzero projection to the single or two photon states, the photon dissipation caused by the imperfection would tend to destroy the dark states. To evaluate the influence of the photon dissipation, the Schro-edinger equation controlled by the effective Hamiltonian

Heff= H共t兲 − i共␬p/2兲a†a 共33兲

needs to be solved. Here, H共t兲 is the Hamiltonian defined in Eq. 共16兲 关or Eq. 共4兲兴, and ␬p the photon decay rate. The dissipation term −i共␬p/ 2兲a†a can be considered as a pertur-bation. As a result of the first-order perturbation theory, the fidelity of the quantum gate operation, i.e., the probability of the ideal finial state, may be expressed as

F⬇ 1 −␬_p

冕

0 T

具n典phdt, 共34兲

where T is the operation time and 具n典ph=具⌿共t兲兩a†a兩⌿共t兲典 is the instantaneous expectation value of photon number. Therefore the condition under which the influence of photon dissipation can be neglected is simply

␬p

冕

0 T

具n典phdtⰆ 1. 共35兲

On the other hand, since our scheme is based on the adia-batic evolution of the quantum states, the adiaadia-batic condition should be satisfied, which can be expressed as⍀˜ TⰇ1, where

⍀˜ is the energy gap between the dark state and other eigen-states of the Hamiltonian关5兴. Here, ⍀˜ has the same order of amplitude as the SQUID-cavity coupling constant g. In prac-tical quantum gate operations, we always have T⬃103_g−1_. Then the condition 共35兲 can be satisfied when g/␬_pⲏ104_. The coupling constant of the SQUID and the cavity available at present is g⬃1.8⫻108s−1关15兴. The high quality factor of the cavity Q = 106_{– 10}8 _{might be achieved experimentally}

关22兴. This will lead to␬p兰0 T_具n典

phdtⱗ10−2and thus the fidelity F⯝1.

Now let us look in some detail into a typical controlled-PHASEgate operation discussed before. In this operation, we assume g共1兲= 2g共2兲= g = 1.8⫻108s−1 and ⌬共1兲=⌬共2兲= 0. The amplitudes of the Rabi frequencies⍀₀共1兲and⍀₀共2兲are varied following the Gaussian functions of time:

⍀0共1兲= 2.5ge−关共t − 3␶兲/␶兴 2 , 共36兲 ⍀0共2兲= ge −关共t − 3␶兲/␶兴2_ei␾₀共2兲共t兲_, where␶= 144 g−1_{. The phase␾}

0

共2兲_{共t兲 is set to be a hyperbolic} tangent function of time:

␾0共2兲共t兲 =␲⫻

冋

1 + tanh

冉

t

0.75␶

冊

册

. 共37兲 As in the above discussions, the controlled gate operation can be written as a unitary transformation

ZHANG et al. PHYSICAL REVIEW A 71, 042301共2005兲

U = ei␩兩a0典2具a0兩_ei␾兩a0a0典具a0a0兩_, 共38兲 where␾⬇␲/ 6 and␩⬇4. The operation time of this gate is about 8⫻102_g−1_{. We estimate the fidelity of this operation} using Eq. 共34兲. In Fig. 2, the fidelity of the quantum gate operation with four possible initial states are plotted as a function of the ratio g /␬p. It is seen that when g /␬p ⬃103_{– 10}4_{, the fidelity is larger than 90%. In particular,} whenever g /␬_p⬃105_{, the fidelity is improved to reach 99%.} As we have shown, during the adiabatic evolution dis-cussed above, the SQUID is in the state兩a1典, 兩a0典, and 兩g典. Since the corresponding three energy levels are the lowest three, we neglected the dissipation of the state. In practical cases, the lifetimes of the two substrates 兩a₁典 and 兩a₀典 are always much longer than the operation time and can be re-garded to be infinite. The decay of the state兩g典 may influence

our scheme. We estimate this influence using the same method as that used to estimate the influence of the photonic decay. Thus the formula共34兲 should be rewritten as

F⬇ 1 −␬p

冕

0 T 具n典phdt −␬g

冕

0 T 具n典gdt, 共39兲

where ␬g is the decay rate of state 兩g典 共for simplicity we assume that the␬_gs of the two SQUIDs are identical兲 and

具n典g=具⌿共t兲兩关兩g典共1兲具g兩 + 兩g典共2兲具g兩兴兩⌿共t兲典 共40兲 is the probability that the SQUIDs are in the state兩g典. It is straightforward to show that 具n典g=具n典ph. The fidelity of the quantum logical gate can be expressed as

F⬇ 1 − 共␬p+␬g兲

冕

0

T

具n典phdt. 共41兲

Therefore to analyze the influence of the dissipation of state

兩g典, we can replace␬p by␬p+␬gin the last two paragraphs. Finally, it is pointed out that if the decay rate ␬_p is too large, we can change the device parameters so that the effec-tive potential in Eq. 共1兲 is of a triple well, rather than a double well关23兴. In this case we can treat each of the three lowest levels to be localized in one well. The lifetimes of these three states are always sufficiently long.

We thank P. Zanardi and Y. Li for useful discussions. We are also grateful to Siyuan-Han for his suggestion to use an effective triple well for the rf SQUID. The work was sup-ported by the RGC grant of Hong Kong 共Grant No. HKU7114/02P兲, the CRCG grant of HKU, the NSFC, the Knowledge Innovation Program共KIP兲 of the Chinese Acad-emy of Sciences, and the National Fundamental Research Program of China共Grant No. 001CB309310兲.

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FIG. 2. The fidelities of a controlled-PHASE gate for the initial states兩a0典1兩a0典2共solid line兲, 兩a0典1兩a1典2共dashed line兲, 兩a1典1兩a0典2 共dot-ted line兲, and 兩a1典1兩a1典2共dashed-dotted line兲, where the quantity n is defined by the relation g /␬p= 10n.