Physica C 477 (2012) 24–31

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Physica C

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Multiqubit quantum phase gate using four-level superconducting quantuminterference devices coupled to superconducting resonator

Muhammad Waseem, Muhammad Irfan, Shahid Qamar ⇑Department of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad 45650, Pakistan

a r t i c l e i n f o

Article history:Received 22 September 2011Received in revised form 10 February 2012Accepted 19 February 2012Available online 1 March 2012

Keywords:Quantum phase gateSuperconducting quantum interferencedevices (SQUIDs)Superconducting resonatorQuantum Fourier transform

0921-4534/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.physc.2012.02.024

⇑ Corresponding author. Tel.: +92 51 2207381; fax:E-mail address: shahid_qamar@pieas.edu.pk (S. Qa

a b s t r a c t

In this paper, we propose a scheme to realize three-qubit quantum phase gate of one qubit simulta-neously controlling two target qubits using four-level superconducting quantum interference devices(SQUIDs) coupled to a superconducting resonator. The two lowest levels j0i and j1i of each SQUID areused to represent logical states while the higher energy levels j2i and j3i are utilized for gate realization.Our scheme does not require adiabatic passage, second order detuning, and the adjustment of the levelspacing during gate operation which reduce the gate time significantly. The scheme is generalized for anarbitrary n-qubit quantum phase gate. We also apply the scheme to implement three-qubit quantumFourier transform.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Quantum information processing has the potential ability tosimulate hard computational problems much more efficiently thanclassical computers. For example factorization of large integers [1],searching for an item from disordered data base [2], and phaseestimation [3]. Quantum logical gates based on unitary transfor-mations are building blocks of quantum computer. The schemesfor realizing two-qubit quantum logical gates using physical qubitssuch as atoms or ions in cavity QED [4,5], superconducting deviceslike Josephson junctions [6], cooper pair boxes [7] have been pro-posed. In earlier studies, strong coupling with charge qubits [8]and flux qubits [9] was predicted in circuit QED. In some recentstudies, experimental demonstration of strong coupling in micro-wave cavity QED with superconducting qubit [10–12] has beenrealized. The results of these experiments make superconductingqubit cavity QED an attractive approach for quantum informationprocessing.

Among the superconducting state qubits, SQUIDs are promisingcandidate to serve as a qubit [13]. They have long decoherencetime of the order of 1–5 ls [14,15], design flexibility, large-scaleintegration, and compatibility to conventional electronics[14,16,17]. They can be easily embedded in the cavity while atomsor ions require trapping techniques. Some interesting schemes forthe realization of two-qubit quantum controlled phase gates based

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+92 51 2208070.mar).

on a cavity QED technique with SQUIDs have been proposed[18–21]. These studies open a way of realizing physical quantuminformation processing with SQUIDs in cavity QED.

Recently, physical realization of the multiqubit gates has gaineda lot of interest [22–25]. Algorithms for quantum computing be-come complex for large qubit system. However, multiqubit quan-tum phase gate reduces their complexity and can lead to fastercomputing. Multiqubit quantum controlled phase gate has greatimportance for realizing quantum-error-correction protocols [26],constructing quantum computational networks [27] and imple-menting quantum algorithms [28].

In this paper, we present a scheme for the realization of three-qubit quantum controlled phase gate of one-qubit simultaneouslycontrolling two qubits using four-level SQUIDs coupled to a super-conducting resonator. It may be mentioned that in an earlier study,a proposal for multiqubit phase gate of one qubit simultaneouslycontrolling n qubits in a cavity has been presented [24], which isbased upon system-cavity-pulse resonant Raman coupling,system-cavity-pulse off-resonant Raman coupling, system-cavityoff-resonant interaction and system-cavity resonant interaction.In another study [25], a multiqubit phase gate based upon the tun-ing of the qubit frequency or resonator frequency is proposed.These proposals are quite general which can be applied to flux qu-bit systems or SQUIDs too. The present scheme is based on system-cavity-pulse resonant and system-cavity off-resonant interactionswhich can be realized using flux qubit (SQUID) system. In thisproposal, two lowest levels j0i and j1i of each SQUID are used torepresent the logical states while higher energy levels j2i and j3iare utilized for gate realization. A single photon is created by

M. Waseem et al. / Physica C 477 (2012) 24–31 25

resonant interaction of cavity field with j2iM j3i transition of thecontrol SQUID. In the presence of single photon inside the cavity,off-resonant interaction between the cavity field and j2iM j3itransition of each target SQUID induces a phase shift of eihn to eachnth target SQUID. Our scheme has following advantages:

(1) Controlled phase gate operation can be performed withoutadjusting level spacing during gate operation, thus decoher-ence due to tuning of SQUID level spacing is avoided.

(2) The proposal does not require slowly changing Rabi frequen-cies (to satisfy adiabatic passage) and use of second-orderdetuning (to achieve off-resonance Raman coupling betweentwo relevant levels), thus the gate is significantly faster.

(3) During the gate operation, tunneling between the levels j1iand j0i is not needed. The decay of level j1i can be madenegligibly small via prior adjustment of the potential barrierbetween the levels j1i and j0i [29]. Therefore, each qubit canhave much longer storage time.

(4) We do not require identical coupling constants of eachSQUID with the resonator. Similarly, detuning of the cavitymodes with the transition of the relevant levels in every tar-get SQUID is not identical, therefore, our scheme is tolerableto inevitable non-uniformity in device parameters.

The scheme is generalized to realize n-qubit quantum controlledphase gate. Finally, it is shown that the proposed scheme can beused to implement three-qubit quantum Fourier transform (QFT).

2. Quantum phase gate

The transformation for three-qubit quantum phase gate withone qubit simultaneously controlling two target qubits is given by:

U3jq1; q2; q3i ¼ eðih2dq1 ;1dq2 ;1

Þeðih3dq1 ;1dq3 ;1

Þjq1; q2; q3i; ð1Þ

where jq1i, jq2i, and jq3i stand for basis states j1i or j0i for qubits 1,2, and 3, respectively. Here dq1 ;1, dq2 ;1, and dq3 ;1 are the Kroneckerdelta functions. It is clear from Eq. (1) that in three-qubit quantumphase gate when control qubit jq1i is in state j1i, phase shift eih2 in-duces to the state j1i of the target qubit jq2i and phase shift eih3 tothe state j1i of target qubit jq3i. When control qubit jq1i is in statej0i nothing happens to the target qubits. Quantum phase gate oper-ator in Dirac notation can be written as:

U3 ¼ j000ih000j þ j001ih001j þ j010ih010j þ j011ih011jþ j100ih100j þ eih3 j101ih101j þ eih2 j110ih110jþ eih2 eih3 j111ih111j: ð2Þ

The schematic circuit diagram for quantum phase gate with onequbit simultaneously controlling two target qubits is shown by cir-cuit-1 in Fig. 1. The circuit-2 in Fig. 1 shows the two successive

Fig. 1. Circuit-1 shows quantum phase gate with one-qubit jq1i simultaneouslycontrolling two target qubits j q2i and jq3i. Circuit-2 shows the two successive two-qubit controlled phase gate with shared target qubit jq1i. The elements U2 and U3

represent two qubit controlled phase gate of phase shift eih2 and eih3 , respectively.These circuits are equivalent to each other.

two-qubit controlled phase gate represented by U2 and U3 withshared target qubit (i.e., qubit jq1i) but different control qubitsjq2i and jq3i (as shown by filled circles). The circuit-2 is knownas gate decomposition method. The elements U2 and U3 representcontrolled phase gate having phase shift eih2 and eih3 , respectively.These circuits are equivalent to each other [24] which can providefast implementation of QFT as discussed in Section 6.

3. Dynamics of the system

Here we consider rf-SQUIDs which consists of Josephson junc-tion enclosed by superconducting loop. The corresponding Hamil-tonian is given by [30]:

HS ¼Q 2

2Cþ ð/� /xÞ

2

2L� EJ cos

2p//0

� �; ð3Þ

where C and L are junction capacitance and loop inductance, respec-tively. Conjugate variables of the system are magnetic flux /threading the ring and total charge Q on capacitor. The static exter-nal flux applied to the ring is /x and EJ � Ic/0

2p is the Josephson cou-pling energy. Here Ic is critical current of Josephson junction and/0 ¼ �h

2e is the flux quantum. We consider the interaction of SQUIDwith cavity field and microwave pulses as discussed in the forth-coming subsections.

3.1. Control SQUID interaction with resonator

Control SQUID is biased properly to achieve desired four-levelstructure by varying the external flux [21] as shown in Fig. 2.The single-mode of the cavity field is resonant with j2i1 ? j3i1transition of control SQUID, however, it is highly detuned fromthe transition between the other levels which can be achieved byadjusting the level spacing of SQUID [18,29]. Using interaction pic-ture with rotating wave approximation one can write the Hamilto-nian of system as [18]:

H1 ¼ �hðg1ayj2i1h3j þ H:cÞ; ð4Þ

where a� and a are photon creation and annihilation operators forthe cavity field mode of frequency xc. Here g1 is the coupling con-stant between cavity field and j2i1 ? j3i1 transition of the controlSQUID. The evaluation of initial states j3i1j0ic and j2i1j1ic underEq. (4) can be written as:

j3i1j0ic ! cosðg1tÞj3i1j0ic � isinðg1tÞj2i1j1ic;j2i1j1ic ! cosðg1tÞj2i1j1ic � isinðg1tÞj3i1j0ic;

ð5Þ

where j0ic and j1ic are vacuum and single photon states of the cav-ity field, respectively.

Fig. 2. Level diagram of control SQUID and target SQUIDs with four levels j0i, j1i,j2i, and j3i. The levels j2i, and j3i of control SQUID interact resonantly to resonatorwhile levels j2i and j3i of each target SQUID interact off-resonantly to the resonator.The difference between level spacing of each SQUID can be achieved by choosingdifferent device parameters for SQUIDs.

(a) (b) (c)Fig. 3. Illustration of control SQUID (1) interacting with the resonator mode and/orthe microwave pulses during the gate operation.

(a) (b) (c)Fig. 4. Illustration of target SQUID (2) interacting with the resonator mode and/orthe microwave pulses during the gate operation.

(a) (b) (c)Fig. 5. Illustration of target SQUID (3) interacting with the resonator mode and/orthe microwave pulses during the gate operation.

26 M. Waseem et al. / Physica C 477 (2012) 24–31

3.2. Target SQUIDs interaction with the resonator

Suppose cavity field interacts off-resonantly with j2it ? j3it(t = 2, 3, . . . , n) transition of each target SQUID (i.e., Dc,t = xc �x32,t� gt) while it is decoupled from the transition between anyother levels as shown in Fig. 2. Here Dc,t is the detuning betweenj2it ? j3it transition frequency x32,t of the target SQUID (t) and fre-quency of resonator xc and gt is corresponding coupling constant.The effective Hamiltonian for the system in interaction picture canbe written as [31,32]:

Ht ¼�hg2

t

Dc;tðj3ith3j � j2ith2jÞaya: ð6Þ

In the presence of single photon inside the cavity, the evolution ofinitial states j2itj1ic and j3itj1ic is given by:

j2itj1ic ! eig2t t=Dc;t j2itj1ic;

j3itj1ic ! e�ig2t t=Dc;t j3it j1ic:

ð7Þ

It is clear that phase shift ei

g2t

t

Dc;t and e�i

g2t

t

Dc;t is induced to the state j2itand j3it of the target SQUID. However, states j2itj0ic and j3itj0ic re-main unchanged.

3.3. SQUID driven by microwave pulses

Let two levels jii and jji of each SQUID are driven by classicalmicrowave pulse. The interaction Hamiltonian in this case is [18]:

Hlw ¼ Xijeiujiihjj þ H:c; ð8Þ

where Xij is the Rabi frequency between two levels jii and jji and uis the phase associated with classical field. From Eq. (8) one can getthe following rotations:

jii ! cosðXijtÞjii � ie�iusinðXijtÞjji;jji ! cosðXijtÞjji � ieiusinðXijtÞjii:

ð9Þ

In our case jii? jji transition corresponds to j0i? j2i, j1i? j2i andj1i? j3i as shown in Figs. 3–5. It may be noted that the resonantinteraction of microwave pulse with SQUID can be carried out ina very short time by increasing the Rabi frequency of pulse, i.e.,intensity of the pulse.

4. Three-qubit controlled phase gate

Let us consider three-qubit controlled phase gate using threefour-levels SQUIDs coupled to a superconducting resonator. Forthe notation convenience, we denote the ground level as j1i andfirst excited state as j0i for each target SQUID as shown in Figs. 4and 5. Here we assume that the resonator mode is initially in a vac-uum state. The notation xn

i;j represents the microwave frequency in

resonance with the transition frequency between jii? jji level ofSQUID (n = 1, 2, 3). Here we discuss a quantum phase gate withh3 = p/4 and h2 = p/2.

The procedure for realizing the three-qubit controlled phasegate is divided into the following four stages of operations:

4.1. Forward resonant operation on control SQUID (1)

Step 1: Apply microwave pulse with frequency f ¼ x13;1 and phase

u = p to the control SQUID (1). Choose time intervalt1 ¼ p

2X13to transform the state j1i1 to ij3i1 as shown in

Fig. 3a. The level j3i1 of control SQUID (1) is now occupied.Cavity field interacts resonantly to the j2i1 ? j3i1 transi-tion of control SQUID (1) shown in Fig. 3b. Wait for timeinterval t01 ¼ p=2g1 such that the transformationj3i1j0ic ? �ij2i1j1ic is obtained. The overall step can be

written as j1i1j0ic!t1 ij3i1j0ic!

t01 j2i1j1ic . However, the statej0i1j0ic remains unchanged.

Step 2: Apply microwave pulse with frequency f ¼ x12;0 and phase

u = p/2 to the control SQUID (1) as shown in Fig. 3c. Wechoose pulse duration t2 = p/2X02 to obtain the transfor-mation j2i1(j0i1) ? j0i1(�j2i1). Here j2i(j0i) ? j0i(�j2i)means the transition from j2i? j0i and j0i? �j2i.

4.2. Off-resonant operation on target SQUID (2)

Step 3: Apply microwave pulse with frequency f ¼ x22;1 and phase

u = �p/2 to the SQUID (2) as shown in Fig. 4a. We choosethe pulse duration t3 = p/2X12 to obtain the transforma-tion j1i2(j2i2) ? j2i2(�j1i2). It is important to mentionhere that, steps 2 and 3 can be performed simultaneously,

M. Waseem et al. / Physica C 477 (2012) 24–31 27

by setting X02 = X12, which makes the implementationtime of both steps equal. This condition can be achievedby adjusting the intensities of the two pulses.

Step 4: After the above operation, the cavity field is in a singlephoton state j1ic, while levels j2i and j3i of the controlSQUID (1) and target SQUID (3) are unpopulated.Therefore, there is no coupling of the cavity field withcontrol SQUID (1) and target SQUID (3). The cavity fieldnow interacts off-resonantly with j2i2 ? j3i2 transitionof the SQUID (2) shown in Fig. 4b. It is clear from Eq. (7)that for time t4 ¼ ðpDc;2Þ=2g2

2 state j2i2j1ic evolves to eip/

2j2i2j1ic, where Dc,2 represents the detuning of therelevant levels of SQUID (2) with the cavity field.However, the states j0i2j0ic, j0i2j1ic, and j2i2j0ic remainunchanged.

Step 5: Apply microwave pulse with frequency f ¼ x22;1 and phase

u = p/2 to the SQUID (2) as shown in Fig. 4c. The transfor-mation j1i2(j2i 2) ? �j2i2(j1i2) is obtained by choosing thepulse duration t5 = p/2X12.

4.3. Off-resonant operation on target SQUID (3)

Step 6: Apply microwave pulse with frequency f ¼ x32;1 and phase

u = �p/2 to the target SQUID (3) as shown in Fig. 5a. Thetransformation j1i3(j2i3) ? j2i3(�j1i3) is obtained bychoosing the pulse duration t6 = p/2X12. One can see thatsteps 5 and 6 can be performed simultaneously, in a sim-ilar fashion as mentioned earlier.

Step 7: After the above operation, when the cavity field is in a sin-gle photon state j1ic, the levels j2i and j3i of control SQUID(1) and target SQUID (2) are unpopulated. Under this con-

j000ij0icj001ij0icj010ij0icj011ij0icj100ij0icj101ij0icj110ij0icj111ij0ic

!1

j000ij0icj001ij0icj010ij0icj011ij0icj200ij1icj201ij1icj210ij1icj211ij1ic

!2

�j200ij0ic�j201ij0ic�j210ij0ic�j211ij0icj000ij1icj001ij1icj010ij1icj011ij1ic

!3

�j200ij0ic�j201ij0ic�j220ij0ic�j221ij0icj000ij1icj001ij1icj020ij1icj021ij1ic

!4

�j200ij0ic�j201ij0ic�j220ij0ic�j221ij0icj000ij1icj001ij1iceip=2j020ij1iceip=2j021ij1ic

!5

�j200ij0ic�j201ij0ic�j210ij0ic�j211ij0icj000ij1icj001ij1iceip=2j010ij1iceip=2j011ij1ic

!6

�j200ij0ic�j202ij0ic�j210ij0ic�j212ij0icj000ij1icj002ij1iceip=2j010ij1iceip=2j012ij1ic

!7

�j200ij0ic�j202ij0ic�j210ij0ic�j212ij0icj000ij1iceip=4j002ij1iceip=2j010ij1iceip=4eip=2j012ij1ic

!8

�j200ij0ic�j201ij0ic�j210ij0ic�j211ij0icj000ij1iceip=4j001ij1iceip=2j010ij1iceip=4eip=2j011ij1ic

!9

j000ij0icj001ij0icj010ij0icj011ij0icj200ij1iceip=4j201ij1iceip=2j210ij1iceip=4eip=2j211ij1ic

!10

j000ij0icj001ij0icj010ij0icj011ij0icj100ij0iceip=4j101ij0iceip=2j110ij0iceip=4eip=2j111ij0ic

ð10Þ

dition, SQUIDs (1) and (2) no longer interact with the cav-ity field. However, cavity field interacts off-resonantly toj2i3 ? j3i3 transition of the target SQUID (3) as shown inFig. 5b. It is clear from Eq. (7) that for t7 ¼ ðpDc;3Þ=4g2

3 statej2i3j1ic evolves to eip/4j 2i3j1ic, where Dc,3 represents the

detuning between the relevant levels of SQUID (3) andcavity field. However, the states j0i3j0ic, j0i3j1ic, andj2i3j0ic remain unchanged.

Step 8: Apply microwave pulse with frequency f ¼ x32;1 and phase

u = p/2 to the SQUID (3) as shown in Fig. 5c. The transfor-mation j1i3(j2i3) ? �j2i3(j1i3) is obtained by choosing thepulse duration t8 = p/2X12.

4.4. Backward resonant operation on control SQUID (1)

Step 9: Apply microwave pulse with frequency f ¼ x12;0 and

phase u = �p/2 to the control SQUID (1) shown inFig. 3c. We choose pulse duration t9 = p/2X02 to trans-form state j2i1(j0i1) ? �j0i1(j2i1) for SQUID (1). Againsteps 8 and 9 can be performed, simultaneously.

Step 10: Now control SQUID (1) is in state j2iwhile levels j2i andj3i of target SQUIDs are unpopulated when cavity field isin a single photon state j1ic. Under this condition, bothtarget SQUIDs no longer interact with the cavity field.Perform an inverse operation of step (1), i.e., wait fortime interval t01 ¼ p=2g1 during which resonator inter-acts resonantly to the j2i1 ? j3i1 transition of controlSQUID (1) such that transformation j2i1j1ic ? �ij3i1j0icis obtained (Fig. 3b). Then apply microwave pulse withfrequency f ¼ x1

3;1 and phase u = p to the control SQUID(1). Choose the time interval t1 ¼ p

2X13to transform the

state j3i1 to ij1i1 as shown in Fig. 3a. The over all stepcan be written as j2i1j1ic!

t01 �ij3i1j0ic!t1 j1i1j0ic . How-

ever, the state j0i1j0ic remains unchanged. The states ofthe whole system after each step of the above mentionedoperations can be summarized as

Here, state jabci is the abbreviation for states jai1, jbi2, jci3 forSQUIDs (1,2, and 3) with a, b, c 2 [0, 1, 2, 3]. It is clear from Eq.(10) that three-qubit phase gate of one qubit, simultaneously con-trolling two qubits can be achieved using three SQUIDs after theabove mentioned process. The total number of steps appeared to

Fig. 6. Circuit-1 is a schematic network for three-qubit QFT. The states jqni and jkni(n = 1, 2, 3) represent inputs and outputs, respectively. Here H represents Hadamardtransformation and Un two-qubit conditional phase transformation. The filledcircles represent the control qubits. Using the equivalent circuit shown in Fig. 1,circuit-1 reduces to circuit-2.

28 M. Waseem et al. / Physica C 477 (2012) 24–31

be ten, however, they can be reduced to seven by performing someof the steps, simultaneously as discussed earlier.

5. Generalization to n-qubit controlled phase gate

Here we discuss the generalization of our scheme to n-qubitquantum controlled phase gate. The n-qubit quantum controlledphase gate of one qubit simultaneously controlling n qubits is de-fined by the following transformation:

Unjq1; q2; . . . ; qni ¼ eih2dq1 ;1dq2 ;1

� eih3dq1 ;1dq3 ;1�; . . . ;�eihndq1 ;1

dqn ;1 jq1; q2; . . . ; qni:ð11Þ

Here dq1 ;1; dq2 ;1; . . . ; dqn ;1 are the Kronecker delta functions. First of allwe define the operators corresponding to the application of micro-wave pulse and cavity field interaction to each SQUID in appropriatecomputational basis.

(1) Operator corresponding to the microwave pulse of phase p

applied to control SQUID (1) in the basis j1i1 ¼01

� �and

j3i1 ¼10

� �is Uð1ÞlwðpÞ ¼

0 ii 0

� �, where lw stands for

microwave.(2) Operator corresponding to the resonant interaction of cavity

field to j2i? j3i transition of control SQUID (1) in the basis

j3i1j0ic ¼10

� �and j2i1j1ic ¼

01

� �is

Uð1Þr ðp=2g1Þ ¼0 �i�i 0

� �, where r stands for resonant

interaction.(3) Operator corresponding to the application of microwave

pulse of phase �p/2 for control SQUID (1) in the basis

j2i1 ¼01

� �; j0i1 ¼

10

� �and for each target SQUID in the

basis j2it ¼01

� �; j1it ¼

10

� �is UðnÞlwð�p=2Þ ¼ 0 �1

1 0

� �,

where n = 1 corresponds to the SQUID (1). In case whenphase is equal to p/2, we have UðnÞlwðp=2Þ ¼ ðUðnÞlwð�p=2ÞÞy.

(4) Operator corresponding to the off-resonant interaction ofthe cavity field with j2i? j3i transition of each target SQUID

in the basis j2it j1ic ¼01

� �; j3it j1ic ¼

10

� �is

UðtÞðhtÞ ¼ e�iht 00 eiht

� �, where ht ¼ g2

t tDc;t

.

In order to achieve n-qubit controlled phase gate with one qu-bit, simultaneously controlling n qubits, apply forward resonantoperation on control SQUID (1). As a result, single photon is createdin the cavity. Then apply off-resonant operations on target SQUID(2), target SQUID (3), up to target SQUID (n) to induce phase shiftof eih2 , eih3 , up to eihn , respectively. At the end, we apply backwardresonant operation on first control SQUID (1). Finally, single pho-ton is absorbed and cavity returns to its original vacuum state.Hence n-qubit controlled phase gate can be achieved with resona-tor mode returning to original vacuum state through a sequence ofoperations given by:

Un ¼ Uð1ÞlwðpÞ � Uð1Þr ðp=2g1Þ � Uð1Þlwðp=2Þ� �y

�Qnt¼2

UðtÞlwðp=2Þ � UðtÞðhtÞ � ðUðtÞlwðp=2ÞÞyh i

� Uð1Þlwðp=2Þ

� Uð1Þr ðp=2g1Þ � Uð1ÞlwðpÞ; ð12Þ

where t = 2, 3, . . . , n stands for target SQUID andQn

t¼2UðtÞ ¼UðnÞ � . . .� Uð3Þ � Uð2Þ shows off-resonant operation on each targetSQUID.

The two-qubit controlled phase gate can easily be obtainedfrom Eq. (12) by choosing n = 2. In this case Eq. (12) reduces tothe following:

U2 ¼ Uð1ÞlwðpÞ � Uð1Þr ðp=2g1Þ � ðUð1Þlwðp=2ÞÞy � Uð2Þlwðp=2Þ

� Uð2Þðh2Þ � ðUð2Þlwðp=2ÞÞy � Uð1Þlwðp=2Þ � Uð1Þr ðp=2g1Þ

� Uð1ÞlwðpÞ: ð13Þ

It is clear that we need to apply forward resonant operation on con-trol SQUID to create single photon in the cavity. Then phase shift ofeip is induced on target SQUID by choosing interaction time t ¼ pDc;2

g22during off-resonant operation. After backward resonant operation

on control SQUID, we obtain two-qubit controlled phase gate in fivesteps [21]. The number of steps required to implement n qubitphase gate are 2n + 1. It may be mentioned that the number of stepsrequired to implement an equivalent n-qubit decomposed phasegate are 5(n � 1).

6. Quantum Fourier transform (QFT)

The factorization of composite number via Shor’s algorithm [1]is an interesting example of quantum information processing.Quantum Fourier transform lies at the heart of Shor’s algorithm.Quantum Fourier transform is a linear operator that transformsan orthogonal basis jki into superposition given by

jki ! 1ffiffiffiffiffi2np

P2n�1

j¼0ei2pjk2�n jji; ð14Þ

where n is the number of qubits. The combination of single-qubitrotations and two-qubit quantum phase gate form a complex net-work to implement QFT for higher qubits. For example, see cir-cuit-1 in Fig. 6 for three-qubit QFT. However, the implementationscheme can be simplified by using equivalent circuit as shown inFig. 1. For three-qubit QFT, single-qubit Hadamard transformation,two-qubit controlled phase gate, and three-qubit quantum con-trolled phase gate of one qubit, simultaneously controlling two tar-get qubits are needed as shown by circuit-2 in Fig. 6. The QFT can beaccomplished in following five stages:

Stage (1) Apply the Hadamard gate on SQUID (3). The Hadamardrotation brings logical qubit j0i and j1i into superposi-tion state, i.e., j0i ! 1ffiffi

2p ðj0i þ j1iÞ and j1i ! 1ffiffi

2p

ðj0i � j1iÞ. Hadamard gate can be accomplished

Fig. 7. Plot of the gate implementation time against the number of qubits.

M. Waseem et al. / Physica C 477 (2012) 24–31 29

through two step process that involves an auxiliarylevel j3i via method described in Ref [34]. We needtwo microwave pulses of different frequencies. One isresonant to j0i? j3i transition frequency and other isresonant to j1i? j3i transition frequency. Hadamardtransformation can be realized through following threesteps:

Step (a) Apply microwave pulse with frequency f ¼ x31;3 and

phase u = p/2 to SQUID (3). We choose pulse durationt = p/2X13 to transform state j1i? �j3i while statej0i remains unchanged.

Step (b) Apply microwave pulse with frequency f ¼ x30;3 and

phase u = �p/2 to SQUID (3). We choose pulse durationt = p/4X13 to obtain the transformation j0i ! 1ffiffi

2p

ðj0i þ j3iÞ and j3i ! 1ffiffi2p ð�j0i þ j3iÞ.

Step (c) Repeat the operation described in step (a) on SQUID (3)to obtain the transformation j3i? j1i while state j0iremains unchanged. The above operations are summa-rized as

j0i!1 j0i!2 1ffiffiffi2p ðj0i þ j3iÞ!3 1ffiffiffi

2p ðj0i þ j1iÞ

j1i!1 �j3i!2 1ffiffiffi2p ðj0i � j3iÞ!3 1ffiffiffi

2p ðj0i � j1iÞ:

ð15Þ

Stage (2) Adjust the level spacing of SQUID (3) so that transitionbetween levels j2i3 and j3i3 is resonant to the cavityfield. It acts as a control qubit. Then apply resonant for-ward operation on SQUID (3). A single photon is createdin the cavity which induces a phase shift of eip2 on SQUID(2) through off-resonant operation.

Stage (3) Repeat the same operations described in stage (1) onSQUID (2).

Stage (4) Readjust the level spacing of SQUID (3), so that its rel-evant levels becomes off-resonant to the cavity fieldto apply three-qubit controlled phase gate with SQUID(1) simultaneously controlling SQUID (2) and SQUID (3)through operations described in Section 4.

Stage (5) Repeat the same operations described in stage (1) onSQUID (1).

Proceeding in a similar way, the proposed scheme can also begeneralized up to arbitrary n-qubit by placing n SQUIDs in a cavity.It is clear from circuit-1 in Fig. 6 that the implementation of n-qu-bit QFT requires n-Hadamard gates and n(n � 1)/2 two-qubit phasegates. This shows the complexity involved in implementing QFT.The situation may become much more complicated, if complexityof calibrating and operating 4-level SQUIDs is also taken into ac-count. However, the complexity can be reduced using quantumcontrolled phase gate of one qubit simultaneously controlling ntarget qubits [24,33]. This is claimed only in terms of number ofsteps involved and implementation time required for our proposalas compared to the corresponding decomposed method.

It must be pointed out, that we need level adjustment for theimplementation of three-qubit QFT. However, this level adjust-ment is only required before implementing stage 2 and 4 (seeFig. 6) and there is no need for level adjustment during the imple-mentation of phase gates. Level adjustment can be controlled byvarying external flux /x or critical current Ic [29]. Thus individualSQUID can be tuned in or out of resonance with cavity field.

7. Discussion

The total estimated operational time for three-qubit controlledphase gate is

s ¼ 2t1 þ 2t01 þ t2 þ t4 þ t5 þ t7 þ t8: ð16Þ

On substituting the values of interaction times given in Section 4,we obtained

s ¼ p 1=X13 þ 1=g1 þ 3=2X02 þ Dc;2=2g22 þ Dc;3=4g2

3

� �: ð17Þ

Here we consider without loss of generality g1 � g2 � g3 �3 � 109s�1 [22]. On choosing Dc,2 = Dc,3 = 10g3, X02 �X13 � 10g1,we have operational time s � 9.16 ns. For n-qubit controlled phasegate, we have gm = g (m = 1, 2, . . . , n) and total operational time is

sn ¼ pg

22þn20 þ

10ð2n�1�1Þ2n�1

� �. However, if we follow the gate decomposi-

tion method then total operation time for the equivalent circuit is

given by sn ¼ pg

6ðn�1Þ5 þ 10ð2n�1�1Þ

2n�1

� �. The time required to implement

an equivalent decomposed three-qubit phase gate can easily be ob-tained from this expression, which comes out to be s � 10.36 ns. Inorder to make a quantitative estimate on the speed of the two ap-proaches, next we show the plot of the operation time as a functionof number of qubits n in Fig. 7. It is clear that implementation timefor the decomposed circuit increases rapidly with n as compared tothe multiqubit gate. This shows that our approach is significantlyfaster than performing two separate two-qubit controlled phasegates which is quite interesting.

Here, we would like to make a comparison of our scheme withthe earlier proposal for n-target-qubit control-phase gate (NTCP)[25]. In the earlier study [25], the phase induced on each target qu-bit is the same, i.e., h1 = h2 = � � � = hn = p, whereas in our proposaldifferent phases are induced on each target qubit. However, NTCPgate can easily be realized in our scheme by using the followingsteps:

1. Apply step 1 of Section 4 on control SQUID (1) which generatesa single photon in the cavity.

2. Apply step 2 on SQUID (1), step 3 on SQUID (2), and step 6 onSQUID (3), simultaneously.

3. Apply off-resonant operation, i.e., step 4 on SQUID (2) and step7 on SQUID (3), simultaneously, to obtain phase shift of eip. Thiscan be achieved by choosing interaction time t ¼ pDc;t

g2t

such thatDc;t

g2t

is the same for all target SQUIDs.4. Apply step 5 on SQUID (2), step 8 on SQUID (3), and step 9 on

SQUID (1), simultaneously.5. Finally, apply step 10 on control SQUID (1), as a result single

photon in the cavity is absorbed.

Here we have considered 3 qubits, however, the scheme isapplicable to arbitrary number of qubits. Thus, NTCP gate can berealized in five steps which is independent of the total number ofqubits. The implementation time comes out to be 12 ns.

30 M. Waseem et al. / Physica C 477 (2012) 24–31

In another study [24], the implementation of a multiqubit phasegate having different phases on all target qubits is proposed. Thenumber of steps required to implement this scheme are indepen-dent of the number of qubits, however, the phase induced are con-ditional. Our proposal can also be modified to implement amultiqubit phase gate of different phases with fixed number ofsteps for arbitrary number of qubits. These modifications are givenby the following:

1. Apply step 1 on SQUID (1) as given in Section 4.2. Apply step 2 on SQUID (1), step 3 on SQUID (2), and step 6 on

SQUID (3), simultaneously.3. Allow both the target qubits to interact off-resonantly with the

cavity mode, i.e., steps 4 and 7 in Section 4. The evolution isgoverned by Eq. (7). In order to induce a phase of p/2 on SQUID(2) and p/4 on SQUID (3), we have to adjust the interaction timeas pDc;2=2g2

2 and pDc;3=4g23, respectively. To implement these

two steps, simultaneously, one needs both these times to beequal which can be achieved if, Dc;2=g2

2 ¼ 12 Dc;3=g2

3. However, thiscondition becomes more complex with the increase in the num-ber of qubits. For example, for a four-qubit gate it becomesDc;2=g2

2 ¼ 12 Dc;3=g2

3 and Dc;3=g23 ¼ 1

2 Dc;4=g24, where Dc,4 and g4 cor-

respond to the detuning and coupling of the fourth SQUID.4. Apply step 5 on SQUID (2), step 8 on SQUID (3) and step 9 on

SQUID (1), simultaneously.5. Apply step 10 on control SQUID (1), as a result single photon in

the cavity is absorbed.

It is clear that we can implement a mutiqubit phase gate havingdifferent phases on each target qubit in five steps for arbitrarynumber of qubits.

Here, we discuss different issues related to the gate operations.The Level j3i of control SQUID (1) is occupied in the steps 1 and 10as discussed in Section 4, which involve the microwave pulse offrequencies x13, and SQUID resonator resonant interaction. Thecorresponding operational time for three-qubit controlled phasegate in step 1 or 10 is given by

s1 ¼ pð1=2X13 þ 1=2g1Þ: ð18Þ

It is clear that s1 can be shortened sufficiently by increasing the Rabifrequencies and coupling constant. Control SQUID (1) can be de-signed so that energy relaxation time of level j3iðc�1

3 Þ is sufficientlylong as compared to the operational time. Thus decoherence due toenergy relaxation of level j3i is negligibly small under the conditionc�1

3 � s1 [21].The effect of dissipation during the gate operations can be ne-

glected by considering a high-Q resonator. The direct interactionbetween SQUIDs can be negligible under the conditionHs,s Hs,rHs,lx [22]. Here Hs,s is the interaction energy betweentwo nearest neighbor SQUIDs, Hs,r is the interaction energy betweenresonator and SQUID and Hs,lx is the SQUID microwave interaction.

When levels j2i and j3i are manipulated by microwave pulses,resonant interaction as well as off-resonant interaction betweenresonator mode and j2i? j3i transition of each SQUID is un-wanted. This effect can be minimized by setting the conditionXi,j� g1 for control SQUIDs and Xi;j � g2

t =Dc;t for target SQUIDs.The level j3i of each target SQUID interacts off-resonantly to thecavity field during steps 4 and 7. Its occupation probability needsto be negligibly small in order to reduce the gate error [21]. Theoccupation probability of level j3i for target SQUID is given by [19]:

p3 �1

1þ D2c;t

4g2t

: ð19Þ

For the choice of Dc,t = 10gt, we have p3 � 0.04 which is furtherreducible by increasing the ratio of Dc,t/gt.

The Hadamard transformation is accomplished by two micro-wave pulses of different frequencies in three steps described inSection 6. These steps can be implemented faster by increasingthe Rabi frequency of pulses. The operation time for Hadamardgate is around 5 ns [34].

8. Conclusion

In conclusion, we have presented a scheme for the realization ofthree-qubit controlled phase gate with one qubit, simultaneouslycontrolling two target qubits using four-level SQUIDs coupled toa single-mode superconducting microwave resonator. The schemeis based on the generation of a single photon in the cavity mode byresonant interaction of cavity field with j2i? j3i transition of con-trol SQUID and introduction of phase shift eihn to each target SQUIDby off-resonant interaction of the cavity field with j2i? j3i transi-tion. Finally, backward resonant operation is applied which ab-sorbs the single photon as a result field inside the microwavecavity reduces to its original vacuum state.

The proposed scheme for quantum controlled phase gate hassome interesting features, for example, it does not require adjust-ment of level spacing during gate operation which reduces thecause of decoherence. The present scheme does not require theadiabatic passage and second order detuning which makes the gateimplementation time faster. During the gate operation, tunnelingbetween the level j1i and j0i is not employed. Prior adjustmentof the potential barrier between level j1i and j0i can be made suchthat the decay of level j1i is negligibly small. Therefore, each qubitcan have much longer storage time [29]. The scheme can readily begeneralized to realize an arbitrary multiqubit quantum phase gateof one qubit simultaneously controlling n qubits. We have also ap-plied the scheme to implement three-qubit quantum Fouriertransform.

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