Efficient protocol of N-bit discrete quantum Fourier transform via transmon qubits coupled to a resonator

Transcription

1 Quantum Inf Process (014) 13: DOI /s z Efficient protocol of N-bit discrete quantum Fourier transform via transmon qubits coupled to a resonator A.-S. F. Obada H. A. Hessian A.-B. A. Mohamed Ali H. Homid Received: 9 May 013 / Accepted: 19 October 013 / Published online: 5 November 013 Springer Science+Business Media New York 013 Abstract Based on the one- and two-qubit gates defined and generated via superconducting transmon qubits homogeneously coupled to a superconducting stripline resonator, we present a new physical protocol for implementing an N-bit discrete quantum Fourier transform. We propose and illustrate a detailed experimental feasibility for realizing the algorithm. The average fidelity is computed to prove the success of this algorithm. Estimated time for implementing the protocol using the proposed scheme is compared with previous schemes. Estimates show that the protocol can be successfully implemented within the present experimental limits. Keywords Solid quantum computer Quantum algorithms Quantum gates Superconducting qubits Average fidelity 1 Introduction There are complex problems that cannot be solved in a reasonable time with high accuracy in the outputs using classical computers. Therefore, there were some solutions to these complex problems, such as parallel programming, but this needs a number A.-S. F. Obada Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt H. A. Hessian A.-B. A. Mohamed Faculty of Science, Assiut University, Assiut 71516, Egypt A. H. Homid (B) Faculty of Science, Al-Azhar University, Assiut 7154, Egypt alihimad@yahoo.com

2 476 A.-S. F. Obada et al. of computers and a large place. For that, some researchers of quantum mechanics took refuge in the recent years to propose the solution for these complex problems via quantum computers [1]. Quantum computers are devices that process information in a way that preserves quantum coherence and a computational system based on the interaction of twolevel quantum mechanical systems called quantum bits (qubits) []. They depend on the fundamental quantum mechanical superposition principle of two states say, 0 and 1 at once. For instance, quantum computers when constructed can factor large numbers more rapidly [3], search data basis more quickly [4], and simulate quantum systems more efficiently [5] than the classical ones. Such computers use the quantum superposition and the interference effect to perform certain complex tasks, such as Deutsch Jozsa algorithm [6], Shor s quantum factoring algorithm [7], phase estimation problem [8], hidden subgroup problem [9], counting solution problem [10], quantum error correction circuits [11], superconducting (SC) qubits [1,13], spin qubits in a quantum-dot-microcavity coupled system [14], and so on. The quantum algorithms are important parts of quantum computers. They give enormous speed in solving some complicated problems compared to the classical computer. We devote our study only to implement discrete quantum Fourier transform (DQFT). The DQFT is completely based on the classical Fourier transform. Moreover, it is a linear unitary transform in the Hilbert space and is a key element for solving some quantum problems. Until now, only a few physical schemes have been proposed to implement DQFT based on nuclear magnetic resonance (NMR) system [15], the cavity quantum electrodynamics (QED) systems [16 19], the optical system [0,1], the solid-state qubits [], and the virtual-photon-induced process in separate cavities [3]. Superconducting Josephson junction is one of the most promising devices for the realization of a quantum computer. SC-qubits are designed and fabricated using standard techniques from conventional electronics thus allowing for large-scale integration [4]. Moreover, SC-qubits have gained substantial interest as an attractive option for quantum information processing []. Recently, a new quantum computing scheme based on Josephson qubits coupled through a cavity field was proposed [6,7]. Now, it seems, after years of painstaking labor, that superconducting qubits [5] may be considered viable elements for the construction of a scalable solid quantum computer. Following the recent work of the realization of DQFT from different systems, and the idea that superconductors are viable elements and feasible within current experimental technology, we propose and display in the current study the implementation of DQFT protocol through such system. Therefore, we consider N transmon qubits capacitively coupled to a superconducting 1D transmission line resonator to implement the N-bit DQFT. This article is organized as follows: In Sect., we furnish the theoretical description of the fundamental model and realizations of the gates as well as a new -qubit gate, which are used for implementing the N-bit DQFT. Section 3 is devoted to a description of an implementation of the DQFT. Section 4 is devoted to numerical results and a discussion to show the success of this algorithm. In Sect. 5, we present our conclusion.

3 Efficient protocol of N-bit discrete quantum Fourier 477 Fig. 1 Simplified schematic of the transmon device design. Transmon consists of a SQUID loop, characterized by a Josephson energy E J and a parallel junction capacitance C J, shunted by an external large capacitance C B and coupled to the external circuit voltage V g through a large gate capacitance C g.the value of E J can be tuned by applying a magnetic flux bias Φ to the SQUID loop. In order to induce Φ in the SQUID loop, a current (I) biased coil is used The fundamental model for the qubits and building of gates Here, we introduce a theoretical description of superconducting transmon qubits capacitively coupled to a superconducting resonator. These are used to generate some useful gates in order to implement the N-bit DQFT. The transmon qubit consists of two superconducting islands connected to a superconducting electrode through a two identical Josephson tunnel junctions with capacitance C J, see Fig. 1. We assume that the Josephson energy E J can be tuned by applying a magnetic field to the circuit which threads an external magnetic flux Φ through the d-c SQUID formed by the two junctions E J ( ) = E J,max cos( π 0 ), with the flux quantum Φ 0 = h/e. The effective offset charge n g on the island, measured in units of the Cooper pair charge e, may be controlled via a gate electrode V g capacitively coupled to the island whose capacitance C g, such that n g = Q r /e + C g V g /e with Q r an environment-induced offset charge. The transmon is distinguished from the Cooper pair box by a shunting connection of the two superconductors via a large capacitance C B, accompanied by a similar increase in the capacitance C g. By means of the additional capacitance C B, the system e is characterized by the charging energy E C = (C B +C J +C g ) which can be made very small compared to the Josephson energy E J (i.e., E J E C ). The Hamiltonian of the transmon qubits in this basis can be expressed as follows [8,9]: Ĥ tr = 4E C (M n g ) M M E J ( M + 1 M +h.c), (1) M where M the number of transmon transferred between the islands. The scaling of the eigenstates say ψ tr {ψ tr (d, g, f, h,...)} of the transmon (1) is valid in the limit E J /E C 1 (typically E J /E C 100). In this study, we will focus on the so-called charging regime E J /E C 1, and the effective offset charge n g is restricted to a unit charge interval 0 n g 1. Therefore, the Hamiltonian (1) in this case of a single qubit in the transmon regime restricted to two states is given by the following: with ˆσ x and ˆσ z are the usual Pauli operators. M Ĥ q = E C (1 n g ) ˆσ z E J ˆσ x, ()

4 478 A.-S. F. Obada et al. The experiment, using one transmon qubit to explain the reasonable assumption for quantum bits which are near resonance with a mode of the resonator and far detuned from others, is discussed in [30]. If the qubits are capacitively coupled to a single-mode superconducting transmission line resonator, the gate voltage V g involves a d c part and an extra quantum part V (x) = ωr π x L 0 c cos( L 0 )(â +â ) depending on the state of the resonator. In this work, it is assumed that the qubits are placed in different positions along a line resonator, and the lowering and raising operators for the superconducting qubits have a phase shift θ. Therefore, the Hamiltonian of the N qubits in the transmon regime coupled to a one-dimensional superconducting resonator is given by [31]: Ĥ N = hω r â â + N ( ) ξx i ˆσ x i + ξ z i ˆσ z i + N [ )] G i ˆσ z (1 i n i g [â +â ], (3) where â (â ) is the lowering (raising) operator for the resonator field, respectively, with frequency ω r = L π depends on its capacity c, inductance l per unit length, and the 0 lc length of resonator L 0. The coefficients ξx i = Ei J,max cos( π i 0 ) and ξz i = Ei C (1 n i g ) for the ith qubit, and the coupling energies G i = Ei C Ci g ωr e L 0 c cos( π x i L 0 ) depend on the positions x i of the qubits with x i [ L 0, L 0 ], i = 1,,...,N. We note that a term in the interaction with the resonator appears via the effective offset charge gate n g. Each qubit state is expressed in the basis { g i, e i }, and ˆσ x i, ˆσ z i are the Pauli operators of the ith qubit. If all superconducting qubits are placed at nodes along the resonator, that is at the locations: { L 0 4, 3L 0 4,..., (κ 1)L 0 4 ; κ = 1,,...}, see Fig. b, and the classical Fig. The diagram circuits for the positions of qubits along a one-dimensional superconducting stripline resonator. The second mode of the electrical field (arrows) mediate the qubit qubit interaction

5 Efficient protocol of N-bit discrete quantum Fourier 479 magnetic field is switched to Φ i = 0 with n i g = 1, the Hamiltonian (3) apart from the free resonator part is given by: Ĥ Nq = N [ ] ξ i ˆσ + i e iθ + h.c, ξ i = Ei J,max. (4) We start by realizing some of one-qubit operations. For the special case of N = 1, the dynamical evolution of any bipartite initial state (e.g., r, g 1, r, e 1 ) can be found by using the unitary operator û(t) = exp( i Ĥ 1q t ), which is given by the following: h û(θ, t) = ( cos(υt) ie iθ ) sin(υt), υ = E1 J,max. (5) ie iθ sin(υt) cos(υt) h Now, some quantum logic gates may be realized for the one-qubit type from the unitary operator (5). If we take the action of a π/ pulse (υt = π ) with the phase θ after the action of a 3π/ pulse (υt = 3π ) with the phase θ,wehave ( û θ, π ) ( θ û υ, 3π ) e i θ 0 ˆR z (θ) = υ 0 e i θ, (6) which is the elementary rotation gate about the z-axes. Therefore, we can generate the phase shift gate Ŝ(θ), which acts on a one-qubit, from (6) asfollows: ( e i θ û θ, π ) ( θ û υ, 3π ) ( ) 1 0 Ŝ(θ) =. (7) υ 0 e iθ Also, we note that the Hadamard gate is generated from the following operation: ( e i π 3π û, 3π ) ( û π, 3π ) ( ) Ĥ = (8) 4υ υ 1 1 From the above results, the phase factor θ does not dependent on Φ but rather a general phase, and we know that θ plays a very important role in our scheme as long as all superconducting qubits in the transmon regime are placed at nodes along the resonator. However, if there was no phase factor, then it would not be possible to have the general form presented above. To recover the standard form for the system (3), the canonical transformation ˆ = Nk=1 exp[i π 4 ˆσ y k ] is used to give the following:

6 480 A.-S. F. Obada et al. Ĥ = ˆ Ĥ N ˆ = hω r â â+ N ( ) ξz i ˆσ x i ξ x i ˆσ z i + N [ )] G i ˆσ x (1 i n i g [â +â ], In case if all qubits are placed at positions along a resonator away from the nodes, see Fig. a, and the classical magnetic field is switched to Φ i = 0 with n i g = 1,the Hamiltonian (9) in the Heisenberg picture can be expressed as follows: Ĥ (t) = hω r â â + (9) N N ξ i ˆσ z i + ] G i [â(t) +â (t) ˆσ x i (t). (10) In the rotating frame for both the qubits and the resonator, the operators of the system take the form: Â(t) =â(t)e i hω dt and ˆϱ ± i (t) =ˆσ ± i (t)e±i hω dt (so that [Â(t), Â (t)] = 1 and [ˆϱ + i (t), ˆϱ j (t)] = ˆσ z iδij ), where ω d is the frequency of the external drive of the resonator which is described by Ĥ D (t) = d [ε d(t)â e iω d t + εd (t)âe iω d t ], and ε d is the amplitude of the d th external drive. Therefore, the Hamiltonian (10) inthe interaction picture is given by the following: Ĥ I (t) = N } i G i {Â(t)e {ˆϱ } h(ω d ω r )t + h.c + i ξi (t)e iθ i e h( h ω d )t + h.c, (11) For simplicity, we assume in this article that the frequency of the driving field is equal to the Josephson energy (i.e., ξ i / h = ω d ). Then, by using the transformation ˆT = Nk=1 exp{ π 4 [ˆϱk + e iθ ˆϱ k eiθ ] iϑ Â Â} with ϑ [0,π], the effective Hamiltonian of the system (11) in the dispersive limit, δ i G i, can be expressed as follows: Ĥ eff = 1 4 ( ) N Â Â + 1 i, j =1 G i G j (δ i + δ j ) δ i δ j ˆσ i z ˆσ j z, (1) where for any relevant photon number r, the qubits in the transmon regime are in a cavity field (resonator) whose single photons frequencies are far from the resonance frequency, and δ i = Ei J,max ω h r is the detuning between the qubits and the resonator. For the case of two-qubit, the dynamical evolution of the system (1), in the subspace ( r, g 1, g, r, g 1, e, r, e 1, g, r, e 1, e ) with r being the number of photons, is given by the following evolution operator: e iφ 1t Û 0 e iφ t 0 0 (t) = 0 0 e iφ, (13) t e iφ 1t

7 Efficient protocol of N-bit discrete quantum Fourier 481 where φ 1 = 1 (r + 1)[ G 1 G δ G 1G δ 1 δ δ 1 + G δ + G 1G δ 1 δ (δ 1 + δ )] and φ = 1 (r + 1)[ G 1 δ 1 + (δ 1 + δ )]. It is noted that any realized gate from this operator does not introduce changing of the original states. If the resonator is initially in the state 0, the coupling parameters are equal (i.e., G 1 = G ) and the detuning δ 1 = δ,the dynamical evolution (13) for time t = G π 1, can be expressed as follows: e iφ Û (φ) =, φ = 4πG 1. (14) δ e iφ Now, from the unitary operator (5), we can realize the following unitary operator: Hence, we get Ô(θ, t) =û(θ, t) û(θ, t). ( Ê(θ) = Ô θ, π ) ( θ Ô υ, 3π ) = υ e iθ (15) e iθ Therefore, the two-qubit controlled-phase gate can be realized from the following: Ê(φ)Û (φ) ˆP(β) =, β = φ. (16) e iβ On the other hand, the Hamiltonian (10) in the rotating wave approximation and the dispersive limit, δ i G i, can be written as follows: where Ĥ eff = N i, j =1 G i G j (δ i + δ j ) {( } ˆσ + i δ i δ ˆσ j ˆσ j + ˆσ i +ˆσ z i δij) â â +ˆσ + i ˆσ j, (17) j { 0, if i = j; δ ij = 1, if i = j.

8 48 A.-S. F. Obada et al. Therefore, the dynamical evolution of system (17) for the two-qubit in the case of δ 1 = δ, in the subspace ( r, g 1, g, r, g 1, e, r, e 1, g, r, e 1, e ), is given by the following evolution operator: where e iχ 1t 0 ( e iχ 3t Ł Û(t) = 1 χ ) 5 χ 4 Ł iχ6 χ 4 e iχ3t Ł 0 ( ) iχ 0 6 χ 4 e iχ3t Ł e iχ 3t Ł 1 χ, (18) 5 χ 4 Ł e iχ t χ 1 = r(g 1 +G ) δ 1, χ = (r+1)(g 1 +G ) δ 1, χ 3 = (G 1 +G ) δ 1, (r+1) χ 4 = (G 1 G ) +4G 1 G δ 1, χ 5 = i(r+1)(g 1 G ) δ 1, χ 6 = G 1G δ 1, Ł 1 = cos(χ 4 t) and Ł = sin(χ 4 t). It is noted that the evolution operator (18) when operated on bipartite states can entangle much states. Therefore, this operator or any gate realized by this operator can change some of the original states. If the resonator is initially in the vacuum state and the coupling parameters of the two qubits with resonator are equal (i.e., G 1 = G ), the dynamical evolution (18) of the system is given by the following: e iμt cos(μt) ie iμt sin(μt) 0 Û(t) = 0 ie iμt sin(μt) e iμt, (19) cos(μt) e iμt where μ = G 1 δ 1. After the action of a π/4 pulse (μt = π 4 ), we can propose the new gate called i ŜWCZ, that is the square root for the combination of minus ŜWAP and Controlled-Z gates, which is defined as follows: i ŜWCZ = (1 i) 1 (1 + i) (1 + i). (0) 1 (1 i) i It can be seen that this gate is universal for quantum computation and differs substantially from the square root of a conventional ŜWAP gate [3,33], where the difference is in all the diagonal elements and off-diagonal except the upper element in the main

9 Efficient protocol of N-bit discrete quantum Fourier 483 diagonal. It can be easily verified that this gate is symmetric, invertible, not nilpotent, not idempotent, and not hermitian. We use this gate to implement the N-bit DQFT which is discussed in the following section. 3 Quantum schematic and experimental realization for implementing the N-bit DQFT In this part, we give the algorithm for implementing the DQFT using an i ŜWCZ, ˆP(β) and several one-qubit gates. On doing so, the homogeneous interaction of N qubits in the transmon regime with the resonator is used to implement the N-bit DQFT. Further, the case of classical magnetic field with the flux is also considered. We assume that the quantum register has N qubits, whose possible states are 0 = , 1 = ,... N 1 = Hence, we have m = m 1 m...m N, with m = N (m i ) i 1 and m i {0, 1}.The DQFT when applied on the N-bit state = N 1 m=0 d m m with can be expressed by the following equation: ˆF = 1 N N 1 n=0 N 1 m=0 d m e πimn/n n. (1) Here, we assume the N qubits are initially in an arbitrary superposition state d 0 g 1 g... g N 1 g N + d 1 g 1 g... g N 1 e N + +d N 1 e 1 e... e N 1 e N, where N 1 i=0 d i = 1. First, we propose and illustrate the experimental setup for implementing the three-bit DQFT and present the schematic circuit for the algorithm of 3-bit in Fig. 3. Therefore, extending this method for the 3-bit, we can realize the algorithm for N-bit which is shown schematically in Fig. 4. To do this, we need N(N 1) i ŜWCZ gates, N(N 1) ˆR z (θ) gates at different θ, several two-qubit controlled-phase gate at different phases, and several different gates from the type of one-qubit by using the pulses for series of the external magnetic fields. A physical protocol requires N qubits in the transmon regime, where all qubits are simultaneously interacting homogeneously with 1D superconducting transmission line resonator, unless an error will be introduced. This is still a challenge for the qubits to be fabricated in the space between the planes at exact places along the resonator in the experiment. Only two nearest-neighbor qubits interact simultaneously with a resonator during the interaction time for implementing the two-qubit gates. In a special case if the qubits are fabricated close to the ends of the resonator, near the input and output ports, the coupling to the resonator is capacitive and maximized for both qubits.

10 484 A.-S. F. Obada et al. Fig. 3 Quantum schematic circuit for implementing the three-bit DQFT. This quantum circuit contains oneand two-qubit gates: Here, ˆP is the two-qubit controlled-phase gate at different phases. An illustration and experimental setup for implementing the three-bit DQFT runs as follows: The qubits are simultaneously interacting with the resonator mode, where the mode is in the vacuum state. When the three qubits are all individually placed at nodes along a resonator and Φ i = 0 (i = 1,, 3), the interaction between the qubits and the resonator is switched off and results in one-qubit gates with different pulses of classical field at different positions on a three-qubit system. Also, if the three qubits are placed at antinodes along a resonator and Φ i = Φ 0, the interaction between the two qubits (whether it can be between 1, or,3) and the resonator is switched on and results in a two-qubit gates This can be used to independently dc bias the qubits at their charge degeneracy point. The resulting state of the qubits carrying the initial input state denotes the DQFT. 4 Numerical results and discussion In this section, we compute the fidelity measure to quantify the success of the present algorithm and discuss the experimental feasibility setup for our scheme. First, we examine and verify the effect of time errors on the implementation of gates. Here, we assume that the actual times of the superconductors in the cavity stay to t π 1 (Ɣ) = + 16π Ɣ, t G 1 G (Ɣ) = πδ 1 1 4G 1 + 8πδ 1 G Ɣ, Ɣ 1, 1 where the effect of the deviation Ɣ for the times is caused by positioning the qubits at a precise location at the transmission line resonator, and we assume the deviations of the time have the same amount. To elucidate the implementation of the N-bit DQFT of our model for quantum computers, the gate average fidelity is defined according to [34] as follows: F DQFT = ψ in Û DQFT ˆρ outûdqft ψ in = ψ(t) ψ ideal, ()

11 Efficient protocol of N-bit discrete quantum Fourier 485 Fig. 4 Schematic circuit and experimental setup for implementing the multibit DQFT where Û DQFT is the ideal DQFT operation, ˆρ out = ψ(t) ψ(t), with ψ(t) being the final state after the DQFT and ψ ideal = Û DQFT ψ in is the ideal target state. Assume that the N qubits are initially in the general state ψ in = ( 1 ) N { g 1, g,...,g N +ie iϕ g 1, e,...,g N +ie iϕ 1 e 1, g,...,g N + + i N e i(ϕ 1+ϕ + +ϕ N ) e 1, e,...,e N }. Therefore, the F DQFT of the N-bit DQFT operation can be written as follows: F DQFT = ( ) 1 N π dϕ 1 π 0 π 0 π dϕ... dϕ N ψ(t) ψ in. (3) The fidelity of 3-bit DQFT is plotted in Fig. 5 as a function of the deviation amount Ɣ. We can see from Fig. 5i that when Ɣ [0.06, 0.065], the fidelity F DQFT 98.8, and from Fig. 5ii that when Ɣ [0.09, 0.095], the fidelity F DQFT Therefore, the fidelity is considerably high. Looking for the experimental realization of the protocol, we use the available experimental data as given in [30,35,36]. There it is given that L 0 1cm,ω r /π.6 GHz, E 1 J,max /π h GHz, E1 C /π h GHz, G 1/π = 347 MHz and E 1 J,max /E1 C 1.7 for experimental setup. Thus, we have δ 1/π = 0

12 486 A.-S. F. Obada et al. (i) Average Fidelity Γ (ii) Average Fidelity Γ Fig. 5 The variation of F DQFT with the deviation amount Ɣ for implementing a 3-bit DQFT GHz G 1 /π, t cp = 3.10 ns for implementing the two-qubit controlledphase gate and t swz = ns for implementing the new gate. Therefore, the total time to complete the procedure of the two-bit DQFT is µs, for the three-bit is µs and for the four-bit is 0.78 µs, which are shorter than the qubit relaxation time T 1 = 1.8 µs and the dephasing time T = 0.50 µs as estimated in [35,36]. Hence, our algorithm may be a useful and good step for quantum algorithms in the building of solid quantum computer, and it may also be realizable within the limits of the present day experiments. Next, we make a brief comparison between the present scheme and the proposed scheme in [1]. In this article, the used system is considered only nearest-neighbor interaction. Therefore, the interaction through the proposed system to implement twoqubit gates will be possible only between qubits 1 and and between and 3 and so on. However, in the algorithm for implementing 3-bit shown in (Chapter 5 of [1] page 0), one controlled-phase gate ˆP( π 4 ) and one ŜWAP gate operations must be applied to qubits 1 and 3, which cannot be directly realized using the proposed model. As it has been indicated in [], a rearrangement has to be introduced which worked need the application of a large number of two-qubit operation. However, to make qubits 1 and 3 nearest-neighbor, that is, to implement the algorithm for 3-bit through proposed system, we rearrange the circuit of [1] and one needs to perform the following: (1) Ĥ on qubit 1; () ˆP( π ) between 1 and ; (3) Ĥ on qubit ; (4) ŜWAP gate between 1 and ; (5) ˆP( π 4 ) between and 3; (6) ŜWAP gate between and 3; (7) ˆP( π ) between 1 and ; (8) Ĥ on qubit ; (9) ŜWAP gate between 1 and. After performing this series, the 3-bit DQFT is achieved. From the above-mentioned experimental realization of the proposed protocol, we get the time for implementing ŜWAP gate is ns. Therefore, the total time to complete the procedure of the -bit DQFT is µs, for 3-bit is 0. µs and for 4-bit is 0.4 µs. As noted in this article, the total time for implementing -, 3-, and 4-bit DQFT is less than from the total time that mentioned in [1]. For that, our algorithm is better than the mentioned algorithm in [1]. Also, we now make a brief comparison with the results of [18,19], and our protocol has the following merits: First, we can realize from the SC-system the two-qubit and

13 Efficient protocol of N-bit discrete quantum Fourier 487 one-qubit used in the implementation of DQFT in [19]. Therefore, we can implement the algorithm of DQFT which mentioned in [19] through the proposed system. Furthermore, it is found that the interaction time which are used in the implementation of two-qubit gate is ns, and this is the same time for implementing ŜWAP gate in [1] through proposed system. Hence, the total time to complete the procedure of the -bit DQFT via this system is µs, for 3-bit is 0.0 µs, and for 4-bit is 0.40 µs. As noted in this article, the total time for implementing -, 3-, and 4-bit DQFT is less than from the total time that mentioned in [19]. Therefore, our algorithm also is better than the mentioned algorithm in [19]. Second, the interaction times required to implement the two-qubit gates in our system to carry out DQFT are much less than the interaction times required to implement the two-qubit gates for the proposed systems through [18,19] to implement DQFT. Third, our protocol via superconducting qubits that superconductors are viable elements and feasible within current experimental technology. 5 Summary and conclusions In conclusion, we have proposed a new quantum circuit for implementing the physical algorithm DQFT using a one-qubit and two-qubit gates via N superconducting transmon qubits homogeneously coupled to a superconducting stripline resonator. A new gate called the square root of minus ŜWCZ and controlled-z gates is proposed and generated from two identical qubits in the transmon regime fabricated in the space between the planes along the resonator. This gate facilitates implementation of the N-bit DQFT. Schematic representation for experimental setting up for the 3-bit and N-bit is exhibited. The estimated time for implementing this protocol using the proposed gate is found to be less than other schemes presented previously. The DQFT represents a key element for some quantum algorithms. Therefore, such a scheme may be a useful step for implementing more complicated quantum algorithms in a solid quantum computer. Moreover, the proposed experimental device for this model can open wide prospects for more complicated quantum computation with qubits in cavities. It is estimated that the protocol is realizable within nowadays experimental availability. Acknowledgments The authors would like to record their gratitude to the referee for this valuable comments that improved the presentation of the article in many aspects. References 1. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (000). Schumacher, B.: Quantum coding. Phys. Rev. A. 51, (1995) 3. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 6, (1997) 4. Lov K. Grover.: A fast quantum mechanical algorithm for database search. In: Proceedings of 8th Annual ACM Symposium on the Theory of Computing (STOC), pp (1996) 5. Lloyd, S.: Universal quantum simulators. Science 73, (1996)

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