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Implementing discrete quantum Fourier transform via superconducting qubits coupled to a superconducting cavity Abdel-Shafy F. Obada,1 Hosny. A. Hessian,2 Abdel-Basset A. Mohamed,2 and Ali H. Homid3,* 1
Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt 2 Faculty of Science, Assiut University, Assiut 71516, Egypt 3 Faculty of Science, Al-Azhar University, Assiut 71524, Egypt *Corresponding author:
[email protected]
Received October 26, 2012; revised February 12, 2013; accepted March 1, 2013; posted March 12, 2013 (Doc. ID 178681); published April 12, 2013 A new physical scheme for implementing an N-bit discrete quantum Fourier transform (DQFT) is proposed via superconducting (SC) qubits coupled to a single-mode SC cavity. Two-qubit and one-qubit gates as well as a new two-qubit gate are realized. Such gates are used for implementing the algorithm of N-bit DQFT. We propose and analyze a detailed experimental procedure for implementing the algorithm and compute the fidelity measure to quantify the success of this algorithm. Estimates show that the protocol can be successfully implemented within the present experimental limits. © 2013 Optical Society of America OCIS codes: (270.0270) Quantum optics; (270.5580) Quantum electrodynamics; (270.5585) Quantum information and processing. http://dx.doi.org/10.1364/JOSAB.30.001178
1. INTRODUCTION Recently, much attention has been paid to quantum computers, based on the fundamental quantum mechanical superposition principle. A quantum computer is a computational system based on the interaction of two-level quantum mechanical systems called quantum bits (qubit) [1]. It can solve some complex problems, which are not feasible on a classical computer, in a reasonable time with high accuracy in the outputs. Such computer uses quantum parallelism and the interference effect to perform certain complex tasks, such as Deutsch–Jozsa algorithm [2], Shor’s quantum factoring algorithm [3], counting solution problem [4], 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 discrete quantum Fourier transform (DQFT). The DQFT is completely based on the classical Fourier transform. Moreover, it is a key element for solving some quantum problems and is a linear unitary transform in the Hilbert space. We assume that the quantum register has λ qubits, the possible states are j1i j00…01i; …j2λ − 1i j11…1i:
j0i j00…0i;
P Hence, we have jmi jm1 m2 …mλ i, with m λi1 mi i−1 and mi ∈ f0; 1g. The DQFT when applied on the state jmi, can be expressed as follows: λ
jm1 m2 …mλ i →
2 −1 1X λ
22 n0
λ
e2πimn∕2 jni
1 λ
22
j0i e2πi0·mλ j1i
j0i e2πi0·mλ−1 mλ j1i…j0i e2πi0·m1 m2 …mλ−1 mλ j1i;
0740-3224/13/051178-08$15.00/0
where 0 · ms ms1 …ml ms ∕2 ms1 ∕4 ml ∕2l−s1 . The basic elements for implementing the DQFT are Hadamard ˆ gate and the controlled phase gate between the different (H) two qubits. So far, some researchers have proposed only a few physical schemes to implement DQFT, such as the nuclear magnetic resonance system [5–8], the cavity quantum electrodynamics systems [9–12], the optical system [13,14], and solidstate qubits [15]. Superconducting (SC) quantum circuits with Josephson junctions are currently studied for their potential applications in quantum information processing for a scalable quantum computer [16,17]. Therefore, the physical objects have been suggested for implementations of qubits. However, solid-state circuits and SC circuits in particular [18,19] are of great interest as they offer scalability. Moreover, the SC-charge qubit is achieved in a Cooper pair [20], which is a small SC island weakly coupled to a superconductor. Recently, a new quantum computing scheme based on Josephson qubits coupled through cavity field was proposed [21–23]. Motivated by both the recent work in this topic and the idea that superconductors are viable elements and feasible within current experimental technology, therefore, in this paper, we propose the system of an N identical direct current superconducting quantum interference device (SQUID) inside the cavity field to implement the N-bit DQFT. This paper is organized as follows: Section 2 displays the theoretical description of the model and realizations of the gates as well as a new two-qubit gate, which are used for implementing the DQFT. Section 3 is devoted to implement the DQFT and the theoretical description of the experimental implementation of this algorithm. Also we calculate the fidelity as a measure of feasibility of the algorithm. Finally in Section 4, we present our conclusion. © 2013 Optical Society of America
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3π∕2-pulse EJ ∕ℏt 3π∕2 with the phase θ after the action of a π∕2-pulse E J ∕ℏt π∕2 with the phase θ∕2, we have
2. THEORETICAL DESCRIPTION OF THE MODEL AND GENERATED GATES Here, a physical model is used to generate some useful gates which are used to implement the N-bit DQFT. Now, consider a d.c. SQUID-type qubit box with κ excess Cooper-pair charges. It contains a small SC island and two Josephson junctions with the same capacitance C J and Josephson energy E J . A controllable gate voltage V g is coupled to the box via the gate capacitor C g with dimensionless gate charge ng C g V g ∕2e. We assume that the qubit is working in a regime with kB T ≪ E J ≪ E C ≪ Δ, where E C , kB , T, and Δ are charging energy, Boltzmann constant, temperature, and SC gap energies, respectively. When the gate voltage is at the degeneracy point, that is ng 1∕2, only two charge states play the main role. The other charge states with much greater energies can be neglected, which implies that the SC qubit can be reduced to a two-level system [24,25]. If N identical d.c. SQUIDs are placed in the middle of a single-mode SC cavity with full wavelength λ1 , the interaction between the cavity field and the qubits reaches its maximum. In our work, it is assumed that the annihilation and creation operators of the superconductors have a phase shift θ. Thus, the Hamiltonian of the system in the Heisenberg picture can be written as [21,26] N πϕc X σˆ iz − EJ cos σˆ i t ϕ0 i1 x i1 N πE πϕc X ˆ h:cσˆ ix t; η bt J sin ϕ0 ϕ0 i1 i
ˆ Ht ℏωbˆ † bˆ Ez
N X
(1)
p where the parameters jηi j jΛar ℏω∕ϵ0 vc2 j of the superconductors depends on the area Λar of the surface defined by the contour of the SQUID, the wavelength of cavity field, the volume of the cavity, and its position. The third term is the charge-qubits photon interaction, with the flux quantum ϕ0 h∕2e and ϕc the external flux generated by a classical applied magnetic field. The charge energy Ez −e2 1 − 2ng ∕C g 2C J depends on the gate charge ng . The operators σˆ ix and σˆ iz are the Pauli operators, which are defined on the space spanned by the charge excited states jeii and ground states ˆ bˆ † the annihilation (creation) operator for the jgii , and b cavity field. If the classical magnetic field is switched to ϕc 0 while ng 1∕2, the Hamiltonian (1) apart from the free field part, is given by ˆ N −E J H
N X
σˆ ix :
(2)
i1
For the special case of N 1, the dynamical evolution of any bipartite initial state (e.g., jr; g1 i, jr; e1 i) can be found by using ˆ 1 t∕ℏ, which is given by the unitary operator Vˆ t exp−iH the following:
0 Vˆ θ; t @
cos
EJ ℏ
ieiθ sin
t EJ ℏ
t
1 ie−iθ sin EℏJ t A: cos EℏJ t
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(3)
Here, we realize some quantum logic gates for the one-qubit from the unitary operator (3). If we take the action of a
θ 3πℏ ˆ θ πℏ e−i2 V ; → Rˆ z θ Vˆ θ; 2EJ 2 2E J 0
0 ; θ ei2
(4)
which is the elementary rotation gate about the z-axes. Therefore, if this elementary rotation Rˆ z θ is shifted with the phase ˆ θ∕2, the phase shift gate Sθ from (4) can be realized as follows: 3πℏ ˆ θ πℏ θ 1 0 ˆ → Sθ : V ; ei2 Vˆ θ; iθ 0 e 2EJ 2 2E J
(5)
Also, the Hadamard gate can be realized as follows: ei2 Vˆ π
π πℏ ˆ 3πℏ ˆ p1 1 1 : →H ; V 2π; 2 4E J 2EJ 2 1 −1
(6)
From the above results, we know that the phase factor θ plays a very important role in our scheme as long as N identical d.c. SQUID placed in the middle of a single-mode SC cavity. However, if there was no phase factor, then it would not be possible to have the general form presented above. In the rotating frame for both the qubits and the field, −iℏωd t ˆ ˆ the operators of the system take the form: Bt bte † i i 2iℏω t ˆ ˆ d and sˆ t σˆ te (so that, Bt; B t 1 and ˆsi t; sˆ j− t σˆ iz δij ), where ωd is the frequency of the exterˆD nal of the field, which is described by H P drive † iωd t ξ bte −iωd t , and ξ is the amplitude of the ˆ ˆ ξ te b d d d d dth external drive. Besides, if ϕc 1∕2ϕ0 with ng ≠ 1∕2, then the Hamiltonian (1) in the interaction picture, can be expressed as follows: N X iℏωd −ωt h:cg ˆ I t πE J ˆ fη Bte H ϕ0 i1 i 2E z × sˆ i te−iθ eiℏ ℏ −2ωd t h:c :
(7)
For simplicity, we assume in this article that the frequency of the driving field is equal to the charging energy, that is 2E z ∕ℏ Q ωd . Therefore, by using the canonical transformation d ˆ ly , the Hamiltonian (7) becomes CT N l1 exp−iπ∕4σ † ˆ d πE J ˆ 0 t d CT H H I tCT ϕ0
N X iℏδt ˆ fηi Bte h:cgσˆ iz ;
(8)
i1
where δ 2E z ∕ℏ − ω is the detuning between the SC qubits and the SC cavity. Physically, the canonical transformation d CT means that the states of each qubit are flipped and shifted by the elementary rotation Rˆ y π∕2 about the y-axes. In the dispersive limit, where for any relevant photon number r, the SC qubits are in a cavity field whose single photons frequency case are far from resonance, and δi δ 2Ez ∕ℏ − ω ≫ πEJ ∕ϕ0 ; hence, the effective Hamiltonian of the system (8) in the dispersive limit, can be expressed as follows:
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ˆ 0eff − H
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2 X N πEJ p fηi η j η i ηj Bˆ † Bˆ ηi η j gσˆ iz σˆ jz : (9) ϕo 2ℏδ i;j1
For the case of N 2, the dynamical evolution of the system, in the subspace (jr; g1 ; g2 i, jr; g1 ; e2 i, jr; e1 ; g2 i, jr; e1 ; e2 i) with r being the number of photons, is given by the following evolution operator: 0 B ˆ t B W @
eiϕ1 t 0 0 0
0 eiϕ2 t 0 0
0 0
eiϕ2 t 0
0 0 0
1 C C; A
(10)
eiϕ1 t
p where ϕ1 t πE J ∕ϕo 2ℏδ2 2r 1fjη1 j2 jη2 j2 η1 η 2 p η1 η2 gt and ϕ2 t πE J ∕ϕo 2ℏδ2 2r 1fjη1 j2 jη2 j2 − η1 η 2 η 1 η2 gt. It is noted that any realized gate from this operator does not introduce flipping, because it describes disentangled state. If the cavity field is initially in the vacuum state and the parameters of the qubits are real and equal, that is η1;2 η 1;2 jηj, then when the required time is switched to t 2π∕μ0 with μ0 2E2J ∕δℏ2 , the dynamical evolution (10) can be expressed as follows: 0
eiϕ B 0 ˆ ϕ B W @ 0 0
0 1 0 0
0 0 1 0
1 0 0 C C; 0 A eiϕ
3 2 2π jηj ℏ ϕ : ϕ2o
(11)
Therefore, the two-qubit conditional phase gate can be realized from the following: 0
1 B B0 B ˆ ϕ → Pβ ˆ ⨂ Rˆ fz ϕW B B0 f 1 @
0 0
0
1
C 0 C C C; 0 1 0 C A 0 0 0 eiβ −iϕ e 2 0 ; Rˆ fz ϕ ϕ 0 ei 2
2
1 0
(12)
ˆ where β 2ϕ. Then a controlled phase gate Pβ for two-qubit is realized from an SC charge qubits-field system. On the other hand, the Hamiltonian (1) in the rotating wave approximation and the case of ϕc 1∕2ϕ0 with ng ≠ 1∕2, can be written as N X
ˆ N t ℏωbˆ † bˆ Ez H
σˆ iz
i1
N πEJ X ˆ σˆ i te−iθ h:cg: fη bt ϕ0 i1 i
(13) The effective Hamiltonian of the system (13) in the dispersive limit, δi δ ≫ πEJ ∕ϕ0 , can be expressed as follows: ˆ eff H
2πE J p ℏϕo δ
2 X N
fηi η j σˆ i σˆ j− bˆ bˆ †
i;j1
ˆ ηj η i σˆ iz δij − σˆ j σˆ i− bˆ † bg:
(14)
For the case of N 2, the dynamical evolution of the system, in the subspace (jr; g1 ; g2 i, jr; g1 ; e2 i, jr; e1 ; g2 i, jr; e1 ; e2 i), is given by the following evolution operator:
0
1 eia1 t 0 0 0 B C a −ia6 −ia3 t B 0 e−ia3 t ϒ1 − a5 ϒ2 ϒ2 0 C a4 e 4 B C ˆ Ut B C; −ia6 −ia t a5 −ia3 t 3 ϒ B 0 C e e − ϒ ϒ 0 2 1 a4 2 a4 @ A −ia2 t 0 0 0 e (15) p p where a1 f2πE J r ∕ℏϕo δg2 fjη1 j2 jη2 j2 g, a2 p 2 p f2πE J r 1∕ℏϕo δg fjη1 j2 jη2 j2 g, a3 f2πE J ∕ p p ℏϕo δg2 fjη1 j2 jη2 j2 ∕2g, a4 f2πE J ∕ℏϕo 2δg2 p f2r 12 jη1 j2 − jη2 j2 2 4jη1 j2 jη2 j2 g, a5 if2πE J ∕ℏϕo p 2 p δg f2r 1jη1 j2 − jη2 j2 ∕2g, a6 f2πE J ∕ℏϕo δg2 η1 η 2 , ϒ1 cosa4 t and ϒ2 sina4 t. It is noted that the evolution operator (15) describes entangled state; therefore, this operator or any gates realized by this operator can flip any state. If the cavity is initially in the state j0i and the parameters of the qubits are real and equal, that is η1;2 η 1;2 jηj, the dynamical evolution (15) of the system is given by 0
1
0
B B 0 e−iμt cosμt B ˆ Ut B B 0 −ie−iμt sinμt @ 0 0 2 2πE J jηj p : μ ℏϕ0 δ
0 −ie−iμt sinμt e−iμt cosμt 0
0
1
C 0 C C C; 0 C A
e−2iμt
(16)
After the action of a π∕4-pulse (μt π∕4), we can propose the p d new gate called i S W CZ , that is the square root of minus d S W AP and controlled-Z gates, which is defined as follows: 0 1 0 q 1 B0 1 − i d 2 B i S W CZ @ 0 − 12 1 i 0 0
1 0 0 − 12 1 i 0 C C: 1 0 A 2 1 − i 0 −i
(17)
It can be seen that this gate is universal for quantum computation and differs substantially from the square root of a cond ventional S W AP gate [27,28], where the difference is in all the elements except the upper element in the main diagonal. We can use this gate to implement the N-bit DQFT, which is discussed in the following section. One can realize the algebraic properties for this gate easily, where we note that p † p −1 d d • fi S W CZ g fi S W CZ g , which means that this gate is p unitary. p t d d CZ , which means that this gate is • fi S W CZ g i S W symmetric. p p −1 d fi S W d • i S WCZ CZ g Iˆ 4 , which means that this gate is invertible. p l0 d • fi S W CZ g ≠ 0, l0 ∈ Z , which means that this gate is not nilpotent. p l0 1 p d d • fi S W CZ g ≠ i SW CZ , l0 ∈ Z , which means that this gate is not idempotent. p p d d • fi S W CZ g ≠ i S W CZ , which means that this gate is not Hermitian. p d , which means that this gate is periodic. • i S WCZ
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In the following section, we present an algorithm for the DQFT using N-charge qubits inside an SC cavity.
3. ALGORITHM FOR IMPLEMENTING THE DQFT WITH THE CHARGE QUBITS COUPLED TO A SINGLE-MODE SC CAVITY In this part, we give the quantum scheme for the P2implementing N −1 DQFT. The DQFT for a given state jΨi m0 cm jmi with N bits, can be expressed as follows:
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Step (2) The two superconductors are simultaneously interacting with a single-mode SC cavity with the initial state jφ1 i when the flux is originally set to ϕc 1∕2ϕ0 . Therefore, when the interaction time is switched to t1 2π∕μ0 , then the state of the two qubits jφ1 i, with the phase parameter β π, evolves to 1 π jφ1 i → jφ2 i fei4 c0 c1 jgi1 jgi2 jgi1 jei2 2 π e−i4 c0 − c1 jgi1 jgi2 − jgi1 jei2 π
ei4 c2 c3 jei1 jgi2 − jei1 jei2 π
ˆ FjΨi
N −1 2N −1 X 1 2X
22
N
cm e
2πimn∕2N
e−i4 c2 − c3 jei1 jgi2 jei1 jei2 g: jni:
(18)
n0 m0
In our study, we assume the N qubits are initially in an arbitrary superposition state
1 π π jφ2 i → jφ3 i p fei4 c0 c1 jgi1 jgi2 − e−i4 c0 − c1 jgi1 jei2 2 π π − ei4 c2 c3 jei1 jei2 e−i4 c2 − c3 jei1 jgi2 g:
c0 jgi1 jgi2 …jgiN−1 jgiN c1 jgi1 jgi2 …jgiN−1 jeiN … c2N −1 jei1 jei2 …jeiN−1 jeiN ; P2N −1 where i0 jci j2 1. First, we implement the 2 bit DQFT of two-qubit states, which in the binary system, is given by jχi c00 j0i1 j0i2 c01 j0i1 j1i2 c10 j1i1 j0i2 c11 j1i1 j1i2 ; (19) while in an alternate notation is written as follows: jχi c0 j0i c1 j1i c2 j2i c3 j3i. After the action of the DQFT, we have 1 ˆ Fjχi fc00 c01 c10 c11 j0i1 j0i2 2 c00 ic01 − c10 − ic11 j0i1 j1i2 c00 − c01 c10 − c11 j1i1 j0i2 c00 − ic01 − c10 ic11 j1i1 j1i2 g:
(20)
To implement the quantum circuit of the 2 bit DQFT, we consider the two superconductors coupled to an SC cavity, where the cavity field is initially in the vacuum state. The wave state of such model is jφi c0 jgi1 jgi2 c1 jgi1 jei2 c2 jei1 jgi2 c3 jei1 jei2 :
(21)
Now, we offer the physical algorithm for implementing the 2 bit DQFT using charge qubits inside a single-mode SC cavity in the following: Step (1) We assume the qubit-2 is interacting with a set of the ˆ classical fields, where qubit-2 undergoes the operations H, ˆ respectively. Therefore, the evolution of Rˆ z −π∕2, and H, the input state (21) is given by 1 π jφi → jφ1 i fei4 c0 c1 jgi1 jgi2 jgi1 jei2 2 π e−i4 c0 − c1 jgi1 jgi2 − jgi1 jei2 iπ4
e c2 c3 jei1 jgi2 jei1 jei2 π
e−i4 c2 − c3 jei1 jgi2 − jei1 jei2 g:
ˆ ˆ and Sπ Step (3) We then apply the transformations H on qubit-2. After these transformations, the evolution of the state jφ2 i, can be expressed as follows:
Step (4) Starting with jφ3 i the qubit-1 and qubit-2 are simultaneously interacting with the field when ϕc 1∕2ϕ0 . Therefore, when the interaction time is switched to t2 π∕4μ, then the entangled state of the qubit-1 and qubit2 is given by π ei4 jφ3 i → jφ4 i p c0 c1 jgi1 jgi2 ic2 c3 jei1 jei2 2 1 f1 ic0 − c1 − 1 − ic2 − c3 gjgi1 jei2 2 1 f1 − ic0 − c1 − 1 ic2 − c3 gjei1 jgi2 : 2
Step (5) By assuming that the second qubit undergoes the ˆ transformation S3π∕2 and the first qubit undergoes the ˆ transformation S3π∕2 by using the pulses for the classical field, then after these transformations the state jφ4 i, evolves to π ei4 jφ4 i → jφ5 i p c0 c1 jgi1 jgi2 − ic2 c3 jei1 jei2 2 1 f1 − ic0 − c1 1 ic2 − c3 gjgi1 jei2 2 1 − f1 ic0 − c1 1 − ic2 − c3 gjei1 jgi2 : 2
Step (6) With jφ5 i as initial state the two qubits are simultaneously interacting with an SC cavity when ϕc 1∕2ϕ0 . Therefore, when the interaction time is switched to t2 π∕4μ, the entangled state of the two qubits jφ5 i, is given by π
ei4 jφ5 i → jφ6 i p fc0 c1 jgi1 jgi2 − c2 c3 jei1 jei2 2 c2 − c3 jgi1 jei2 − c0 − c1 jei1 jgi2 g:
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Step (7) The two qubits are simultaneously interacting with a single-mode SC cavity with jφ6 i as initial state when ϕc 1∕2ϕ0 . Therefore, when t1 is switched to 2π∕μ0 , then the state of the two qubits jφ6 i, with the phase parameter β 3π∕2, evolves to π
ei4 jφ6 i → jφ7 i p fc0 c1 jgi1 jgi2 ic2 c3 jei1 jei2 2 c2 − c3 jgi1 jei2 − c0 − c1 jei1 jgi2 g: ˆ on qubitStep (8) We apply the transformations Rˆ z −π and H 1. After these transformations, the evolution of the state jφ7 i, is given by π p ei4 p jφ7 i → jφ8 i p i 2c0 jgi1 jgi2 i 2c1 jei1 jgi2 2 c p2 jgi1 jei2 ijgi1 jei2 − jei1 jei2 ijei1 jei2 2 c3 p jgi1 jei2 − ijgi1 jei2 − jei1 jei2 − ijei1 jei2 : 2
Step (9) Also, starting with jφ8 i the two qubits are simultaneously interacting with a single-mode cavity field when ϕc 1∕2ϕ0 . Therefore, when t1 is switched to 2π∕μ0 , the entangled state of the two superconductors, with β π, evolves to π p ei4 p jφ8 i → jφ9 i p i 2c0 jgi1 jgi2 i 2c1 jei1 jgi2 2 c p2 jgi1 jei2 ijgi1 jei2 jei1 jei2 − ijei1 jei2 2 c3 p jgi1 jei2 − ijgi1 jei2 jei1 jei2 ijei1 jei2 : 2
Step (10) Again, assume that the two qubits are individually interacting with a series of the classical fields by using the pulses for these fields, where qubit-1 undergoes the transforˆ ˆ and qubit-2 undergoes the transformation Sπ∕2, mation H respectively. After these transformations, we get π
ei 4 jφ9 i → jφ10 i p fic0 c1 jgi1 jgi2 ic0 − c1 jei1 jgi2 2 c2 ijgi1 jei2 − jei1 jei2 c3 ijgi1 jei2 jei1 jei2 g: Step (11) The two qubits are simultaneously interacting with the field with jφ10 i as initial state when ϕc 1∕2ϕ0 . Therefore, when the interaction time is switched to t1 2π∕μ0 , the state of both the qubit-1 and qubit-2 jφ10 i, with phase β 3π∕2, evolves to π
ei 4 jφ10 i→jφ11 i p fic0 c1 jgi1 jgi2 ic0 −c1 jei1 jgi2 2 c2 ijgi1 jei2 ijei1 jei2 c3 ijgi1 jei2 −ijei1 jei2 g: ˆ on the second quStep (12) By applying the transformation H ˆ on the first qubit, the evolution of bit and the transformation H the state jφ11 i, becomes
Obada et al. 3π
ei 4 jφ11 i → jφ12 i p fc0 c2 jgi1 jgi2 c1 c3 jei1 jgi2 2 c0 − c2 jgi1 jei2 c1 − c3 jei1 jei2 g:
Step (13) Assume that the two charge qubits are simultaneously interacting with a cavity with jφ12 i as initial state, where ϕc 1∕2ϕ0 ; therefore, when t1 is switched to 2π∕μ0 , the state of the two qubits, with the phase parameter β π∕2, evolves to 3π
ei 4 jφ12 i → jφ13 i p fc0 c2 jgi1 jgi2 c1 c3 jei1 jgi2 2 c0 − c2 jgi1 jei2 ic1 − c3 jei1 jei2 g: Step (14) Finally, assume that the first qubit undergoes the ˆ after that, the evolution of the state transformation H, jφ13 i, becomes 3π
jφ13 i → jφ14 i
ei 4 fc0 c1 c2 c3 jgi1 jgi2 2 c0 ic1 − c2 − ic3 jgi1 jei2 c0 − c1 c2 − c3 jei1 jgi2 c0 − ic1 − c2 ic3 jei1 jei2 g:
One can see that the step (4) or (6) corresponds to the p d i SW CZ -gate, and steps (2), (7), (9), (11), and (13) correˆ spond to the two-qubit controlled phase-gate Pβ at the different phases. Therefore, the 2 bit DQFT in a charge qubits coupled to an SC cavity is achieved. The schematic diagram for this algorithm is shown in Fig. 1. Second, we can implement the case of 3 bit and 4 bit DQFT using the detailed steps that have been displayed above. For the 3 bit DQFT, we need to employ six gates from the type of p d ˆ i SW CZ , 14 gates from the type of Pβ at different phases, six gates from the type of Rˆ z θ at different θ, 14 gates from the ˆ ˆ type of Sθ at different θ, and 17 gates from the type of H. Also, for implementing the case of 4 bit DQFT, we need to p d employ 12 gates from the type of i S W CZ , 28 gates from ˆ the type of Pβ at different phases, 12 gates from the type ˆ of Rˆ z θ at different θ, 29 gates from the type of Sθ at differˆ ent θ, and 30 gates from the type of H. Therefore, extending the method for the 4 bit DQFT, the N-bit DQFT scheme is p d presented in Fig. 2. To do this, we need NN − 1i S W CZ gates, NN − 1Rˆ z θ gates at different θ, several two-qubit controlled phase gates at different phases and several different gates from the type of one-qubit by using the pulses for series of the external magnetic fields. Our physical scheme requires an N identical SQUID, where all two qubits are simultaneously interacting with a field, unless an error will be introduced. This is still a challenge to make superconductors simultaneously interact with a field and putting accurately the different switches for classical magnetic field in the experiment. Only two nearest-neighbor qubits interact simultaneously with a field during the interaction time t1 2π∕μ0 for implementing the two-qubit controlled
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Fig. 1. Quantum schematic circuit for implementing the 2 bit DQFT. This quantum circuit contains one- and two-qubit gates: here Pˆ is the two-qubit controlled phase gate at different phases. An experimental setup of a charge qubits-field system for implementing the 2 bit DQFT runs as follows: the superconductors are simultaneously interacting with the SC cavity mode, where the mode in the vacuum state. When the classical magnetic field is switched to ϕc 0, the interaction between the charge qubit and the field is switched off and results in one-qubit gates with different pulses of classical field at different positions on a two superconductors. Also, if the classical magnetic field is switched to ϕc 1∕2ϕ0 , the interaction between the charge qubits and the field is switched on and results in a two-qubit gate, where the interaction time is switched to t1 2π∕μ0 to realize p d the controlled phase gate at different phases and is switched to t2 π∕4μ to realize i S WCZ gate.
phase gate and during t2 π∕4μ for implementing the univerd sal gate square root of minus S W AP and controlled-Z gates. Here, we assume that the actual times for implementing the DQFT via superconductors in the cavity are t01 t1
4πΓ ; μ0
t02 t2
4πΓ ; μ
where we consider Γ to be due to the effect of the deviation in time caused by positioning the qubits at a precise location in the cavity mode. To elucidate the implementation of the N-bit DQFT, we calculate the gate average fidelity according to the following relation [29]:
F hΦin jUˆ †D ρˆ out Uˆ D jΦin i jhΦtjΦideal ij2 ;
(22)
where Uˆ D is the ideal DQFT operation, ρˆ out jΦtihΦtj, with jΦti being the final state after the DQFT, and jΦideal i Uˆ D jΦin i is the ideal target state. Assume that the p N qubits are initially in the general state jΦin i 1∕ 2N fjg1 ;g2 ;…;gN i ie−iϑ2 jg1 ;e2 ;…;gN iie−iϑ1 je1 ;g2 ;…;gN i iN e−iϑ1 ϑ2 …ϑN je1 ;e2 ;…;eN ig. Therefore, the F of the N-bit DQFT operation can be written as follows: F
1 2π
N Z 0
2π
Z dϑ1
0
2π
Z dϑ2 …
2π 0
dϑN jhΦtjΦin ij2 :
Fig. 2. Quantum circuit diagram and experimental setup for implementing the N-bit DQFT.
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1
1
(a)
(b) 0.9
0.8
0.8
0.7
0.7
Average Fidelity
Average Fidelity
0.9
0.6 0.5 0.4 0.3
0.5 0.4 0.3
0.2
0.2
2−bit 3−bit
0.1 0
0.6
2−bit 3−bit
0.1 0
0.05
0.1
0.15
0.2
′
0
0
0.05
Γ / µ t1
0.1
0.15
0.2
Γ / µ t2
Fig. 3. F for implementing 2 bit and 3 bit DQFT with the deviation amount Γ.
Now, we plot F with Γ, as shown in Fig. 3. We can see from Fig. 3(a) that, when Γ 0.0796μ0 t1 , the fidelity F ≈ 100% for the 2 bit and almost the same for the 3 bit case and from Fig. 3(b) that, when Γ 0.2μt2 , the fidelity F ≈ 89.9% for the 2 bit and almost the same for the 3 bit DQFT. We note that, the average fidelity for more than 2 bit DQFT whether three or more is almost the same for the 2 bit case. Therefore, our DQFT algorithm may be realizable for N-bit system. Looking for the experimental realization of the protocol we use the available experimental data as given in [30,31]. There it is given that the SC Cooper-pair box is made from aluminum film, where the gap energy is ∼50 GHz, 0.1 cm ≤ λ1 ≤ 15 cm, volume of the cavity v ∼ 10−2 λ31 , E C ∕h 37.38 GHz, E J ∕h 6.6 GHz, ω∕2π 38 GHz, ng 0.634233, jηj 1.778 × 10−17 J · sec ∕C, ϕ0 2.07 × 10−15 J · sec ∕C, πjηj∕ϕ0 2.7 × 10−2 , and area of SQUID 100 μm × 100 μm. Thus, we get E z ∕h 20.1 GHz, δ 0.35 GHz ≫ πE J ∕ϕ0 , πE J ∕ϕ0 δ 1.9 × 10−17 ≪ 1, πEJ jηj∕ℏϕ0 1.12 GHz, t1 0.64 ps, and t2 54.8 ps. Therefore, the total time to complete the procedure of DQFT is T total ≃ 112.78 ps for the 2 bit, T total ≃ 337.66 ps for the 3 bit, T total ≃ 675.32 ps for the 4 bit and T total ≃ 1.13 ns for the 5 bit, which is much less than the qubit relaxation time T 1 1.3 μs and the dephasing time T 2 5 ns. Hence, our algorithm may be a useful step toward implementing a more complicated quantum algorithm in the solid quantum computer.
4. CONCLUSIONS In our work, N-bit DQFT is achieved from the SC qubits inside an SC cavity, where some quantum logic gates for one and two-qubit are realized from this system to implement N-bit p d DQFT. A new gate called i S W CZ is defined and generated from two identical d.c. SQUIDs placed in the middle of a single-mode SC cavity. This gate is used to implement the N-bit DQFT. The DQFT represents a key element for some quantum algorithms. Schematic representation for experimental setting up for the 2 bit and N-bit are exhibited. Therefore, such a scheme may be a useful step for implementing more complicated quantum algorithms in a solid quantum computer. Moreover, an experimental setup is proposed for implementing the N-bit DQFT for development of a quantum computer by using SC qubits inside an SC cavity. It is estimated that the protocol is realizable within nowadays experimental availability.
ACKNOWLEDGMENT The authors wish to thank the referees for their valuable comments that improved the presentation in many aspects for this manuscript.
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