Simple Scheme for Realizing the General Conditional Phase Shift Gate and a Simulation of Quantum Fourier Transform in Circuit QED

Commun. Theor. Phys. 56 (011 35 39 Vol. 56, No. 3, September 15, 011 Simple Scheme for Realizing the General Conditional Phase Shift Gate and a Simulation of Quantum Fourier Transform in Circuit QED WU Chun-Wang ( Ë, HAN Yang (ô, DENG Zhi-Jiao ( ã, LI Hong-Yi (Ó, CHEN Ping-Xing (í», and LI Cheng-Zu (Ó Ý College of Science, National University of Defense Technology, Changsha 10073, China (Received March 18, 011; revised manuscript received May 0, 011 Abstract We propose a theoretical scheme for realizing the general conditional phase shift gate of charge qubits situated in a high-q superconducting transmission line resonator. The phase shifting angle can be tuned from 0 to π by simply adjusting the qubit-resonator detuning and the interaction time. Based on this gate proposal, we give a detailed procedure to implement the three-qubit quantum Fourier transform with circuit quantum electrodynamics (QED. A careful analysis of the decoherence sources shows that the algorithm can be achieved with a high fidelity using current circuit QED techniques. PACS numbers: 03.67.Lx, 7.50.+r,.50.Pq Key words: circuit QED, conditional phase shift gate, quantum Fourier transform 1 Introduction The quantum Fourier transform (QFT is an efficient quantum algorithm for performing a Fourier transform of quantum mechanical amplitudes, which lies at the heart of quantum factoring [1 and many other interesting quantum algorithms. [ 3 The QFT for N qubits requires N one-qubit Hadamard gates and N(N 1/ two-qubit conditional phase shift gates. [ Up to now, a small-scale QFT has been demonstrated with ensembles of molecules using nuclear magnetic resonance, [5 7 but none of these systems are scalable to a large number of qubits. Scully et al. [8 proposed a scalable scheme to implement QFT in cavity QED, which relies on the passage of a series of suitably chosen atoms through a sequence of classical fields and high-q cavities. However, such a proposal is still challenging using current cavity QED techniques. In the pursuit of a scalable architecture for quantum information processing, Josephson-junction-based superconducting circuits are currently a leading candidate. [9 In 00, Blais et al. suggested coupling superconducting charge qubits with a high-q transmission line resonator (TLR playing the role of a data bus. [10 This architecture, called circuit QED, has been proved to be the most experimentally advanced architecture for coupling superconducting qubits in the long-range case. The coherent transfer of quantum states between two charge qubits has been realized via virtual exchange of photons [11 and sideband transitions, [1 respectively. By adiabatically using the avoided crossing between computational and noncomputational states, Dicarlo et al. [13 implemented a quantum phase gate and demonstrated the two-qubit Grover search and Deutsch Jozsa quantum algorithms. In this paper, we focus on a scheme for the QFT with circuit QED architecture. It is certain that we can construct the algorithm using one-qubit rotations and twoqubit gates implemented in Refs. [11 13 (e.g., a general conditional phase shift gate can be decomposed into two quantum phase gates and four one-qubit gates [1, but such procedures consist of a large number of quantum gates and consume a long operation time. To obtain an efficient procedure, a physical mechanism, which can produce the general conditional phase shift directly, is desirable. Here, we propose a simple scheme for the general conditional phase shift gate between two charge qubits capacitively coupled to a high-q TLR. The distinct advantage of our scheme is that the phase shifting angle can be tuned from 0 to π by simply adjusting several experimental parameters. Based on this gate proposal, we present a detailed procedure for three-qubit QFT and analyze the experimental feasibility. Compared with the conventional gate-decomposition protocols, we succeed in decreasing the number of quantum gates and shortening the computational time. Our procedure for QFT could in principle be extended to the N-qubit case. Two-Qubit General Conditional Phase Shift Gate For the implementation of QFT, the main ingredient is the two-qubit conditional phase shift gate. This gate can be defined through the following input-output relation: ε 1 ε e iε1εη ε 1 ε, (1 Supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant No. 005 and by the Program for New Century Excellent Talents of China under Grant No. 06-090 Corresponding author, E-mail: cwwu@nudt.edu.cn c 011 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

36 Communications in Theoretical Physics Vol. 56 where ε 1 and ε stand for the basis states 0 or 1 of the qubits, and η is the phase shifting angle. In this section, we focus on how to implement a conditional phase shift gate with circuit QED architecture. Fig. 1 Electrical circuit representation of the two-qubit circuit QED device with local controls. Two CPBs are fabricated at the antinodes of the electric field of the full-wave resonator mode. A dc voltage U applied to the gate capacitance C g and a dc current I φ applied to a coil producing a flux Φ in the circuit loop tune the quantum energy levels. Microwave pulses u(t applied to the gate serve for driving the qubit transitions. The system we consider comprises two Cooper pair boxes (CPB strongly coupled to a high-q TLR, and flux bias lines proximal to each qubit (see Fig. 1. For an ideal one-dimensional TLR with the geometric length L and the inductance (capacitance per unit length l(c, the full-wave resonance frequency is w r = 1/(L lc. Introducing the annihilation (creation operator a(a, the resonator can be described by the Hamiltonian H r = w r a a. A CPB consists of one superconducting island connected by two identical Josephson junctions to a segment of a superconducting ring. This apparatus has a number of in situ tunable parameters including the Josephson energy E J and the electrostatic energy E el. A dc current I φ applied to a coil producing a flux Φ in the circuit loop tunes E J. Through the gate capacitance C g, a bias voltage V g = U + u(t is applied to the island, where the dc component U tunes E el by inducing offset charges, and the ac one u(t, as a classical microwave pulse, serves for driving the qubit transitions. In our proposal, the boxes are biased at charge degeneracy (E el = 0, where their energy levels are first-order insensitive to dephasing from offset charge noise. At this working point, the CPB is also sufficiently nonharmonic to be treated as a pseudospin- 1/ particle, whose Zeeman energy w q equals E J. By fabricating CPBs at the antinodes of the electric field of the full-wave mode, strong capacitive coupling between the qubits and photons in the resonator can be induced by the electric dipole interaction. In the rotating frame, the coupled system can be described by the Hamiltonian (assuming = 1, H JC = g i (a σ,i e i it + aσ +,i e i it, ( i=1, where σ + and σ are the spin-flip operators up and down, g is the qubit-resonator coupling strength, = w r w q, and the i subscript is used to distinguish the different qubits and their parameters. Microwave pulses applied to the gates can induce Rabi oscillations of the qubits, then the total Hamiltonian of the system can be written as H = [g i (a σ,i e i it + aσ +,i e i it + Ω i σ x,i, (3 i=1, where Ω i is the Rabi frequency for qubit i and σ x,i = σ +,i + σ,i. Hereafter, we set 1 = = and Ω 1 = Ω = Ω. The Hamiltonian in the form of Eq. (3 has been investigated thoroughly in Ref. [15. It was demonstrated that, in the strong driving regime Ω {, g 1, g }, Hamiltonian H can be approximated by the effective expression H eff = [ gi σ x,i(a e i t + a e i t + Ωσ x,i. ( i=1, To verify the validity of this approximation, we do a numerical calculation to compare the systemic evolutions governed by H and H eff. The parameters we use are g 1 = g = = π 50 MHz, Ω = π 500 MHz, and the initial state is (1/ vac ( 0 1 0 + 0 1 1 + 1 1 0 + 1 1 1 1, where vac is the vacuum state of the resonator. The fidelity between the approximate state and the exact state is plotted as a function of time in Fig.. It is shown that, during one gate operation time ( 0 ns as calculated in this paper later, the deviation of the approximate state is smaller than 1%. Fig. Numerical results for the fidelity between the approximate state and the exact state as a function of time. The parameters we use are g 1 = g = = π 50 MHz, Ω = π 500 MHz, and the initial state is (1/ vac ( 0 1 0 + 0 1 1 + 1 1 0 + 1 1 1 1. Using the Wei Norman algebraic method, [16 the evolution operator of the system governed by H eff can be written in a factorized way as U(t = e iωt(σx,1+σx, e ia(tσx,1σx, e ic(t ( ( e ibi(taσx,i e ib i (ta σ x,i. (5 i=1, i=1,

No. 3 Communications in Theoretical Physics 37 Solving the Schrödinger equation i U(t = H eff U(t for the initial condition A(t = B i (t = C(t = 0, we obtain A(t = g g [ 1 1 i (ei t 1 t, B i (t = g i i (1 e i t, C(t = g 1 + g [ 1 i (ei t 1 t. (6 Note that B i (t is a periodic function of time and it vanishes at t n = nπ/ for integer n. To realize the twoqubit conditional phase shift gate with a shifting angle η, appropriate qubit-resonator detuning η, interaction time t η and Rabi frequency Ω η can be chosen to satisfy g1 g π η = π + η, t η = π η, Ω η t η = π η + kπ, where k is an integer. Substituting these parameters into Eq. (6, we have [ U(t η = exp i η + kπ (σ x,1 + σ x, [ exp i + η x,1 σ x, exp[ig η, (7 σ where the phase factor G η = g 1 + g 8g 1 g (π + η. It can be easily shown that U(t η + 1 + = exp i G η π η exp[iη + 1 +, U(t η + 1 = exp i G η π η + 1, U(t η 1 + = exp i G η π η 1 +, U(t η 1 = exp i G η π η 1, (8 where ± i = ( 0 i ± 1 i / are eigenstates of operator σ x,i with eigenvalues ±1. Then up to a global phase factor G η π/ η/, we obtain a general conditional phase shift gate of two charge qubits in the computational basis { +, }. The phase shifting angle can be tuned from 0 to π by adjusting several tunable experimental parameters. Moreover, this coupling mechanism works at a small detuning ( g, which leads to a faster gate rate than the ones involving a dispersive regime. [11 13 3 Simulation of Quantum Fourier Transform The QFT for N qubits is a unitary transformation in N dimensions. It is defined relative to the orthonormal basis 0,..., N 1 by p QFT 1 N 1 e πipq/n q. (9 N q=0 In Ref. [, Coppersmith proposed an efficient quantum algorithm for N-qubit QFT, which is composed of N onequbit Hadamard gates and N(N 1/ two-qubit conditional phase shift gates. Let S i denote the Hadamard gate acting on the state of qubit i and D ij (η denote the conditional phase shift of η performing on the states of qubits i and j. The quantum circuit for QFT with three qubits is depicted in Fig. 3. In this circuit, the horizontal lines represent the qubits and the boxes represent the quantum logic gates. The circuit is to be read from right to left, so the total operation corresponding to the network is S D 1 S 1 D 0 D 10 S 0. (10 Fig. 3 The quantum circuit for three-qubit QFT. Here, S i is the Hadamard gate acting on qubit i and D ij(η is the two-qubit conditional phase shift gate acting on qubits i and j. In the following, we concentrate on how to perform the three-qubit QFT algorithm with circuit QED architecture. Since the CPB dimension (several micrometer is much smaller than the wavelength of the resonator (centimeter, multiple qubits can be fabricated in the space between the center conductor and the ground planes of TLR. Here, we place three CPBs at the two ends and the central position of the resonator. In this case, all the qubits are located at the antinodes of the electric field of the full-wave resonator mode, so the strong electric dipole interaction between the qubits and TLR can be induced. Note that the conditional phase shift gate U ij (η we implemented in Sec. is defined relative to the eigenbasis { +, }. To obtain the operator D ij (η in sequence (10, several single-qubit rotations of qubits i and j are needed in addition to U ij (η. It is easy to find that D ij (η = M i M j U ij (ηm i M j, (11 where M transforms each qubit as 0 1 ( 0 + 1, M : 1 1 ( 0 + 1. (1 Substituting Eq. (11 into Eq. (10 and simplifying the sequence by removing sequential rotations, which cancel each other out, the final transformation performing the whole QFT algorithm writes O M 1M 0 U 1 U 0 S 1 U 10 M M 1 O 0, (13

38 Communications in Theoretical Physics Vol. 56 where the single-qubit rotation O i = M i S i. We assume that any required initial state for the algorithm can be prepared with close to unit fidelity in experiment. Single-qubit rotations are implemented by pulses of microwave produced by the gate driving lines. Following the results of Ref. [10, different drive frequencies can be chosen to realize rotations around arbitrary axes in the x z plane, and the rotation angle can be changed easily via varying the microwave pulse length. These rotations around x axis and z axis are sufficient to realize any single-qubit logical gate. Since each qubit has its own gate driving line, parallel operation is allowed for our experimental procedure. To implement the two-qubit gate U ij (η in sequence (13, we turn on the interaction H eff by biasing the external magnetic fluxes Φ i to some appropriate values and adding the strong microwave fields at the same time. After a time duration t η = π/ η, H eff is turned off by adjusting Φ i and removing the driving fields. Our procedure is robust to the device parameter variations, which are unavoidable in solid-state systems. The local magnetic fluxes Φ i can be tuned to compensate the difference of the qubit transition frequencies. The homogeneous qubit-resonator coupling strength is not required in our procedure. Therefore, the inevitable nonuniformity in device parameters is tolerable and the nonexact placement of CPBs is allowed. Let us now discuss the practical feasibility of this experimental proposal. This discussion is based on the parameters of the current circuit QED techniques. Coupling strength g/π = 100 MHz has been realized experimentally in Ref. [17. With conservative parameters g i = π 50 MHz (i = 0, 1,, the required time for implementing U ij (η is t η = π/ η 0 ns. To satisfy the strong driving condition of our model, we choose the Rabi frequency to be Ω = 10g i = π 500 MHz. Then we can approximately evaluate the time of one single-qubit rotation as t rot π/ω = 1 ns. Because the sequential singlequbit rotations acting on different qubits can be implemented simultaneously, the total operation time of threequbit QFT algorithm is t tot = 3t η + 3t rot 63 ns. In our procedure, the main decoherence processes are the cavity damping and dephasing of the charge qubits. Relaxation and dephasing of a CPB were measured in Ref. [18. There, γ 1 /π = 0.0 MHz and γ φ /π = 0.31 MHz were reported, where γ 1 and γ φ are the qubit relaxation rate and pure dephasing rate, respectively. A modified version of CPB proposed by Koch et al., [19 called transmon, can be used to reduce the qubit dephasing rate greatly, but the infidelity introduced by its small anharmonicity must be analyzed carefully. The dissipation of the TLR that occurs through coupling to the external leads can be described by the photon decay rate κ = w r /Q, where Q is the quality factor. Moderate-Q resonators with κ π MHz were used in earlier experiments. [1 13 A high-q resonator is advantageous for the coherent quantum operations, but it is adverse to the fast dispersive measurement. Recent progress shows that this problem can be solved by using either two cavities [0 or two modes of a cavity with different quality factors. [1 The high-q resonator modes with κ π 0.1 MHz were used for photon storage and quantum operations in Ref. [0 1, and the photon lifetime can be calculated as 1/κ 1.6 µs. The preceding rough estimates indicate that the decoherence processes occur much slower than the implementation of the three-qubit QFT algorithm. To check the validity of our proposal more strictly, we numerically simulate the time evolution of the system for performing QFT on the initial state 000, of which the ideal target state is 7 m=0 (1/ m. The fixed parameters we choose are g = π 50 MHz, Ω η = (π/ η/+0π/t η, γ 1 = π 0.0 MHz, and the Hamiltonian we use is the exact expression Eq. (3. As shown in Fig., the fidelity for performing QFT on 000 decreases greatly with the increase of the qubit dephasing rate γ φ. And for the same qubit parameters, a high-q resonator can implement the algorithm with a fidelity about 6% higher than the moderate-q case. Using the qubit parameters in Ref. [18 and the resonator parameters in Refs. [0 1, we can obtain the desired state with a fidelity higher than 95%. Fig. Numerical results for the fidelity of performing the QFT on the initial state 000 as a function of the dephasing rate γ φ, of which the ideal target state is 7 m=0 (1/ m. The solid and dashdotted curves represent the cases of high-q (κ = π 0.1 MHz and moderate-q (κ = π MHz resonators, respectively. In conclusion, we have proposed a potential scheme for realizing the general conditional phase shift gate of charge qubits coupled to a high-q TLR. The phase shifting angle can be tuned from 0 to π by simply adjusting several experimental parameters. Based on this gate proposal, a detailed procedure for three-qubit QFT was presented. This procedure is robust to the device parameter variations and could be extended to the N-qubit case. By analyzing the main decoherence sources, we have shown that our proposal could be implemented with a high fidelity using current circuit QED techniques.

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