Efficient scheme for one-way quantum computing in thermal cavities
aa r X i v : . [ qu a n t - ph ] A p r Efficient scheme for one-way quantum computing in thermal cavities
Wen-Xing Yang
1, 2 and Zhe-Xuan Gong ∗ Department of Physics, Southeast University, Nanjing 210096, China State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China (Dated: October 31, 2018)We propose a practical scheme for one-way quantum computing based on efficient generation of2D cluster state in thermal cavities. We achieve a controlled-phase gate that is neither sensitive tocavity decay nor to thermal field by adding a strong classical field to the two-level atoms. We showthat a 2D cluster state can be generated directly by making every two atoms collide in an arrayof cavities, with numerically calculated parameters and appropriate operation sequence that can beeasily achieved in practical Cavity QED experiments. Based on a generated cluster state in Box (4) configuration, we then implement Grover’s search algorithm for four database elements in a verysimple way as an example of one-way quantum computing.
PACS numbers: 03.67.Lx, 03.65.Ud, 42.50.Vk
Over the past few years, the construction of a practical quantum computer has become a challenging goal forexperimentalists. It is well known that the building blocks of a general quantum computer are single-qubit rotationsand two-qubit quantum gates [1]. Recently, Briegel and Raussendorf [2, 3] proposed a new idea for constructingquantum computer, known as one-way quantum computing, which shows that preparation of a particular entangledstate, called cluster state, accompanied with local single qubit measurements, is sufficient for simulating any arbitraryquantum logic operations. Cluster state as a universal resource for general quantum computing has drawn extensiveresearch interests [4]-[8]. Moreover, one way quantum computing by optical elements based on four-qubit clusterstates was recently demonstrated experimentally [9]. It is hoped that experimental difficulties in performing complexquantum gates may be overcome by one-way quantum computing based on the generation of cluster state.A cluster state | ψ i c can be visualized as a collection of qubits positioned at certain sites of a 2D lattice structurewith lines connecting them, which can be specified by the following set of eigenvalue equations: K ( a ) | ψ i c = ( − κ a | ψ i c (1)with the correlation operators K ( a ) = σ ( a ) x ⊗ b ∈ nghb ( a ) σ ( b ) z (2)where nghb(a) is set of all the neighbors of any site a of the lattice, and κ a ∈ { , } . To generate an arbitrary clusterstate, one can first initialize each qubit in state | + i = ( | i + | i ) / √
2, where | i and | i are the computational basis,and then perform controlled-phase operations between all neighboring qubits connected by the lines of the lattice.Cavity QED system is known to be a qualified candidate for quantum information processing [10]. However, up tonow, one-way quantum computing based on Cavity QED techniques has neither been proposed theoretically nor beencarried out experimentally. The main difficulty lies in generating an arbitrary 2D cluster state. Proposed schemesof generating cluster states using Cavity QED methods [11]-[14] are difficult for practical and scalable experimentseither due to the decoherence of the cavity field mode or due to the sensitivity of thermal field. Besides, most ofthe schemes are mainly focused on linear cluster state prepared in one dimension, which are not suitable for use assubstrate for quantum computation since one-way quantum computing based on 1D cluster state can be efficientlysimulated by classical computer, [15, 16] and most proposals for generating 2D cluster state are inefficient, as theyfirst need to generate several 1D cluster states and then collide them into a 2D configuration.In the present work, we propose a practical scheme for one-way quantum computing based on efficient generationof 2D cluster state in thermal cavities. Compared to Ref.[11]-[14], our scheme is neither sensitive to cavity decay norto thermal field, since the evolution of the atomic states is independent of the cavity field mode, which is achieved byadding a strong classical field to our system. In addition, our implementation of controlled-phase gate does not needany auxiliary state, i.e. two-level atoms are used instead of three-level atoms, which further reduces experimental ∗ Electronic address: [email protected] difficulties. On the other hand, the 2D cluster state is generated in a direct and efficient manner in our scheme byappropriately choosing the initial velocity of each atoms as well as the time delay between atom preparations andplacing an array of cavities at certain locations in the path atoms passing through, so that every two atoms can collidein a certain cavity and be subjected to entanglement generation as in Ref.[17]. We give the generation of arbitrary4-qubit cluster state as an example with reasonable parameters and concrete operation sequence, and show that ourscheme can perform one-way quantum computing process such as Grover’s search algorithm for four database elementsin a simple and convenient way that is within the current experimental techniques.Our generation of entanglement in cluster state is based on the interaction between two identical two-level atomsand a single-mode cavity field driven by a classical field. In the rotating wave approximation, the Hamiltonian forsuch a system is given by (assuming ~ = 1) [18] H = P j =1 ω σ z,j + ω a a † a +
12 2 P j =1 [ g ( a † σ − j + aσ + j )+Ω( σ + j e − iωt + σ − j e iωt )] (3)where a and a † are the annihilation and creating operators for cavity mode, σ z,j = | e i jj h e | − | g i jj h g | , σ + j = | e i jj h g | , σ − j = | g i jj h e | , with | e i j ( | g i j ) being the excited (ground) state of the j th atom. ω , ω a and ω are the frequencies foratomic transition, cavity mode, and classical field respectively. g is the atom-cavity coupling strength and Ω is theRabi frequency of the classical field. Assume that ω = ω . Then we can obtain the following interaction Hamiltonianin the interaction picture: H i = X j =1 [ Ω2 ( σ + j + σ − j ) + g e − iδt a † σ − j + e iδt aσ + j )] , (4)with δ = ω − ω a . For the new atomic basis |±i j = ( | g i j ± | e i j ) / √
2, then we make a further transformation withrotation with respect to the terms regarding Ω in Eq.(4), and obtain H I = g P j =1 ( | + i jj h + | − |−i jj h−| + | + i jj h−| e i Ω t −|−i jj h + | e − i Ω t ) e − iδt a † + H.c. (5)Free Hamiltonian H = Ω2 2 P j =1 ( | + i jj h + | − |−i jj h−| ) has been used here for the transformation. Assuming thatΩ ≫ δ, g , we can neglect the fast oscillating terms. Then obtain the effective interaction Hamiltonian, H e = g e − iδt a † + e iδt a ) σ x (6)where σ x =
12 2 P j =1 ( σ + j + σ − j ). The evolution operator for Hamiltonian (6), which was first proposed for trapped-ionsystem [19], can be written as U e ( t ) = e − iA ( t ) σ x e − iB ( t ) σ x a e − iC ( t ) σ x a † (7)By solving the Schr¨odinger equation idU e ( t ) /dt = H i U e ( t ), we can obtain B ( t ) = g ( e iδt − / iδ , C ( t ) = − g ( e − iδt − / iδ , A ( t ) = g [ t + ( e − iδt − /iδ ] / δ . Choosing δt = 2 π , we have B ( t ) = C ( t ) = 0. Thus we get the evolutionoperator of the system, U ( t ) = e − iH t U e ( t ) = e − i Ω tσ x − iλtσ x (8)where λ = g / δ . If we first apply single-qubit Hadamard gate on both atoms, and then set the interacting time t and Rabi frequency Ω appropriately so that λt = π/ t = (2 k + ) π ( k is integer), followed by implementingagain the Hadamard gate on both atoms, then we obtain a controlled quantum phase gate with computational basis | i , | i represented by | e i , | g i : H ⊗ U ( t ) H ⊗ | g i | g i = −| g i | g i H ⊗ U ( t ) H ⊗ | g i | e i = | g i | e i H ⊗ U ( t ) H ⊗ | e i | g i = | e i | g i H ⊗ U ( t ) H ⊗ | e i | e i = | e i | e i (9)In order to generate an arbitrary two-dimensional cluster state using the controlled quantum phase gate given inEq.(9), we first assume that the horizontal velocity v i and time t i of the i th atom emitting from the single-atom sourcehas been pre-selected according to our need, and the vertical position of each atom are slightly different. After theatoms initially in ground state move out horizontally from the source, a π/ | i + | i ) / √
2. Our next aim is to let every two atoms collide ina certain cavity and so that they may undergo the evolution in Eq.(8), and an arbitrary 2D cluster state can thusbe formed. For N atoms forming a N-qubit cluster state, calculations show that this requires us to place (2 N − k th cavity center is at a distance L k from the single-atomsource, the following equations should be satisfied in order to achieve collisions between every two atoms. L i + j − ( 1 v j − v i ) = t i − t j (10)where i = 1 , , ..., N − j = i + 1 , i + 2 , ..., N and we set t = 0 for simplicity. Eq.(10) contains N ( N − / N −
1) variables. Calculations show that it has a group of solutions for N ≤ N ≥
7. For pedagogical reasons, we are aimed at the relatively simple but important case of N = 4,where one can get many reasonable solutions that satisfy Eq.(10). Figure 1(b) shows a possible case qualitatively. Theanalytical solution we obtained is rather complicated, but our numerical calculation gives many quantitative solutionsthat are appropriate in practical experiments. Table I shows one example. TABLE I: One numerical solution for Eq.(10) in the case of N = 4. Part of the quantities are set as constants and others arevariables being solved. v v v v t t t t L L L L L Since we only want the atom pairs representing neighboring qubits in the cluster state to undergo the controlled-phase operation in Eq.(8), in all the other case, the cavity is set far off resonance by a large electric field applied acrossthe cavity mirrors. This field stark-shifts the atomic levels far off resonance, so that the atom-cavity interaction isthen negligible [20]. The interaction time in Eq.(8) is also controlled strictly in the same way together with controlof the length of the strong classical pulse added. In fact, small difference of the interaction time between two atomswill not cause measurable errors. [18, 21]Now we’d like to show how to implement one-way computing based on the generation of cluster state in our scheme.Here we use the Grover’s search algorithm for four unsorted database elements as an example, the circuit model [22]and one-way computing model [9] of which have been shown in Figure 2. The cluster state we need to generate is inBox (4) configuration ( | ψ i c = | i | + i | i | + i + | i |−i | i |−i + | i |−i | i | + i + | i | + i | i |−i ), which entailsus to tune the cavity field to near resonance when atom pairs (4,3), (4,1), (3,2), (2,1) collide in Cavity 1, 3 and 5 attime t , t , t , t as in Figure 1(b), while { v , v , v , v } , { t , t , t , t } and { L , L , L , L , L } are pre-set to valuessatisfying Eq.(10), such as in Table I.After t , the cluster state has been generated and we are then to perform single-qubit measurements in the orderof atom 4-3-2-1. Atom 4 and atom 3 should be measured in the basis ( | i ± e iα | i ) / √ | i ± e iβ | i ) / √
2, whichcan be achieved simply by adding π/ ω r satisfying ( ω r − ω ) T = α ( β ) respectively,( ω is the atomic transition frequency and T is the coherent interaction time between the pulse and atom) and then Cavity ArrayTime Position
Cavity 1 Atom 4Atom 3Atom 2Atom1 t t t t L L L L L Cavity 2 Cavity 3 Cavity 4 Cavity 5
AtomSource
RR RR R * Detector (b)(a) R t t t t t t t' t' t' t' FIG. 1: (a)
Proposed experimental set-up for one-way quantum computing in an array of thermal cavities. R represents forRamsey zone with π/ ∗ denotes for Ramsey zone with detuned π/ (b) Space-time diagram for the sequence of events. Every two atoms are made to cross the center of certain cavitysimultaneously before finally reaching the detector.
Z ZHH ZZ HH
Measurement in basis
Output ( ) z R (cid:68)(cid:16) ( ) z R (cid:69)(cid:16) ( 0 1 ) / 2 i e (cid:68) (cid:114) Measurement in basis ( 0 1 ) / 2 i e (cid:69) (cid:114) ( 0 1 ) / 2 (cid:14) Atom 4Atom 3 Atom 1Atom 2 Read outRead out ( 0 1 ) / 2 (cid:14) (a)(b)
FIG. 2: (a)
Circuit model of Grover’s search algorithm for four database elements. The two encoded qubits pass through aseries of unitary quantum logic gates before being measured in computational basis as output of the search algorithm. α, β aredetermined by the ”Oracle”, and setting αβ to ππ , π π ,00 corresponds to the marked element encoded as 00,01,10,11. (b) Cluster state quantum computing model for the same algorithm. The physical qubits carried by Atom 4,3 are measured in anyorder, changing the states of Atom 2, 1 to states of the encoded qubits in the circuit model at the same stage. measuring the atom in basis {| i , | i} through Field Ionization Detector. Atom 2 and Atom 1 carry the read-outqubits in the cluster state model and should go through a σ z and a Hadamard gate, which can be realized by addingthe same π/ ω r satisfying ( ω r − ω ) T = π , before final measurements by the Detectorin computational basis. If the measurement results of the four flying atom at time sequence t ′ , t ′ , t ′ , t ′ are denotedby r , r , r , r , then the one-way quantum computing process succeeds in giving us the search result encoded as { r ⊕ r , r ⊕ r } if r and r are not both zero [9]. ∆ t - - Ρ FIG. 3: Coherent evolution of two atoms in a thermal cavity according to Eq.(8), with the parameters δ = g , Ω = 5 δ and n th = 1. The two atoms are assumed to be initially in the ground states | g i | g i . The red, blue, green and purple curversrepresent for ρ gg,gg , ρ ee,ee , Re( ρ gg,ee ) and Im( ρ ee,ee ) respectively. The small deviation from ideal situation on the curves aresmall oscillations with a high frequency. We now discuss further the experimental feasibility of our one-way quantum computing scheme. From Eq.(8), wenote that the photon-number-dependent parts in the evolution operator are canceled with the strong resonant classicalfield added, thus our scheme can be done in thermal cavities and is also insensitive to cavity decay, greatly reducingexperimental difficulties. The robustness of our scheme to thermal field is illustrated in Figure 3, where we simulatethe time evolution curves of the two atoms in the cavity by solving an appropriate master equation and allowing forheating in form of quantum jumps described by jump operators √ Γ n th a and p Γ( n th + 1) a † (Γ and n th are the typicalheating rate and the mean excitation of the photon). The Figure clearly shows that we have a coherent evolution ofthe atomic state which is not entangled with the cavity. Besides, in obtaining Eq.(8), there is no requirement thatthe atom-cavity detuning should be much larger than the atom-cavity coupling strength. Operation time can be thusshortened, which is also important in view of decoherence.Moreover, the velocities of the atoms can be selected by Doppler-selective optical pumping techniques, with aprecision of ± m/s , and the timing of each atom preparation can be controlled to a precision of 2 µs [10]. Besides,the Cavity QED experimental apparatus are typically 20cm in length [10]. These validates our choice of v i , t i and L i . It is also worth mentioning that since the waist of the cavity field is at most a few mm, which is much smallerthan the distance between the cavities, it is unlikely that our space-time arrangement of atoms will cause more thantwo atoms simultaneously crossing the center of the cavity field.The radiative time for Rydberg atoms with principal quantum numbers around 50 is T r = 3 × − s , and thecoupling constant is g = 2 π × T c = 1ms [10].In the present scheme, the virtually excited cavities have only a small probability, about 1%, of being excited duringthe passage of the atom pairs through them. Thus the efficient decay time of the cavity T ceff ∼ . δ = g ,direct calculation shows that the interaction time is on the order of 10 − s, which is much shorter than the cavitydecay time T ceff . The time needed for the whole one-way quantum computing process can be controlled within a fewms according to the choice of t i , apparently smaller than the the radiative time of the Rydberg atoms T r , renderingour scheme insensitive to the decoherence of atoms.Errors in our scheme will mainly come from the process of entanglement generation in cavities and the single-qubitmeasurement process after generating the cluster state. In the former case, the errors can be induced by Start shift onthe states | + i j and |−i j as we’ve discarded the fast-oscillating terms in obtaining Eq.(6), and by pulse imperfectionsand initial cavity Fock state. Ref.[23] shows that these errors will only have slight influence on the fidelity of theentangled state obtained. In the latter case, we note that the measurement efficiency of ionization detectors can bemore than 80% with current techniques [24]. However, we’re still expecting higher measurement efficiency in CavityQED experiments.For one-way computing of quantum algorithms that need no more than 6 physical qubits, our scheme does not needmuch change. However, for a large number of qubits, Eq.(10) cannot be satisfied by simply setting { v i } , { t i } , { L i } before the computing process. One of the possible ways to solve this problem is to change the velocity of the atomsin the midway. Although Rydberg atoms are neutral and usually not easy to accelearte/deccelerate, we note thatrecent progress on Cavity QED experiment with optically transported atoms can manipulate the motion of atoms byresorting the optical dipole force [25, 26]. 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