Experimental Realization of Brüschweiler s exponentially fast search algorithm in a 3-qubit homo-nuclear system

Transcription

1 Experimental Realization of Brüschweiler s exponentially fast search algorithm in a 3-qubit homo-nuclear system Li Xiao 1,2,G.L.Long 1,2,3,4, Hai-Yang Yan 1,2, Yang Sun 5,1,2 1 Department of Physics, Tsinghua University, Beijing , P R China 2 Key Laboratory For Quantum Information and Measurements, Beijing , P R China 3 Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing , P R China 4 Center for Atomic, Molecular and NanoSciences, Tsinghua University, Beijing , P R China 5 Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, U.S.A. Searching marked items from an unsorted database is a very important problem and has many practical applications. In a classical computer, this is done by exhaustive search and requires O(N) steps. In a quantum computer, Grover s quantum search algorithm achieves quadratic speedup over classical algorithms. The Brüschweiler algorithm combines the ideas of DNA computing with that of quantum computing. It achieves an exponential speedup. In this paper, we report the experimental realization of the algorithm in a 3 qubit NMR ensemble system. The pulse sequences are given for the algorithms. In particular, we have modified the measurement part of Brüschweiler s original paper. Instead of measuring quantitatively the spin projection of the aucilla bit, we measure the shape of the aucilla bit. By just looking at the downwardness or upwardness of the corresponding peaks in the aucilla bit spectrum, we can read out the bit value of the marked state. The geometric nature of this read-out makes the algorithm robust against errors. I. INTRODUCTION Quantum algorithms are important in quantum computing. Deutsch and Josza s algorithm demonstrated the advantage of quantum computing [1]. Two more quantum algorithms which are closely related to practical applications of quantum computation are: Shor s factoring algorithm [2] and Grover s quantum search algorithm [3]. The factorization of a large number into prime factors is a difficult mathematical problem. Existing classical algorithms require exponential times to complete the factorization in terms of the input. In Shor s algorithm, this becomes polynomial in the digit of the number to be factorized. Grover s quantum search algorithm deals with unsorted database search. Classically it can only be done by exhaustive searching. Many scientific and practical problems can be reduced to an unsorted database search problem. Hence it is very important. Unlike Shor s algorithm, Grover s algorithm achieves only quadratic speedup over classical algorithms, namely the number of searching is reduced from O(N) too( N). However, it has been proven that Grover s algorithm is optimal for quantum computing, and one can not search faster [4]. The strong restriction of the optimality theorem can be broken off so that exponential speedup can be achieved if we go out of quantum computation. Using nonlinear quantum mechanics, Abram and Lloyd [6] have constructed a quantum algorithm that achieves exponential speedup. However the applicability of nonlinear quantum mechanics is still under investigation, and the realization of this algorithm is not known. Brüschweiler put forward a hybrid quantum search algorithm that combines DNA computing idea with the quantum computing idea using multiple-quantum operator algebra [5]. The new algorithm achieves an exponential speedup in searching an item from an unsorted database. It requires the same amount of resources as effective pure state quantum computing. This algorithm is particularly suitable for the NMR system where ensembles of quantum nuclear spin system are used. There are several known schemes for quantum computers, such as cooled ions [7], cavity QED [8], nuclear magnetic resonance [9] and so on. NMR technique is sophisticated and many experimental realizations of quantum algorithms have been realized using NMR system [10 16]. Comparing with effective-pure state quantum computation, Brüschweiler algorithm enjoys the freedom from the debate about the quantum nature of the NMR quantum computation using effective-pure state, a subject that is being argued [17,18]. Because the algorithm is exponentially fast, it tales much shorter time to finish a search problem and this makes the algorithm more robust again errors and decoherence. In this paper, we report the experimental realization of this algorithm in a 3-qubit homo-nuclear system. In our realization we have modified the measurement procedure in Brüschweiler s original algorithm. Instead of measuring the aucilla bit s spin projection, we measure the aucilla bit s spectrum. By looking at the downwardness or upwardness of the corresponding peaks in the spectrum, the bit value of the marked state can be read out. Since the shape of the spectrum is a geometric property, it is easy to recognize. This makes the algorithm more tolerant to errors. The paper is organized as follows. After this introduction, we briefly describes Brüschweiler algorithm in Section II. In section III, we give the modification of the read-out. In Section IV, we give the details of the pulse sequences of the algorithm and the results of the experiment. In section V, we give a summary. 1

2 II. THE BRÜSCHWEILER S ALGORITHM As is well-known, the preparation of the effective pure state is one of the most troublesome part in a NMR quantum computing experiment. The effective pure state also sets a restriction on the number of qubits [19,20] for use in quantum computation. The effective pure state is represented by the density operator At room temperature, under the high temperature approximation we have ρ =(1 ε)2 n 1 + ε (1) ε = nhv 2 n kt. (2) In eq. (1), the first part has no contribution to the final outcome and the second part s contribution to the output is scaled by the factor ε, which decreases exponentially with n, the number of qubit. Though it takes a lot of effort in preparing an effective pure state, the computation speed is the same as that of a true pure state quantum computer. There is no exploitation of the mixed state nature of the NMR system. Brüschweiler s algorithm takes this advantage and achieves an exponential speedup. In NMR ensemble system, the state can be represented by density operators which are linear combinations of direct products of spin polarization operators [9,21]. In a strong external magnetic field, the eigenstates of the Zeeman Hamiltonian are mapped on states in the spin Liouville space φ in = = ααβ...αβ, (3) σ in = φ φ = I α 1 Iα 2 Iβ 3...Iα n 1 Iβ n, (4) where I α k = α k α k = 1 2 (1 k +2I kz )= [ ] 1 0, (5) 0 0 I β k = βk β k = 1 2 (1 k 2I kz )= [ ] (6) represent respectively spin up and spin down state of the spin. Usually the oracle or query is a computable function f: f(x) = 0 for all x except for x = z which is the item that we want to find out for which f(z) = 1. Usually the oracle can be expressed as a permutation operation which is a unitary operation U f, implemented using logic gates [5]. In Brüschweiler algorithm, an extra bit(also called the aucilla bit) is used and its state is represented by I 0.The output of the oracle is stored on the aucilla bit I 0 whose state is prepared in the α state at the beginning. The output of f can be represented by an expectation value of I 0z for a pure state f = F (I α 0 σ in )= 1 2 Tr(U f I α 0 σ in U + f I 0z). (7) If σ in happens to satisfy the oracle, then I0 α is changed to Iβ 0. This gives the value of the trace equal to 1/2, and hence f equals to 1. The input of f can be an mixed state of the form ρ = M j=1 Iα 0 σ j where σ j is one of the form in eq. (4): f = M M F (I0 α σ j )=F( I0 α σ j )+ M 1. (8) 2 j=1 j=1 The oracle is applied simultaneously to all the components in the NMR ensemble. The oracle operation is quantum mechanical. Brüschweiler put forward two versions of search algorithm. We adopt his second version. The essential of the Brüschweiler algorithm is as follow: suppose that the unsorted database has N =2 n number of items. We need n qubit system to represents these 2 n items. The algorithm contains n oracle queries each followed by an measurement: (1) Each time, I0 αiα k (k=1,2,...,n) is prepared. In fact, the input state Iα 0...I...Iα k...i... is a highly mixed state [21]. In the following text, the identity operator will be omitted. This Liouville operator actually represents the 2 n 1 number of items encoded in mixed state: 2

3 I0 α Iα k = Iα 0 (Iα 1 + Iβ 1 )(Iα 2 + Iβ 2 )...ɛ(iα n + Iβ n ) = I0 α Iγ1 1 Iγ2 2...Iγ k 1 k 1 Iα k Iγ k+1 k+1...iγn n = γ 1,γ 2,...,γ k 1,γ k,γ k+1,...,γ n=α,β i 1,i 2,...,i k 1,i k+1,...,i n=0,1 0i 1 i 2...i k 1 0i k+1...i n 0i 1 i 2...i k 1 0i k+1...i n. (9) This mixed state contains half of the whole items in the database. The k-th bit is set to α. The other half of the database with k-th bit equals to β(or 1) is not included. (2) Applying the oracle function to the system. As seen in eq. (8), the operation is done simultaneously to all the basis states. If k-th bit of the marked state is α, then the marked state is contained in eq. (9). One of the 2 n 1 terms in equation (9) satisfies the oracle and the oracle changes the sign of the aucilla bit from α to β. If one measures the spin of aucilla spin after the oracle, the value will be (2 n 1 1) 1/2 1/2 =N/4 1. If the k-th bit of the marked state is 1, then the state (9) will not contain the marked item. Upon the operation of the oracle, there is no flip in the aucilla bit. A measurement on the aucilla bit s spin I 0z will yield 1/2 (2 n 1 )=N/4. Therefore by measuring the aucilla bit s spin, one actually reads out the k-th bit of the marked state. (3) By repeating the above procedure for k from 1 to n, one can find out each bit value of the marked state. In the following, we give a simple example with N = 4 for illustrating the algorithm, and the example is realized in an experiment. The example is used for demonstration. The advantage of the algorithm will be seen if the number of qubit becomes large. Suppose the unsorted database with four items {00, 01, 10, 11} is represented by Zeeman eigenstates of the two spins I 1, I 2. The item z=10 is the one which we want. That is to say, f = 1 for z =10, which is expressed as I β 1 Iα 2. For the other three items, {00(I1 α I2 α ), 01(I1 α I β 2 ), 11(Iβ 1 Iβ 2 )}, f = 0. Function f can be realized by a permutation illustrated in Fig.1. The extra qubit I0 α is included in the permutation. First we prepare the a mixed state I0 αiα 1, which is the sum of Iα 0 Iα 1 Iα 2 + Iα 0 Iα 1 Iβ 2. Then the permutation described above is operated on this mixed state. Since the first bit of the marked state is 1, the permutation will have no effect on the aucilla bit. I0 αiα 1 Iα 2, Iα 0 Iα 1 Iβ 2 each contributes 1/2 to the spin of the aucilla bit. Upon measurement of the aucilla bit on its spin, the intensity will be 2 1/2 = 1 unit. That tells us that the first bit of the marked item is 1( in state I β 1 ). Secondly, we prepare another state, Iα 0 Iα 2 = Iα 0 Iα 1 Iα 2 + Iα 0 Iβ 1 Iα 2. We get output Iα 0 Iα 1 Iα 2 + Iβ 0 Iβ 1 Iα 2 after the action of permutation f. Measuring the spin of aucilla bit, we get 0, since I0 αiα 1 Iα 2 and Iβ 0 Iβ 1 Iα 2 contribute to the spin measurement equally but with opposite signs. Then this tells us that the second bit is 0( in state I2 α ). After these two measurement, we have obtained the marked state. In the actual experiment, we have modified the measuring part of the algorithm. We read out the bit values by looking at the shape of the aucilla bit. III. AN MODIFICATION TO THE ORIGINAL ALGORITHM We have not measured the I0 z in our experiment. We use the shape of the spectrum to distinguish the state of the aucilla bit. Because different initial state I0 α Ik α has the same form, except for a the difference in the k subscript, it is natural that the spectrum I 0 will have similar shapes for I γ 0 Iδ k 1 and I γ 0 Iδ k 2. We use the shape of the spectrum of the state I0 α Ik α as a reference where k =1, 2. First, the phase of Iα 0 I1 α is determined as making peaks of the spectrum up. In this NMR system, the I 0 bit has J coupling to both I 1 and I 2. When we measure the spectrum of I 0 in I 0 I k, we decouple the other qubits. Then there are only two peaks in the I 0 spectrum. The spectrum of I 0 I1 α before the operation of the permutation is given in Fig.4. After the permutation operation, we measure the spectrum of I 0 I k α again. If the shape of the spectrum is the same as the one before the oracle, i.e., two peaks are up still, then the permutation operation has not changed the spin of I 0 qubit, and this means that I k is 1, that is to say, the k-th bit value of the marked states z is 1. If the k-th bit of the marked state is 0, the aucilla bit will flip after the operation of the permutation U f. We can see from density matrice before and after the query operation. Before the query is evaluated on the mixed state I0 αiα 1, the density matrix(apart from a multiple of the identity matrix and a scaling factor) is After the query, the matrix at the acquisition is ρ 01in =, (10)

4 ρ 01out =. (11) When we measure the spectrum of aucilla bit I 0, the left peak, corresponding matrix element 51 and the right peak, corresponding the matrix element 62, do not change. This indicates that the shape of the spectrum does not change. As for the second step, before the query is evaluated on the mixed state I0 αiα 2, the outcome matrix is ρ 01in =, (12) and after the query, the matrix becomes ρ 02out =. (13) The left peak( (51) matrix element) does not change, but the right peak, ((72) matrix element) changes sign. Thus the right peak of the spectrum will be downward. This method of reading out the bit of the marked state is effective. Since it depends on the shape of the spectrum, a topological quantity, it is insensitive to errors as compared to the quantitative measurement of the spin of the aucilla bit. IV. THE REALIZATION OF THE ALGORITHM IN NMR EXPERIMENT We implemented the Brüschweiler algorithm in a 3 qubit homonuclear NMR system. The physical system used in the experiment is 13 C labeled alanine 13 C 1 H 3 13 C 0 H(NH 2 + ) 13 C 2 OOH whose structure is given in Fig.2. The solvent is D 2 O. The experiment is performed in a Bruker Avance DRX500 spectrometer. The parameters of the sample were determined by experiment to be: J 02 =54.2Hz, J 01 =35.1Hz and J 12 =1.7Hz. In the experiment, 1 H is decoupled throughout the whole process. 13 C 0, 13 C 1 and 13 C 2 are used as the 3 qubits, whose state are represented by I 0, I 1, I 2 respectively. 13 C 0 is used as the aucilla bit and the result of the oracle is stored on it. We assume the marked item is 10. Firstly, the state I0 αiα 1 is prepared. It is achieved by a sequence of selective and non-selective pulses, and J-coupling evolution. We begin our experiment from thermal equilibrium state. This thermal state is expressed as, In order to get the input state, the pulse sequence σ(0 )=I 0 z + I1 z + I2 z. (14) ( π 2 )2 y Grad ( π 4 )0,1 x τ ( π 6 )0,1 y Grad, (15) is used to get I0 αiα 1. Here the subscripts denote the directions of the radio frequency pulse, and the superscripts denote the nuclei on which the radio frequency are operated. Two numbers at the superscript mean that the pulse are applied simultaneously to two nuclei(in actual experiment, the pulses are applied in sequence. Because the duration of the pulse is very short, they can be regarded as simultaneous). Grad refers to applying gradient field. τ = 1 2J 01 is free evolution time during which nuclear 13 C 2 is decoupled. 4

5 The oracle, represented as a permutation f is applied to this initial state: I0 αiα 1. Then result of the oracle operation is stored on the aucilla bit I 0, that is, the state of the 13 C 0 indicates the state of the first bit of the marked item. Specifically the expression of the unitary operation corresponding to the permutation f is U f =. (16) The permutation U f can be completed using a sequence logic gates given in Fig.3. In the NMR, the pulse sequence for realizing this network is ( π 2 )1 y τ ( π 2 )1 x, (17) where τ = 1 2J 01. After the operation of the oracle, we measure the spectrum of the aucilla bit. Secondly, the initial state I0 αiα 2 is prepared. There are two ways to prepare this initial state. One method is to use a pulse sequence as in eqn. (15) by exchanging 1 with 2 in the superscripts. Another method is to use the swap operator ( π 2 )1,2 y τ 1 ( π 2 )1,2 x τ 1 ( π 2 )1,2 y, (18) onto the initial state I α 0 Iα 1 and the state Iα 0 Iα2 will be obtained. The swap operator is important in generalizing the experiment into more qubit system and we will discuss this later. Then we apply the permutation U f again, and the result of the oracle is stored in the aucilla bit 13 C 0. The spectra for I 0 I α 1 and I 0 I α 2 after the oracle query are given Fig.5 and Fig.6 respectively. We can see clearly that the one has the same shape as the reference spectrum and the other one has flipped the right peak. This tells us that the first bit and the second bit of the marked state are 1 and 0 respectively. Thus the marked state is 10. We also notice that there are small differences between the spectra before and after the permutation operations for I 0 I 1. These are expected due to imperfections caused by the inhomogeneous field, the errors in the selective pulse and in the evolution of chemical shift. V. SUMMARY To generalize the searching machine to more qubit system, there are several issues to be addressed. First, one must find a suitable molecule to act as the quantum computer. According to Brüschweiler s algorithm, the aucilla qubit I0 α must interplay with every other qubit. However, usually in a molecule, the interaction between remote nuclear spins is very weak. This can be overcome by the swap operation as given in Ref. [21]. Using swap operation, we can prepare any initial state I0 αiα k without the direct interplay between spin I 0α and spin I k α. At the same time, the qubit can be read out easily from the shape of the spectrum. The number of iteration required for this algorithm is very small. This is particularly advantageous in combating with decoherence, especially for NMR system at room temperature. Another advantage of the algorithm is its robustness against errors. The shapes of the spectrum in reading the k-th bit value of the marked state are very different. One can easily distinguish the two values. Another advantage of the algorithm is its exactness. The probability of finding the marked state is 100%. In summary, we have successfully demonstrated the Brüschweiler algorithmin a 3 qubit homo-nuclear NMR system. Pulse sequences are given. A method for reading out the bit value of the marked state is proposed and tested. The authors thank Prof. X. Z. Zeng, J. Luo and M.L. Liu for help in preparing the NMR sample and the use of selective pulses. This work is supported in part by China National Science Foundation, the Fok Ying Tung education foundation, major state basic research development program contract no. G , the HangTian Science foundation. [1] D. Deutsch, R. Jozsa, 1992, Proc. R. Soc. Lond. A., 439:553. 5

6 [2] P. Shor,in the Proc. of the 35 th Annual Symp. on the Found of Computer Science, edited by S. Goldwasser (IEEE Computer Society, Los. Alamitos):16; [3] L.K.Grover, 1997, Phys. Rev. Lett., 79:325; [4] C. Zalka,1999, Phys. Rev. A60:2746. [5] R.Brüschweiler, 2000, Phys, Rev. Lett. 85:4815 [6] D. Abrams and S. Lloyd, 1998, Phys.Rev.Lett,81: 3992 [7] C. Monroe et al., 1995, Phys, Rev. Lett.75: 4714 [8] Q. A.Turchette, et al., ibid., 4710 [9] R. R.Ernst, G. Bodenhausen, A. Wokaun, Princeples of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford University Press, 1987) [10] D.G.Cory, et al., 1997, Proc. Nat. Acad. Sci. USA, 94:1634 [11] N. A.Gersenfeld, I. L.Chuang, 1997, Science, 275:350 [12] J. A.Jones and M. Mosca, 1998, J. Chem. Phys., 109:1648 [13] I. L. Chuang, et al., 1998, Nature, 393:143 [14] J. A. Jones, NMR Quantum Computing in Quantum Computation and Quantum Information Theory (World Scientific, in press) [15] J. A.Jones, Quantum computing and NMR in The Physics of Quantum Information (Springer-Verlag, Berlin, 2000) [16] G. L. Long, H. Y. Yan et al, 2001, Phys. Lett. A286: 121. [17] S. L. Braunstein et al, 1999, Phys. Rev. Lett. 83: [18] R. Laflamme and D. G. Cory, NMR quantum information processing and entanglement, quant-ph/ [19] W. S.Warren, 1997, Science, 277:1688 [20] N. A.Gershenfeld, I. L.Chuang, 1997, Science, 277:1689 [21] Z. L. Madi, R.Brüschweiler, R. R.Ernst, 1998, J. Chem. Phys., 109:10603 FIG. 1. The permutation for the oracle FIG. 2. The structure of the 13 C labeled alanine FIG. 3. Network for realizing the oracle U f FIG. 4. The spectrum of the aucilla qubit in state I α 0 I α 1 FIG. 5. The spectrum of the aucilla qubit in state I 0I α 1 after the oracle FIG. 6. The spectrum of the aucilla qubit in state I 0I α 2 after the oracle 6

7 1 H O 1 H 13C 2 1 H 13 C0 13 C1 1H NH 2 OH

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