Data Search Algorithms based on Quantum Walk

Transcription

1 Data Search Agorithms based on Quantum Wak Masataka Fujisaki, Hiromi Miyajima, oritaka Shigei Abstract For searching any item in an unsorted database with items, a cassica computer takes O() steps but Grover s quantum searching agorithm takes ony O( ) steps. However, it is aso known that Grover s agorithm is effective ony in the case the initia ampitude distribution of dataset is uniform, but is not aways effective in the nonuniform case. In this paper, we propose some quantum search agorithms. First, we propose an agorithm in anaog time based on quantum wak by soving the schrodinger equation. The proposed agorithm shows best performance in optimum time. ext, we wi appy the resut to Grover search agorithm. It is shown that the proposed agorithm shows better performance than the conventiona one. Further, we propose the improved agorithm by introducing the idea of the phase rotation. The agorithm shows best performance compared with the conventiona ones. Index Terms quantum search agorithm ; Grover search agorithm; initia ampitude distributions of dataset; observed probabiity I. ITRODUCTIO With quantum computation, many studies have been made. Shor s prime factoring and Grover s data search agorithms are we known[], [], [4]. Further, Ventura has proposed quantum associative memory by improving Grover s agorithm [3], [5]. Data search probem is to find any data effectivey from unsorted dataset. For searching any item in an unsorted database with items, a cassica computer takes O() steps, but Grover s agorithm takes ony O( ) steps. However, it is aso known that Grover s agorithm is effective ony in the case the initia ampitude distribution of dataset is uniform, but is not aways effective in the nonuniform case[3]. Further, Ventura has proposed the quantum searching agorithm but it is effective ony in the specia case for the initia ampitude distribution[6], [7]. Therefore, it is necessary to find effective agorithms even in the case the initia ampitude distribution of dataset is not uniform. For exampe, associative memory needs non-uniform initia data distribution [3]. In this paper, we propose some quantum search agorithms. First, we propose an agorithm in anaog time based on quantum wak by soving the schrodinger equation. The proposed agorithm shows best performance in optimum time. ext, we wi appy the resut to Grover search agorithm. It is shown that the improved agorithm shows better performance than the conventiona one. Further, we propose the agorithm by introducing the idea of the phase rotation[8]. The proposed agorithm shows best performance compared with the conventiona ones. This work is supported by Grant-in-Aid for Scientific Research (C) (o.57) of Ministry of Education, Cuture, Sports, Science and Technoogy of Japan. * Graduate Schoo of Science and Engineering, Kagoshima University, --4 Korimoto, Kagoshima 89-65, Japan emai: k765487@kadai.jp corresponding author to provide emai: miya@eee.kagoshima-u.ac.jp emai: shigei@eee.kagoshima-u.ac.jp II. PRELIMIARY The basic unit in quantum computation is a qubit c + c, which is a superposition of two independent states and corresponding to the states and in a cassica computer, c and c are compex numbers such that c + c =. We use the Dirac bracket notation, the ket i is anaogous to a coumn vector. Let n be a positive integer and = n. A system with n qubits is described using independent state i ( i ) as foows: i= c i i () c i is a compex number, i= c i = and c i is the probabiity of state i.the direction of c i on the compex pane is caed the phase of state i and the absoute vaue c i is caed the ampitude of state i. In quantum system, starting from any quantum state, the desired state is formed by mutipying coumn vector of the quantum state by unitary matrix. Finay, we can obtain the desired state with high probabiity through observation[]. The probem is how we can find unitary matrix. Grover has proposed the fast data search agorithm. Let us expain the Grover s agorithm shown in Fig.. Grover has proposed an agorithm for finding one item in an unsorted database. In the conventiona computation, if there are items in the database, it woud require O() queries to the database. However, Grover has shown how to perform this using the quantum computation with ony O( ) queries[]. Let Z = {,,, }. Let us define the foowing operators. I a = Identity matrix except for which inverts any state ψ and I(a +, a + ) =, a Z () W (x, y) = ( ) x y + +x y for x = i= x i i, y = i= y i i, (3) which is caed the Wash or Hadamard transform and performs a specia case of discrete Fourier transform. We begin with the state and appy W operator to it, means that a states are and the number of s for is. As a resut, a the states have the same ampitude /. ext, we appy the I τ operator, τ is the searching state. Further, we appy the operator G = W I W (4) Foowed by the I τ operator T = (π/4) times and observe the system[]. G operator has been described as inverting each of the state s ampitudes around the average

2 Fig.. Grover search agorithm. Initia state ψ. ψ = W = 3. Repeat T times 4. ψ = I τ ψ 5. ψ = G ψ 6. Observe the system h is Pank constant and H is Hamitonian which means a the energy of the system. Then H = H hods, H is the transposed matrix of compex conjugate for H. Then the soution is represented by ψ(t) = U(t) ψ(), (7) one of a states. Exampe [6]: Let n = 4 and = 6. Let searching data τ = 6 = and the number of stored (memorized) data = 6. Then, the desired data is obtained with the probabiity.96 by using Grover s agorithm. We can get the searching data with high probabiity. ext, assuming that stored data are, 3, 6, 9,, and 5, and searching data is 6, that is = 6. The initia state ψ is as foows: { for any i of stored date ψ i = (5) otherwise, Then, the desired data is obtained with the probabiity.44. It shows that Grover s agorithm does not aways give a good resut in the case. Therefore, it is needed to find effective agorithms even in the case the initia ampitude distribution of dataset is not uniform. III. QUATUM SEARCH I AALOG MODEL BASED O i j Fig.. i j m- i m- j m- QUATUM WALK - - A the data i - Description of data search probem pieces of memorized data (stored) m pieces of search data (marked) In the foowing, we introduce a quantum search agorithm using anaog mode of quantum wak. In order to carify the probem, we wi expain the Fig.. It is assumed that pieces of data are memorized (stored) in the system of pieces of data. ow we want to find any data in m pieces of data (marked) with high probabiity. Grover has shown the effective agorithm in the case of =, m = (see Fig.). A. Schroedinger equation and quantum wak In this chapter, we propose an agorithm based on quantum wak in anaog time mode. As the state of system in quantum mode is determined by the Schroedinger equation, we can obtain the resut by soving the Schroedinger equation under the specia condition[9],[]. The Schroedinger equation for the state of system is represented as foows[9]: i h d ψ dt = H ψ, (6) U(t) = exp( ith) (8) and U(t) is caed time expansion operation of the state and ψ() is the initia state of system. Therefore, the state of system is determined by Hamitonian H. Let P w (t) be defined as the observed probabiity of system at time t as foows: P w (t) = w U(t) ψ(). (9) The state of system to search is caed the marked one and the other is caed the unmarked state. The number of marked states is m(see Fig.). Hamitonian H ρ corresponding to the potentia energy is represented by an identity matrix except for H ρ (j i, j i ) =, () i Z m. Let L be the state assignment over the graph. Then the Hamitonian H of the system is defined using the mobiity r as foows[9]: H = γl + H ρ () Let G = (V, E) be the perfect graph to act for system, V is the set of vertexes and E is the set of edges. Then L is represented as foows: L = I = I + x y x y = I + s s, () s = x (3) x= Finay, Hamitonian H is represented as foows: H = γi γ s s + H ρ (4) It is known that any quantum arrives at any pace (state) in O( ) steps by using quantum wak. Assuming that the potentia energy of the pace to arrive is ow, the high probabiity of the state is performed. Then, et U be defined as foows: U = exp( itγi) exp(ita), (5) A = γ s s H ρ γi (6)

3 B. Derivation of the time expansion operator U Let us compute the Eq.(5). Let C be any matrix. Then the foowing reation hods: exp(c) = r! Cr. (7) r= In order to compute the Eq.(5), A r must be computed. Here, in order to understand the computation of A r easiy, we wi consider the case of = 8. A r = ζ(r) δ(r) α(r) α(r) δ(r) δ(r) α(r) α(r) β(r) ɛ(r) β(r) β(r) η(r) η(r) β(r) β(r) α(r) δ(r) ζ(r) α(r) δ(r) δ(r) α(r) α(r) α(r) δ(r) α(r) ζ(r) δ(r) δ(r) α(r) α(r) β(r) η(r) β(r) β(r) ɛ(r) η(r) β(r) β(r) β(r) η(r) β(r) β(r) η(r) ɛ(r) β(r) β(r) α(r) δ(r) α(r) α(r) δ(r) δ(r) ζ(r) α(r) α(r) δ(r) α(r) α(r) δ(r) δ(r) α(r) ζ(r) (8) From the reation that A r+ = A r A and A is known, the foowing recursion formua are obtained: { β(r + ) = ( m)γ } β(r) +(m )γη(r) + γɛ(r) (9) { η(r + ) = ( m)γβ(r) + (m )γ + } η(r) +γɛ(r) () ɛ(r + ) = ( m)γβ(r) + (m )γη(r) α(r + ) = +(γ + )ɛ(r) () { ( m )γ } α(r) +mγδ(r) + γζ(r) () δ(r + ) = ( m )γα(r) + (mγ + )δ(r) +γζ(r) (3) ζ(r + ) = ( m )γα(r) + mγδ(r) +(γ )ζ(r) (4) ow, et v (r), v (r), K andk be defined as foows: K = K = v (r) = v (r) = β(r) η(r) ɛ(r) α(r) δ(r) ζ(r) (5) (6) γ( m) γ(m ) γ γ( m) γ(m ) + γ γ( m) γ(m ) γ + γ( m ) γm γ γ( m ) γm + γ γ( m ) γm γ (7) (8) Then, the foowing reation hod: v (r + ) = K v (r) (9) v (r + ) = K v (r) (3) By diagonaizing the matrixes K and K, v and v are obtained. As a resut, the operator U is obtained as foows: ζ δ α α δ δ α α β ɛ β β η η β β α δ ζ α δ δ α α U = α δ α ζ δ δ α α β η β β ɛ η β β (3) β η β β η ɛ α β α δ α α δ δ ζ α α δ α α δ δ α ζ β = P R exp ( i γmt) Q L λ λ η = m + R λ exp ( i γmt) L λ (3) (33) ɛ = m m + R exp ( i γmt) L λ λ α = m exp( it) + R λ exp ( i γmt) L λ δ = P R exp ( i γmt) Q L λ λ ζ = m m exp( it) + R λ exp ( i γmt) L λ λ = γm λ = + γm γm P = Q = R = L = γm γm γm+ γmq + Q γm m(q P ) γmp+ P γm m(q P ) λ = γm λ = + γm γm P = Q = R = γm γm+ γm γ( m)( Q)+ Q ( m)(p Q ) L = γ( m)( P )+ P ( m)(p Q ) IV. SEARCH ALGORITHMS BASED O QUATUM WALK A. Appication to quantum search probem in anaog time (34) (35) (36) (37) (38) (39) The resut of the chapter III is appied to quantum search probem. Remark that t is in anaog time. The method is

4 caed the method (in anaog time)[9]. The probabiity P w (t) is computed by the Eq.(9). Then the initia ampitudes ψ a () and ψ b () of memorized and non-memorized data of time is as foows, respectivey: ψ a () = (4) ψ b () = (4) From the Eq.(7), the foowing resut is obtained ψ a (t) = [( m)α m + (m )β m + δ m ] = cos γmt + i sin γmt (4) m As a resut, the observed probabiity of search data P w (t) is obtained from the Eq.(9) as foow: P w (t) = m cos t + m m sin t (43) Let us show the exampe of four cases, () m =, =, () m =, =, (3) m =, = /, (4) m =, = / for = 4. The resut shows P w ((π/) /m) = /(m), t = (π/) /m is the optimum time. If =, then P w (t) = /m and P w (t) = for m =. Fig.3 shows the simuation resut. The case and case for = shows high probabiity. Observed probabiity Fig case case case3 case time The simuation resut of the method for four cases Then, how is the case. In this case, the maximum probabiity is computed by soving dp w (t)/dt =. Fig.4 shows the resuts which are P w (t) except for = and P w (t).5 for < /. B. Improved search agorithm in anaog time The state ψ a (t ) of memorized data after t step and the state ψ b (t ) of non-memorized data after t step are represented as foows: ψ a (t ) = cos t + i sin t (44) ψ b (t ) = i sin t (45) When t = (π/) /, it hods ψ a (t ) = ψ b (t ). Therefore, the states except for marked data at the step t Observed probabiity Fig maximum probabiity umber of stored date Maximum probabiity for the number of memorized data are identica, so the method is possibe to appy at the time t = t. The use of the method for the time interva t eads to the foowing probabiity: i ψ w (t + t ) = [( m)α m + (m )β m + δ m ] m = i cos t m sin m t (46) P a (t + t ) = cos m t + m sin m t (47) By taking t = (π/) /m, it hods P a (t + t ) = /m. The method is caed the proposed method in anaog time. Fig.5 shows the numerica exampe the comparison between method and proposed method for = 4, m =, = 5. Probabiity method_ The proposed method_ time Fig. 5. The comparison between the method and the proposed method C. The appication to Grover search agorithm of the proposed methods In this section, et us appy the proposed method obtained in the section B to Grover search agorithm shown in Fig.. Fig.6 shows the proposed method 3 (in digita time) corresponding to the proposed method Let us compute the

5 . Initia state ψ. Repeat T times 3. ψ = I ρ ψ 4. ψ = G ψ 5. Repeat T g times 6. ψ = I τ ψ 7. ψ = G ψ 8. Observe the system Fig. 6. The agorithm of the proposed method 3 time T as the same method used in the section B. ψ a (T ) = cos ω T (48) ψ b (T ) = sin ω T, (49) ( ) ω = arccos. (5) Therefore, we can find the time when ψ a (T ) = ψ b (T ) as foows: ( ) π arctan T = CI arccos ( ), (5) CI(x) means rounding of x. Fig.7 shows the resut As T =, I ρ and G to the Eq.53 are iterated one more time. As a resut, the foowing state is obtained: ψ = (,,,,,,,,, 4,,,,,, ) t (54) Further, iterating the steps 5 to 8 of the proposed method 3 to the Eq.54, the desired data is obtained with the probabiity.96. ext, supposing that the stored data are,, 3, 6, 7,,, as foows: ψ = (,,,,,,,,,,,,,,, )t (55) When the steps to 4 of the proposed method 3 are iterated for the Eq.55, the foowing state is obtained: ψ = (,,,,,,,,,,,,,,, )t (56) Then, the desired data is obtained with the probabiity.5 after appying the steps 5 to 8. It shows that the proposed method 3 does not aways give a good resut. As shown in Fig.7. we can not aways get the maximum resut, because the maximum vaue of T is rea number in anaog mode. Therefore, in order to improve the resut we can introduce the phase rotation[8]. These are represented as the foowing unitary matrixes: W = I ( e iα ) ρ k ρ k (57) k= V = ( e iβ ) s s + e iβ I (58).9 Observed probabiity Fig Grover Proposed method 3 Proposed method umber of stored data The comparison among the proposed agorithms of numerica simuation for = 56. Exampe : Let n = 4 and = 6. Let, 6,, be stored data. ψ = (,,,,,,,,,,,,,,, )t (5) The suffix t means the transpose of the vector The foowing reut is obtained by appying I ρ and G to the initia state of the Eq.5. ψ = (,,,,,,,,,,,, 4,,, ) t (53). Initia state ψ. Repeat T times 3. ψ = W ψ 4. ψ = V ψ 5. Repeat T g times 6. ψ = I τ ψ 7. ψ = G ψ 8. Observe the system Fig. 8. The agorithm for the proposed method 4 The method is caed the proposed method 4 The Fig.8 shows the agorithm for the proposed method 4. Matrixes I ρ and G used before are the specia case for α = β = π of W and V, respectivey. Let us compute the time T. Let us consider the case of T =. Then, as the imaginary parts of ψ a () and ψ b () are agree, we wi find the condition that the rea parts of ψ a () and ψ b () are agree. The foowing reation is satisfied with the condition: ( α = arccos ) (59) (/4) Fig.7 shows the resuts of Grover and two proposed agorithms. Exampe 3: Let n = 4 and = 6. Let the initia state be defined as the Eq. 55. In the proposed method 4, α is set to π/.

6 TABLE I A SUMMARY OF THE PROPOSED METHODS. Anaog mode Digita mode Schroedinger Method Grover agorithm equation Proposed method Proposed method 3 Proposed method 4 As T =, the steps 3 to 4 of the proposed method 4 are iterated two times. Then, the foowing state is obtained: ψ = 4 ( + i, + i, + i, + i, + i, + i, + i, + i + i, + i, + i, + i, + i, + i, + i, + i) t (6) Grover agorithm shows good performance ony in the case of =. The proposed method 3 shows better performance compared with Grover agorithm, but does not aways show good performance in the case of. The proposed agorithm 4 shows best performance of three agorithms in digita mode. V. COCLUSIOS AD FUTURE WORK The resut in this paper is summarized in Tabe I. The method and the proposed method are obtained by soving the schrodinger equation. The method and the proposed method in anaog time ead to Grover agorithm and the proposed method 3 in digita time. The proposed method in anaog time gives the optimum soution, but the proposed method 3 in digita method 3 does not give the optimum soution, because the optimum time is rea number. Therefore, we propose the method 4 and show that it gives the optimum soution. As the future work, we wi consider the reation between the proposed method and 4. REFERECES [] P. Shor, Poynomia-Time Agorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, SIAM Journa of Computing, vo.6, no.5, pp , 997. [] L. Grover, A Fast Quantum Mechanica Agorithm for Database Search, Proc. of 8th ACM Symp. on Theory of Computing, pp.- 9, 996. [3] D. Ventura and T. Martinez, Quantum Associative Memory, Information Science, vo.4, pp.73-96,. [4] D. Biron, O. Biham, E. Biham, M. Grass and D.A. Lidar, Generaized Grover Search Agorithm for Arbitrary Initia Ampitude Distribution, Lecture otes In Computer Science, Vo.59, pp.4-47, 998. [5] D. Ventura and T. Martinez, Initiaizing the Ampitude Distri- bution of A Quantum State, Foundations of Physics Letters, Vo., o.6, pp , 999. [6] K. Arima, H. Miyajima,.Shigei and M. Maeda, Some Properties of Quantum Data Search Agorithms, The 3rd Int. Technica Conf. on Circuits/Systems, Computers and Commu- nications, no.p-4, pp.69-7, 8. [7] K. Arima,. Shigei, H. Miyajima, A Proposa of a Quantum Search Agorithm, Fourth Internationa Conference on Computer Sciences and Convergence Information Technoogy, pp , 9. [8] P. Li and K. Song Adaptive Phase Matching in Grover s Agorithm, Journa of Quantum information Science, Vo., o., pp.43-49,. [9] K. Akiba, T. Miyadera and M. Ohya, Search by Quantum Wak on the Compete Graph, Technica Report of IEICE, IT4-7(4-7), pp [] L. K. Grover, From Schroedinger Equation to the Quantum Search Agorithm, Winter Institute on the Foundations of Quantum Theory, S Bose Center, Cacuttu, India, in Jan, pp-6.