Return Probability of the Fibonacci Quantum Walk

Transcription

1 Commun. Theor. Phys. 58 (0 0 Vol. 58 o. August 5 0 Return Probability of the Fibonacci Quantum Walk CLEMET AMPADU 3 Carrolton Road Boston Massachusetts 03 USA (Received February 7 0; revised manuscript received May 7 0 Abstract In this paper the return probability of the one-dimensional discrete-time quantum walk is studied. We derive probabilistic formulas for the return probability related to the quantum walk governed by the Fibonacci coin. PACS numbers: Lx 05.0.Fb 0.50.Cw Key words: return probability Fibonacci transformation quantum random walk Introduction The quantum walk is regarded as the analogue of the classical random walk. The walk has been intensely investigated in the literature due to its connection with quantum computing. [ Like the classical random walk the quantum walk has two parts the discrete-time quantum walk [3 and the continuous time quantum walk. [ In this paper we focus on the discrete-time quantum walk on the line. Localization see Ref. [5 and the references therein and recurrent properties [6 9 of the quantum walk are intensely investigated in the literature. The return probability [30 33 and long-time limiting probability [3 36 play a key role in understanding these properties of the quantum walk. In this paper we obtain the necessary expressions for the return probability and long-time limiting probability of the quantum walk in the case the walk is operated by the Fibonacci operator. This paper is organized as follows. In Sec. we briefly review the discrete-time quantum walk on regular graphs there the Fibonacci operator and its Fourier analysis are presented. In Sec. 3 we present the analytic formula for the return probability via the method of stationary phase approximation (SPA; the formula for the long-time limiting probability. Section is devoted to the conclusions there we propose studying the recurrent behavior and localization of the Fibonacci quantum walk. Definitions Let us begin by recalling some facts in the general setting concerning discrete-time quantum walks (DTQW on a d-regular graph which is a regular graph each vertex has exactly d edges. The DTQW on the d-regular graph happens on the coin Hilbert space H C and the position Hilbert space H P the total Hilbert space is given by H = H C H P. If the d-regular graph has vertices then H P = span{ i : i =...} and H C = span{ e i : i =...d}. We should remark that e i corresponds to the chirality state of the particle it is the column vector with one in the i-th position and zero everywhere else. For example on the D lattice we have e = [ 0 = L and e = [ 0 = R where we have let e denotes the left chirality state of the particle L; e denotes the right chirality state of the particle R. The evolution of the system at each step of the walk is governed by the operator U = S(I C where S C are the shift and coin operators respectively. The coin operator randomly chooses the direction the particle moves and the shift operator moves the particle according to the coin state. For every vertex all the outgoing edges are labeled as... d. The conditional shift operator S moves the particle from v to w if the edge (v w is labeled by j on v s side: { ej w if e j v = (v w S e j v = 0 otherwise. For example on the D lattice we have ( S = i i e e i ( + i + i e e. i In terms of the initial state ψ(0 the total state of the particle is given by ψ(t = U t ψ(0. The probability distribution of the particle is given by P(x t = = d e i x ψ(t i= d x e i ψ(t. i= We should remark that the definitions above can be found in Ref. [37. In this paper we will take the coin operator to be the Fibonacci transformation. [38 39 The transfor- drampadu@hotmail.com c 0 Chinese Physical Society and IOP Publishing Ltd

2 o. Communications in Theoretical Physics mation is given by ( i e iφ cosθ i e iφ sinθ M(φ θ = i e iφ sin θ i e iφ cosθ where φ [ π. It should be noted that ( cosθ sin θ im(π θ = sin θ cosθ and this coin operator is intensely investigated in the literature in various settings. It is intimately related to the Hadamard gate in the following way: ( im π π = (. The Fourier transformation of the state in the particle s Hilbert space is given by the discrete Fourier transform ψ (k t = x {... }. x= e πix/ ψ (x t The time evolution of the walk can be expressed by the difference equation ψ (k t = Ũ(k ψ (k t. By induction on t we can write the time evolution of the walk as ψ (k t = Ũ(kt ψ (k 0 where ψ (k 0 is the initial state in the Fourier picture. We should remark that Ũ(k = Diag (e ik/ e πik/ M(φ θ. Let λ j (k be the eigenvalues of Ũ(k and let v j(k be the normalized eigenvectors of Ũ(k for j = then we can express ψ (k 0 by ψ (k 0 = v j (k ψ (k 0 v j (k. So j= ψ (k t = U(k ψ (k 0 = λ j (k v j (k ψ (k 0 v j (k. j= The discrete Fourier transform of the initial state is given by ψ (k 0 = e πikx0/ ψ(0. In this paper we will set ψ(0 = [ a a e iφ where a [0 and φ [0 π. By the inverse discrete Fourier transform we have ψ (x t = = k= k= e ikx/ ψ (k t e ik(x x0/ λ t j (k v j(k ψ(0 v j (k. j= The probability distribution is given by P(x t = m= k= e ik(x x0/ λ t j (k e m v j (k v j (k ψ(0. j= We should remark that if we let the above expressions for P(x t and ψ (x t immediately above can be written in integral form. The integral form will be useful later on as we derive the return probability for the quantum walk. 3 Return Probability and Long-Time Behavior According to Ref. [37 we can define the return probability as P(X = x 0 t which means the probability of finding the walker at the initial node x 0. Let θ(k = πk/ then as we can write the return probability in integral form as P(X = x 0 t = { m } dθ e m v j (θ v j (θ ψ(0 λ t j π (θ. m= j= The explicit calculation of the above integral is done using the SPA. The SPA is an approach for solving integrals analytically by evaluating the integrands in regions where they contribute the most. [0 Suppose we want to evaluate the behavior of the function I(λ for large λ I(λ = (/π g(xe λf(x dx. The SPA says that the main contribution to the integral comes from those points where f(x is stationary. If there is only one point x 0 for which f (x 0 = 0 and f (x 0 0 then the integral is approximated asymptotically by I(λ πλf (x 0 g(x 0e λf(x0. If there are more than one stationary point then the integral I(λ is approximately given by the sum of the contributions each having the same form as I(λ πλf (x 0 g(x 0e λf(x0 for all the stationary points. [ In Ref. [39 we showed that the eigenvalues of Ũ(k are given by λ (k = e iw(k λ (k = e iw(k where w(k is determined by sinw(k = sin θ(kcosθ and θ(k = πk/. The eigenvectors of Ũ(k are given compactly for j by [ i(sin(θ(kcotθ λj e iθ(k cscθ+cotθ v j (k = j where j is an appropriate normalization factor. Recall that the return probability can be written in integral form since as ( should not be confused with j he values of θ(k = πk/ are almost continuous. The SPA says we can write the return probability as P(x = x 0 t = I + I where

3 Communications in Theoretical Physics Vol. 58 I = π I = π e v (θ v (θ ψ(0 e iw(θt dθ + e v (θ v (θ ψ(0 e iw(θt dθ + e v (θ v (θ ψ(0 e iw(θt dθ e v (θ v (θ ψ(0 e iw(θt dθ. Since sin w(k = sin θ(kcosθ we have w(θ = arcsin( sin θ cosθ. So the stationary points of the above integrals satisfy w cos(θ (θ = = 0. sin(θ sin (θ It follows that the contribution of each integral should come from the stationary points θ = π/ θ = /. otice that w sin(θ + cos(θ 3 (θ = ( sin(θ(sin(θ. 3/ However w (π/ 0 is a real number compared to w (/. In particular since the phase is a real-valued function the only contribution of each integral comes from the stationary point θ = π/. The SPA says we can write I z + z upon substituting θ = π/ into the integral formula above similarly we have I f + f. Let w(θ = arcsin( sin θ cosθhen the SPA says z πλ w (π/ g (π/e λw(π/ where g (θ = e v (θ v (θ ψ(0 λ = it z πλ w (π/ g e λw(π/ where g (θ = e v (θ v (θ ψ(0 λ = it f πλ w (π/ g 3 e λw(π/ where g 3 (θ = e v (θ v (θ ψ(0 f πλ w (π/ g e λw(π/ g (θ = e v (θ v (θ ψ(0. Explicitly we have z = [ i( 3 + (3 3 [ i ( 3 + (3 3 ( a + ae iφ πit i 3 where z has been left in terms of the normalization constant {( i( 3 + (3 3 } / = + in the special case θ = π/. z = [i( [( i( a + ( a e iφ πit i 3 where z has been left in terms of the normalization constant = {(i( } / in the special case θ = π/. f = [ i( 3 + (3 3 ( a + ae iφ π it i 3 where f has been left in terms of the normalization constant [( i( 3 + (3 3 / = + in the special case θ = π/. f = {( i( a + ( a e iφ } πit i 3 where f has been left in terms of the normalization constant = {(i( } / in the special case θ = π/. We make the following observations { i( 3 + (3 3 } z = f z = {i( }f. So the return probability is given by { i( 3 + (3 3 P(x = x 0 t = I = I + I = } f + {i( }f + f + f.

4 o. Communications in Theoretical Physics 3 As the figure below shows the return probability closely behaves like t. It should be noted that in this figure the return probability is a function of three parameters the time step a [0 and φ [0 π. The values have been discretized and the chart was reproduced using Microsoft Excel 00. The x-axis denotes the time step and the y-axis denotes the return probability. ow let us consider the long-time average of the probability distribution. The time-averaged distribution is defined by P(x t = (/T T P(x t while the long-time probability is given by χ(x = lim P(x t. Substituting the t= T eigenvalues λ = e iw(θ(k and λ = e iw(θ(k into P(x t = m= [ ik(x x0 exp λ t j(k e m v j (k v j (k ψ(0 k= we have the following P(x t = [ ik(x x0 exp (e iw(kt e m v (k v (k ψ(0 + e iw(kt e m v (k v (k ψ(0 = m= k= m= kk = If w(k = w(k hen [ i(k k (x x 0 exp [(e iw(kt e m v (k v (k ψ(0 + e iw(kt e m v (k v (k ψ(0 { e iw(k t ψ(0 v (k v (k e m + e iw(k t ψ(0 v (k v (k e m }. lim T T So the long-time limiting probability is given by X(x = [ i(k k (x x 0 δ w(kw(k exp m= kk = j= T e ±i(w(k w(k t = δ w(kw(k. t= + e m v (k v (k ψ(0 ψ(0 v (k v (k e m. ( e m v (k v (k ψ(0 ψ(0 v (k v (k e m limiting probability. In a related work [37 for a generalization of the Hadamard walk given by ( ρ ρ C(ρ = ρ ρ Fig. Return Probability P(x = x 0 t as a function of the initial parameters. Discussion and Concluding Remarks In this paper we have studied the discrete-time quantum walk on the one-dimensional lattice with the Fibonacci operator governing the walk and obtained expressions for the return probability as well as the long-time where 0 ρ it is shown that the coherent dynamics of the walk depends on the initial states and coin parameters. The return probability is shown to have scaling behavior P(0 t t. In the long-time limit the probability distribution exhibits various patterns depending on the initial states coin parameters and lattice size. In terms of the return probability the future interesting problem is to study the recurrent nature of the walk. In particular we can say the following about the Fibonacci quantum walk: Between the two dominant peaks the probability is roughly independent of the position and decays like /t On the other hand outside the decay is exponential as we depart from the peaks. In terms of the long-time limiting probability the future interesting problem is to investigate localization-type phenomena for the Fibonacci quantum walk near different initial nodes.

5 Communications in Theoretical Physics Vol. 58 References [ D. Aharonov A. Ambainis J. Kempem et al. Quantum Walks on Graphs in Proceedings of the 33 rd Annual ACM Symposium on Theory of Computing 50 (00. [ A. Ambainis E. Bach A. ayak et al. One Dimensional Quantum Walks in Proceedings of the 33 rd Annual ACM Symposium on the Theory of Computing 37 (00. [3 E. Bach S. Coppersmith M.P. Goldschen et al. [ A.M. Childs E. Farhi S. Gutmann et al. Quantum Information Processing ( [5 W. Dur R. Raussendorf V.M. Kendon et al. Phys. Rev. A 66 ( [6 J. Kempe Probability Theory and Related Fields 33 (005. [7. Konno J. Math. Soc. Japan 57 (005. [8. Konno et al. Interdisciplinary Information Sciences 0 (00. [9 T.D. Mackay S.D. Bartlett L.T. Stephenson et al. J. Phys. A: Math. Gen. 35 ( [0 C. Moore and A. Russell Auantum Walks on the Hepercubes quant-ph/0037. [ B.C. Travaglione and G.J. Milburn Phys. Rev. A 65 ( [ T. Yamasaki H. Kobayashi and H. Imai Phys. Rev. A 68 ( [3 A. ayak and A. Vishwanath Quantum Wall on the Line quant-ph/0007. [ E. Fahri and S. Gutmann Phys. Rev. A 58 ( [5 C. Ampadu arxiv:08.9 (0. [6 B. Kollar M. Stefanak T. Kiss and I. Jex Phys. Rev. A 8 ( [7 M. Stefanak T. Kiss and I. Jex ew J. Phys. ( [8 C.M. Chandrashekar Cent. Eur. J. Phys. 8 ( [9 M. Stefanak T. Kiss and I. Jex Phys. Rev. A 78 ( [0 M. Stefanak B. Kollar T. Kiss and I. Jex Phys. Scr. T 0 ( [ Z. Darazs and T. Kiss Phys. Rev. A 8 ( [. Konno and E. Segawa Localization of Discrete-Time Quantum Walks on a Half Line via the CGMV Method Quantum Information and Computation (0 85. [3 E. Bach and L. Borisov arxiv: (009. [ M.J. Cantero F.A. Grunbaum L. Moral and L. Velazquez Communications on Pure and Applied Mathematics 63 (00. [5 O. Muelken V. Bierbaum and A.Blumen J. Chem. Phys. ( [6 D. Ben-Avraham E. Bolt and C. Tamon One- Dimensional Continuous-Time Quantum Walks Quantum Information Processing 3 ( [7. Konno Quantum Walks and Elliptic Integrals Mathematical Structures in Computer Science 0 ( [8 M. Stefanak I. Jex and T. Kiss Phys. Rev. Lett. 00 ( [9 M. Stefanak T. Kiss and I. Jex ew J. Phys. ( [30 X. Xu Phys. Rev. E 79 ( [3 X. Xu Phys. Rev. E 77 ( [3 Y. Ide. Konno T. Machida and E. Segawa Return Probability of one-dimensional Discrete-Time Quantum Walks with Final-Time Dependence Quantum Information and Computation (0 76. [33 A. Anishchenko A. Blumen and O. Muelken arxiv:.7065 (0. [3 V. Kargin J. Stat. Phys. 0 ( [35 X. Xu and F. Liu Phys. Lett. A 37 ( [36 E. Agliari A. Blumen and O. Muelken J. Phys. A ( [37 Xin-Ping Zu Eur. Phys. J. B 77 ( [38 A. Romanelli Physica A 388 ( [39 C. Ampadu arxiv: (0. [0 C.M. Bender and S. Orzag et al. Advanced Mathematical Methods for Scientists and Engineers McGraw Hill ew York (978. [ O. Mulken and V. Pernice Phys. Rev. E 77 ( [ J. Scales Theory of Seismic Imaging Springer Berlin Heidelberg (985.