Two-Dimensional Quantum Walks with Boundaries

Transcription

1 WECIQ 26 - Artigos Two-Dimensional Quantum Walks with Boundaries Amanda C. Oliveira 1, Renato ortugal 1, Raul Donangelo 2 1 Laboratório Nacional de Computação Científica, LNCC Caixa ostal etrópolis RJ Brazil 2 Universidade Federal do Rio de Janeiro UFRJ Caixa ostal Rio de Janeiro RJ Brazil {amanda,portugal}@lncc.br Abstract. We present here the quantum walk in two-dimensional lattices in an infinite and in a finite lattice with rectangular and diamond boundaries. These boundaries are generated by permanently breaking the links apart from the domain. We present here the probability distribution and the evolution of the standard deviation for the Hadamard, Fourier and Grover coins for many finite lattices. 1. Introduction In a seminal paper, Aharanov, Davidovich, and Zagury [Aharonov et al. 1993] introduced the discrete-time quantum walk model,which has new features when compared to the classical random walk. In special, the quantum walk spreads quadratically faster then the classical one. Many authors have used this fact to propose new quantum algorithms based on quantum walks [Farhi and Gutmann 1998, Shenvi et al. 23]. In this work we focus our attention on quantum walks in two dimensional lattices. We present the formalism for the quantum walk on an infinite line and for the quantum walk on a finite lattice with rectangular boundaries and lozenge shape boundaries. Ref.[Mackay et al. 22] was one of the first to analyze 2-D quantum walks.the authors concluded that the entanglement has a negative influence on the rate of spread. [Tregenna et al. 23] pointed out that this conclusion is not true in general, because it depends on the initial condition. They analyzed the full range of possible coin initial states of quantum walks starting at the origin and concluded that there are 1 types of non- equivalent coins. The Hadamard, Fourier, and Grover coins are of different types, the Grover coin being the one that produces the maximum spreading rate. The main goal of this work is to study the evolution of quantum walkers on lattice regions of arbitrary shape, through the procedure of permanently breaking the appropriate links in order to define its boundary. A possible application of this method is to study the transmission of quantum walkers through open billiards [Berggren et al. 22], or in a region where the corresponding classical motion would be chaotic. Other applications that could be considered are the problem of quantum percolation, and the propagation of the walkers in inhomogeneous regions, such as the interface of two regions with different conductivities. 211

2 2. Quantum walks in a two-dimensional infinite lattice A coined quantum walk on an infinite two-dimensional lattice has a Hilbert space H 4 H, where H 4 is the coin space and H is the lattice space. The coin consists of two qubits with basis { j,k,j,k {, 1}}. The links of the lattice are either in the main diagonal or in the secondary diagonal, then the basis for H is { m,n,m,n integers} such that m + n is even. The links have length 2. The generic state of the quantum walker is ψ(t) = j,k= m,n= A j,k; m,n (t) j,k m,n. (1) The evolution operator for the one step of the walk is U = S (C I), where C = j,k= j,k = C j,k; j,k j,k j,k (2) is the coin operator, I is the identity matrix, and S is the shift operator given by S j,k m,n = j,k m + ( 1) j,n + ( 1) k. (3) The walker moves along the main diagonal if the value of the coin is, or 1, 1 ; and along the secondary diagonal if the value of the coin is, 1 or 1,. Note that S does not entangle the first qubit of the coin with direction n nor the second qubit with direction m. Only the combined action of the coin and shift operators can perform such entanglement. Applying the evolution operator on state (1) we get A j,k; m,n (t + 1) = j,k = C j,k; j,k A j,k ; m ( 1) j,m ( 1) k(t). (4) The probability distribution for the walker at position m,n at time t is m,n (t) = A j,k; m,n (t) 2. (5) j,k= We have analyzed this walk with the Hadamard, Grover and Fourier coins. The initial state for each coin is the one which gives the maximum spread rate. Fig. 1 shows the probability distribution for the Hadamard coin (C = H H), where after steps, taking the initial state H = 1 2 ( ) ψ() = 1 ( i 1 )( + i 1 ),, (7) 2 which produces a symmetric walk. The Hadamard coin does not entangle the coin-qubits and the shift operator does not entangle the two directions. The result is the Hadamard walk in both directions. 212 (6)

3 Figure 1. The probability distribution of the 2-D QW after iterations using the Hadamard coin and the initial state (7) Figure 2. The probability distribution of the 2-D QW after iterations using the Fourier coin and the initial state (9). We observe that the density plot reveals details that are hardly seen in the 3-D plot. The walk is symmetric in the following sense. Take any line passing through the origin, the distributions are the same in both directions. This is equivalent to saying that the plot is invariant under a rotation of π. The initials states (7), (9), and (11), were chosen to guarantee a maximum of spreading when the walk starts at the origin [Tregenna et al. 23]. This situation is the most interesting in decoherence analysis. Fig. 2 shows the probability distribution for the Fourier coin F 4 = i 1 i i 1 i, (8) after steps, taking the initial state ψ() = 1 2 ( + 1 i i ) 11,. (9)

4 Figure 3. The probability distribution of the 2-D QW after iterations using the Grover coin and the initial state (11). In Fig. 3 we show the probability distribution for the Grover coin G = , (1) after steps, taking the initial state ψ() = 1 ( 1 )( 1 ),. (11) 2 The density plot shows remarkable properties of the Grover coin with the initial state (11). The walk is highly symmetric, since it is apparently invariant under a rotation of π/2. The walk is delocalized having a central region of about 1/3 of the reachable radius with almost zero probability distribution. In the infinite 2-D lattice the standard deviation for the Hadamard coin is the same for all choices of initial state that produces a symmetric distribution, even maximally mixed, and is 2 larger than the standard deviation for the walk on the line which is linear in the number of steps. However with the appropriated initial conditions both the Fourier and Grover coins have larger spreading rate than the Hadamard coin.even so the most differences between these three coins are not in the degree of spread, but in the extent to which this can be varied simply trough varying the initial coin state [Tregenna et al. 23] leading to many other applications than the Hadamard walk. 3. Quantum walks in a two-dimensional finite lattice Now we restrict the walk to a finite portion of the lattice. Let us analyze the evolution of the quantum walker considering explicitly the four links of each position m,n, as in fig. 4, so eq.(4) must be generalized yielding A 1 j,1 k; m,n (t + 1) = C j+link1 (j,k; m,n),k+link 2 (j,k; m,n); j,k j, k = A j,k ;m+link 1 (j,k; m,n),n+link 2 (j,k; m,n) (12) 214

5 m 1,n + 1 m 1,n 1 link 1,2 (1,;m,n) link 1,2 (1,1;m,n) m,n link 1,2 (,;m,n) link 1,2 (,1;m,n) m + 1,n + 1 m + 1,n 1 Figure 4. The four links at site m,n where link 1 (j,k; m,n) = { ( 1) j, if link to site m + ( 1) j,n + ( 1) k is closed,, if link is open, (13) link 2 (j,k; m,n) = { ( 1) k, if link to site m + ( 1) j,n + ( 1) k is closed,, if link is open, (14) and j,k 1. We easily see that eq.(12) reduces to eq.(4) if there are no broken ( links. When implementing this equation, one must impose that link 1 1 j, 1 k; m + ( 1) j,n + ( 1) k) = if link 1 (j,k; m,n) =, and similarly with link 1. The evolution equation for quantum walks in n-dimensional lattices is a generalization of Eqs. (12), (13), and (14). In this case one needs to use n link functions defined analogously to Eqs. (13) and (14). Eq. (12) must be modified accordingly, adding each link function to its corresponding index. With these equations in hand, it is possible not only to analyze broken-link-type decoherences in n-dimensional lattices, but also to analyze the decoherence-free walks in lattices with reflecting boundary conditions. In fact, one can choose a variety of lattice topologies by permanently breaking the relevant links. So, here we definitely break the appropriated links according to the size of the square. When the particle reaches some broken link to its neighbor site it is forced to remain at that site and the walk starts to be different from the infinite lattice. This is an unitary process and we do not make measurements, so the particle is just prohibited to advance, then it is reflected. A coined quantum walk on a finite two-dimensional square lattice has a Hilbert space H 4 H s, where H 4 is the coin space and H s is the lattice space. Thus the particle is confined to a square lattice whose positions are { m,n,m,n { M,M}} such that m + n is even. We analyze numerically the boundaries effects in the 2-D walk for the Hadamard, Fourier and Grover coins for some square s sizes. Fig.5 shows the evolution of the standard deviation of the Hadamard walk for many values of M. The standard deviation is linear in the number of steps until the walk encounters some boundary, At this time the standard deviation is maximum. Then the particle is reflected and the standard deviation decreases until the walk returns to the origin at this moment σ has a local minimum, so 215

6 9 M=1 M=2 M=4 M= σ Figure 5. Evolution of the standard deviation of the Hadamard walk with square lattice for an initial state (7), in the cases M = 1, M = 2, M = 4, and M =. t Figure 6. The probability distribution of the 2-D QW after iterations using the Grover coin, initial state (11) and square boundaries at M =. the walk spreads again but with another initial state and this process continues indefinitely. In fig.6 we show the probability distribution for the Grover walk after steps, the walk starts to be reflected by the boundaries, in fig. 7 we show the probability after 14 steps, at this time the walk is almost totally reflected and in fig. 8 the standard deviation is minimum and the particle can be found near the origin. In all the moments the probability distribution assumes a very intrincated pattern which is the signature of the quantum world. We observe some different results in the case of finite lattices when compared to infinite lattices. While in the infinite lattice the standard deviation of the Hadamard, Fourier and Grover walks are different when the walker starts at the origin with initial states (7), (9), and (11), respectively, which gives the maximum spreading rates for each case. We still have, in all cases, σ = αt with σ H =.77,σ F =., and σ G =

7 Figure 7. The probability distribution of the 2-D QW after 14 iterations using the Grover coin, initial state (11) and square boundaries at M = Figure 8. The probability distribution of the 2-D QW after 1 iterations using the Grover coin, initial state (11) and square boundaries at M =. [Oliveira et al. 26], respectively with the Grover walk leading to the largest diffusion rate among all the coins considered [Tregenna et al. 23]. In the finite square lattices we observe that all the coins have an oscillatory behavior, besides that the Hadamard walk leads to the largest diffusion rate. Showing that the coins have different sensitivity to boundaries. Now the quantum walk has a finite two-dimensional diamond lattice with a Hilbert space H 4 H s, where H 4 is the coin space and H s is the lattice space. Thus the particle is confined to a diamond lattice whose vertices are { M,, M,,,M,, M } such that M is integer. In fig. 9 we show the evolution of the standard deviation for the Fourier walk in a diamond lattice. In the diamond lattice the Grover and Fourier walks have the greatest spreading rates while the Hadamard walk looses much acessible space thus its standard deviation decreases. As in the square lattices we observe that the standard deviation oscillates reaching the maximum value when the walk beats the boundaries and the minimum when the walk passes towards the origin. 4. Conclusions We have analyzed the the 2-D quantum walks in the infinite and finite lattices. We have used the Hadamard, Fourier and Grover coins, taking as initial condition the one 217

8 6 M=1 M=2 M=4 M= 5 4 σ Figure 9. Evolution of the standard deviation of the Fourier walk with diamond lattice for an initial state (9), in the cases M = 1, M = 2, M = 4, and M =. t that leads to a maximum rate of spread. We observed that the particle is reflected when it reaches some boundary and the walk presents an oscillatory behavior. We have obtained until now only the standard deviation of the walks, in order to comprehend fully the behavior of the quantum walk in finite lattices we will still study the mixing times and the average standard deviation of these walks. References Aharonov, Y., Davidovich, L., and Zagury, N. (1993). Quantum random walks. hys. Rev. A, 48(2): Berggren, K., A.F.Sadreev, and Starikov, A. A. (22). Crossover from regular to irregular behavior in current flow through open billiards. hysical Review E, 66: Farhi, E. and Gutmann, S. (1998). Quantum computation and decision trees. hys. Rev. A, 58: Mackay, T. D., Bartlett, S. D., Stephenson, L. T., and Sanders, B. C. (22). Quantum walks in higher dimensions. J. hys. A, 35:2745. Oliveira, A. C., ortugal, R., and Donangelo, R. (26). Decoherence in 2-d quantum walks. hysical Review A, 74: Shenvi, N., Kempe, J., and Whaley, K. (23). A quantum random walk search algorithm. hys. Rev. A, 67(5):5237. lanl-arxive quant-ph/264. Tregenna, B., Flanagan, W., Maile, R., and Kendon, V. (23). Controlling discrete quantum walks: coins and intitial states. New J. hys., 5:83. lanl-arxive quant-ph/