Spatial Search on Sierpinski Carpet Using Quantum Walk
aa r X i v : . [ qu a n t - ph ] J u l Spatial Search on Sierpinski Carpet Using Quantum Walk
Shu Tamegai, Shohei Watabe, and Tetsuro Nikuni
Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-9601, Japan
Abstract
We investigate a quantum spatial search problem on a fractal lattice. A recent study for the Sierpinskigasket and tetrahedron made a conjecture that the dynamics of the search on a fractal lattice is determined byspectral dimension. We tackle this problem for the Sierpinski carpet, and our simulation result supports theconjecture. We also propose a scaling hypothesis of oracle calls for the quantum amplitude amplification. ffi ciently find a marked objectfrom a huge database. The quantum spatial search algorithm evolves the state by alternating theoracle and quantum walk operators. In order to solve this spatial search problem, the number oforacle calls should be optimized to concentrate the probability amplitude at the marked vertex.Understanding the scaling behavior of the optimal oracle calls is an important problem, which hasbeen studied in a variety of regular lattice geometries [1–4]. The number of interval steps betweenthe peaks of the marked vertex probability is given by DN / D or π √ N / D is thedimension, and N the number of sites.In fractal geometry, there are three characteristic dimensions—the Euclidean dimension d E ,fractal (or the Hausdor ff ) dimension d f , and spectral (or the fracton) dimension d s —, so the ques-tion is which dimension determines the scaling behavior of the optimal number of oracle calls ina fractal lattice. Recently, an interesting conjecture for a fractal lattice was made by Patel andRaghunathan [5] that the scaling behavior of the spatial search is determined by the spectral di-mension d s , and neither by the Euclidean dimension d E nor by the fractal dimension d f . Thisconjecture was derived based on numerical studies for Sierpinski gasket ( d E =
2) and Sierpinskitetrahedron ( d E = d E = ff ective number of oracle calls for the quantum amplitude amplification.In the quantum spatial search, we employ the flip-flop walk [6] as the quantum walk, wherethe state | ψ ( t ) i ≡ P x , l a x , l ( t ) | ~ x i ⊗ | ˆ l i is constructed in the Hilbert space H search ≡ H N ⊗ H k . Here, | ~ x i ∈ H N is associated with the position degree of freedom, and | ˆ l i ∈ H k is associated with the link → → → FIG. 1. (Color online) Sierpinski carpets at various stages. The red vertex pointed by the arrow at eachstage represents the marked vertex. IG. 2. (Color online) Probability distribution for quantum spatial search on a Sierpinski carpet in the casewhen the amplitude distribution is concentrated toward the marked vertex. The data are for the stage S = P t o FIG. 3. (Color online) Time evolution of the marked vertex probability P ( ~ x = ~ , t ) for the quantum spatialsearch on the Sierpinski carpet. The data are for the stage S = degree of freedom [5]. In the case of Sierpinski carpets, we take the dimension of H k as k =
4, forsimplicity.The quantum spatial search algorithm is performed by the time evolution with alternately oper-ating the oracle R and the quantum walk W , i.e., | ψ ( t ) i = ( WR ) t | ψ ( t = i . The oracle R = R N ⊗ I k with R N ≡ I N − | ~ i h ~ | gives the maximum contrast between a marked vertex at the origin ~ x = ~ I N and I k are the identity operators in H N and H k , respectively. Thequantum walk operator W is composed of a Grover di ff usion operator G [7] and a shift opera-tor S , i.e., W = S G . The Grover di ff usion operator works as the inversion operator, given by [5] a x , l G −−→ (2 / k ) P l ′ a x , l ′ − a x , l ; in the case of the marked vertex, we do not operate the Grover di ff usionoperator, but the sign of the amplitude is flipped according to the oracle R [5]. The operator S shiftsthe amplitude along its link direction and reverses the link direction[5]: | ~ x i ⊗ | ˆ l i S −−→ | ~ x + ˆ l i ⊗ |− ˆ l i .If there is no vertex in the link direction ˆ l , we take | ~ x i ⊗ | ˆ l i S −−→ | ~ x i ⊗ | ˆ l i . For the time evolutiongoverned by the unitary operator W = S G [5], one may expect oscillation of the probability dis-tribution as a function of the time step. In the present quantum spatial search, it is customary tochoose the initial state to be a uniform superposition state: | ψ ( t = i = ( Nk ) − / P x , l | ~ x i ⊗ | ˆ l i .3
10 100 1000 10000 100000 10 100 1000 10000 100000 1×10 Q N FIG. 4. (Color online) Scaling of the optimal steps of the oracle calls Q for the spatial search on theSierpinski carpet. The linear fit is for the data from the stage S = S = P t FIG. 5. (Color online) Time evolution of return probability for a classical random-walker on a Sierpinskicarpet. The linear fit is for the data of the stage S = We numerically simulate the quantum spatial search algorithm on the Sierpinski carpet forstages S = N = P ( ~ x , t ) = P l | a x , l ( t ) | can be con-centrated at the marked vertex in the Sierpinski carpet (Fig. 2). We reasonably find the oscillationof the probability at the marked vertex P ( ~ x = ~ , t ) = P l | a , l ( t ) | (Fig. 3). Since the amplitudeis small, one may need the quantum amplitude amplification. We identify the periodicity as theoptimal time steps Q to the oracle calls, which is extracted by using the Fourier transformation ofthe oscillatory data of the marked vertex probability P ( ~ x = ~ , t ). In our numerical simulation, weemployed the number of steps to be approximately 1 000 000. The size dependence of the optimalnumber of oracle calls Q is shown in Fig. 4. Assuming Q ∝ N b , our fit gives the scaling behavior, Q = . N . , which gives the scaling exponent b = . Q . Assuming P ∝ N − a ,the fit of our data at stages S = P = . N − . with thescaling exponent a = . d E =
2. The fractal dimension4s defined as d f ≡ log M ( s ) / log s . Here, M ( s ) is the number of the self-similar pieces, and s thescale factor, where a line segment is broken into s -self-similar intervals with the same length. Inthe Sierpinski carpet, one has d f = ln 8 / ln 3 = .
892 789 · · · . The spectral dimension d s can bedefined by the scaling behavior of the return probability of the classical random walk [8], givenby P c ( ~ x = ~ , t ) ∝ t − d s / , where the classical random-walker starts from ~ x = ~ t =
0. Althoughthe spectral dimension for the Sierpinski gasket is simply given by d s = d E + / ln( d E + d s of the Sierpinski carpet. We,therefore, determine the spectral dimension from the numerical simulation of the classical randomwalk on the Sierpinski carpet. From a simple scaling fit of our numerical simulation, we find P c ( ~ , t ) = . t − . (Fig. 5), and thus, the spectral dimension of the Sierpinski carpet isgiven by d s = . .
673 7 · · · ≤ d s ≤ .
862 0 · · · [11].We compare the spectral dimension with the scaling behavior of the optimal number of oraclecalls Q . Using our numerical data, we find the approximate relation b ≃ / d s = . / d f = .
528 3 · · · .)Finally, we propose the scaling hypothesis for the e ff ective number of oracle calls [5] Q / √ P ∝ N b + a / ≡ N c for the quantum amplitude amplification. Our hypothesis is given by the relation c = d s / ( d E − + d f − s . (1)Using our data for the Sierpinski carpet ( s = . . s = d E = . .
950 17 · · · ,and for the Sierpinski tetrahedron ( d E = . .
773 70 · · · .Although no mathematically rigorous arguments exist, numerical results imply that the relevantscaling for the quantum spatial search may be given by Q ∝ N / d s , and Q / √ P ∝ N d s / ( d E − + d f − s .In this study, we investigated the spatial search problem on a Sierpinski carpet using quantumwalk. Our numerical simulation supports the recent conjecture that the scaling behavior of thequantum spatial search on a fractal lattice is determined by the spectral dimension, and not bythe fractal dimension for the optimal oracle calls. We also proposed the scaling hypothesis of thee ff ective number of oracle calls for the quantum amplitude amplification in a fractal lattice, whichholds in the Sierpinski carpet, gasket, and tetrahedron.5 CKNOWLEDGMENTS
S.W. was supported by JSPS KAKENHI Grant No. JP16K17774. T.N. was supported by JSPSKAKENHI Grant No. JP16K05504. [1] N. Shenvi, J. Kempe, and K. B. Whaley, Phys. Rev. A , 052307 (2003).[2] A. Patel, K. S. Raghunathan, and M. A. Rahaman, Phys. Rev. A , 032331 (2010).[3] G. Abal, R. Donangelo, F. L. Marquezino, and R. Portugal, Math. Struct. Comp. Sci. , 999 (2010).[4] A. Patel and M. A. Rahaman, Phys. Rev. A , 032330 (2010).[5] A. Patel and K. S. Raghunathan, Phys. Rev. A , 012332 (2012).[6] A. Ambainis, J. Kempe, and A. Rivosh, in Proceedings of ACM-SIAM SODA05 (ACM Press, NewYork, 2005), p. 1099.[7] L. K. Grover, Phys. Rev. Lett.
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