arXiv:quant-ph/0608019v1 1 Aug 2006 Classical search algorithm with resonances in √ N cycles A. Romanelli * and R. Donangelo † Instituto de F´ ısica, Facultad de Ingenier´ ıa Universidad de la Rep´ ublica C.C. 30, C.P. 11000, Montevideo, Uruguay (Dated: October 8, 2018) In this work we use the wave equation to obtain a classical analog of the quantum search algorithm and we verify that the essence of search algorithms resides in the establishment of resonances between the initial and the serched states. In particular we show that, within a set of N vibration modes, it is possible to excite the searched mode in a number of steps proportional to √ N . Keywords: search algorithm; quantum optics; quantum information I. INTRODUCTION It has been shown that a quantum search algorithm is able to locate a marked item from an unsorted list of N elements in a number of steps proportional to √ N , instead of proportional to N as is the case for the usual algorithms employed in classical computation. The most well studied quantum search algorithm is the one due to Grover [1], where the search is performed by alternatively shifting the phase of the searched for state, and amplify- ing its modulus. A continuous time version of the original Grover algorithm has been described by Farhi and Gut- mann [2]. In a recent work [3], we have presented an alternative continuous time quantum search algorithm, in which the search set is taken from the set of eigenvec- tors of a particular hamiltonian. The search is performed through the application of a perturbation which leads to the appearance of a resonance between the initial and the searched state. That study also provided a new in- sight on the connections between discrete and continuous time search algorithms. Our search algorithm can be im- plemented using any Hamiltonian with a discrete energy spectrum. However, we would like to emphasize that the possibility of establishing a resonance between two states is an intrinsic property of oscillatory motions in general and not an exclusive property of quantum mechanics as described through the Schr¨ odinger equation, as in our algorithm. A link between quantum computation and classi- cal optical waves has been well established by several authors[4, 5, 6, 7, 8, 9]. It is possible to simulate the behavior of some simple quantum computers using clas- sical optical waves. Although the necessary hardware grows exponentially with the number of qubits that are simulated, and thus these simulations are not efficient, optical simulations could still be very useful. In fact, * Corresponding author.E-mail: alejo@fing.edu.uy † Permanent address: Instituto de F´ ısica, Universidade Federal do Rio de Janeiro C.P. 68528, 21945-970 Rio de Janeiro,Brazil as some simulations of quantum algorithms employ op- tical simulations, a classical analogue of such a quantum search algorithm might be a valuable tool to test the functioning of the optical system as a computer. In this work we present a continuous time search algo- rithm, which is controlled by a classical wave equation, showing explicitly, once again, that the search algorithm is essentially a resonance phenomenon between the initial and the searched states [3, 10]. The paper is organized as follows: in the next section we briefly describe our quan- tum search algorithm. Then, in section 3, we develop the search model using the ordinary wave equation. In section 4 we present results obtained with this model. We end by discussing these results and extracting con- clusions, in the last section of this work. II. QUANTUM SEARCH ALGORITHM We consider a continuous time quantum search algo- rithm which is controlled by a time dependent Hamil- tonian H . The wavefunction |Ψ(t)〉 satisfies the Schr¨ odinger equation i ∂ |Ψ(t)〉 ∂t = H |Ψ(t)〉, (1) where H = H 0 + V (t) and we have taken the Planck constant  = 1. Here H 0 is a known nondegenerate time-independent Hamiltonian with a discrete energy spectrum {ε n } and eigenstates {|n〉}. V (t) is a time- dependent potential that shall be defined below. We then consider a subset N of {|n〉} , formed by N states, which will be our search set. Let us call |s〉 the unknown searched state in N whose energy ε s is given. We assume that it is the only state in N with that value of the energy. Therefore, knowing ε s is, in our algorithm, equivalent to the action of “marking” the searched state, in Grover’s algorithm. The potential V that produces the coupling between the initial and the searched states, is defined as [3] V (t)= |p〉〈j | exp (iω sj t)+ |j 〉〈p| exp (−iω sj t) (2)