1 Introduction

In the past, database search for an unsorted database of *N* items has been
object of intense investigation [^{1}, ^{6}]. The efficiency of the algorithm consists in
finding the desired item in the minimum number as possible of queries made to the
database. Within conventional approaches a query is a binary oracle with outcomes
called YES and NOT. If one employs Boolean logic then to find the desired item in an
unsorted database takes on the average 〈*Q*〉=(*N*+1)/2
queries providing the search mechanism has a memory with which an item rejected once
is not picked up again.

With the help of the superposition of states of quantum mechanics, Grover discovered a search algorithm that reduces the number of necessary queries up to in a time [^{1}]. One can conclude that Grover’s algorithm is strictly of quantum nature and constitutes one of the greatest achievements of Quantum Computation. Such an algorithm is the optimal one for an unsorted database search [^{3}].

So far, several different experimental implementations of the algorithm have been done. For
instance in Ref. [^{7}] nuclear magnetic
resonance techniques with a solution of chloroform molecules were employed. In
[^{8}] lasers techniques were used for the
implementation of the algorithm while in [^{9}]
its optical implementation was made. The number of necessary queries
(*Q*) in the Grover’s algorithm is determined by the following
equation [^{1}]:

Since Eq. (1) does not have an integer solution, the quantum algorithm stops when this is
sufficiently close to the desired state which represents the target item. Thus, one
says that the desired item is found with high *probability*. In the
*n*−qubit implementation of the algorithm it is chosen
*N* = 2* ^{n}* while the items in the
unsorted database are labeled with binary digits. By the way, there exists a
Hamiltonian version of the algorithm where the discrete unitary oracle is replaced
with a continuous time interaction Hamiltonian that prevails along the entire
duration of the algorithm being the respective number of queries represented by the
time one has to wait for before finding the desired state encoding the target item
[

^{4}]. According to Grover’s algorithm [

^{2}] the starting and the target state are such that

where the index *a* labels the different items. Thus, the algorithm evolves the
starting state |*s*〉 towards the target state
|*t*〉 through the operators

in such a way that

The operator *U _{t}* plays the role of a binary oracle that flips the
sign of the target state. On the other hand, −

*U*accomplishes the reflection-in-the-mean operation. The number of necessary queries of Eq. (4) is given by the solution of Eq. (1) [

_{s}^{1}].

It is worth emphasizing that there is a semiclassical version of Grover’s algorithm called
wave database search [^{6}]. In the wave version
of the algorithm, the n qubits are replaced by *N* =
2* ^{n}* distinct wave modes without involving quantum
entanglement at any stage [

^{5}]. As we shall see lines below the absence of entanglement could be a serious deficiency of the wave model. The above is due that entanglement is a necessary ingredient for the quantum information processing [

^{10}]-[

^{11}]. By the way, the wave version of the unsorted database search has been already partially implemented from the experimental point of view through the use of classical optics where the respective oracle is constituted by a phase-shift plate [

^{12}].

An interesting version of the wave database search based on a coupled simple harmonic oscillators which is soluble within semiclassical domain was introduced in Ref. [^{6}]. However, the approach of [^{6}] does not include an essential quantum resource as it is entanglement.

The above contradicts the spirit of the original formulation of Grover’s algorithm while in such algorithm the initialization process starts with the following uniform superposition state

which is entangled. In fact, in Ref. [^{13}] the amount of
multipartite entanglement associated to multiqubit quantum states employed in the
Grover algorithm was calculated. Furthermore, the authors of [^{13}] prove that genuine multipartite entanglement depending on N
= 2* ^{n}* in a non trivial way is present at each
computational step of Grover’s algorithm.

In spite of the above, in the wave version of the unsorted database search algorithm based on a coupled simple harmonic oscillators introduced in [^{6}], entanglement is not taken into account. In the present paper we find that a consequence of the absence of entanglement in the wave version of the database search algorithm based on a coupled simple harmonic oscillators is that its prediction for the time of execution of Grover’s algorithm is . However, Grover’s algorithm predicts that such a time must be .

In order to be consistent with the time of execution of Grover’s algorithm, in the present work we introduce both a qualitative argument based on probabilistic considerations and a quantitative argument where entanglement is incorporated in an effective way in the coupled simple harmonic oscillators approach. As a consequence of the above we eliminate the contradiction between the times of execution of the wave version of the unsorted database search algorithm with Grover’s algorithm. We also remark that our effective approach is simpler than that of Ref. [^{13}] and it gives good results. In our quantitative argument, within the wave version of the unsorted database search algorithm we identify a quantity that can be interpreted as an effective measure of entanglement and to explore its different values until a consistency with the original algorithm is found.

The present effective approach suggests methods and strategies for future experimental implementations of the wave version of the unsorted database search inspired in the coupled simple harmonic oscillators approach. Our suggestion is to experimentally implement such an approach through a system of N quantum dots with a harmonic oscillator potential of confinement for each one of them.

On the other hand, each of the N vibrating quantum dots must be coupled to a central vibrating quantum dot. In order to conciliate the value of the respective time of execution as predicted by the wave version of the unsorted database search algorithm with the value of the time of execution of Grover’s algorithm, the parameters of the present wave model should be adjusted in the way indicated in the present work. The paper is organized as follows: In Section 1 we explain the wave version of the unsorted database search algorithm. Meanwhile, in Section 2 we discuss the validity of our arguments. Finally, in Section 3 we give a conclusions of the present formalism.

2 The Wave Version of the Database Search Algorithm Based on a Coupled Simple Harmonic Oscillators without Entanglement

According with the wave version of the data base search, the *N* different
items of the unsorted database are associated with *N* identical
harmonic of constant k and mass *m*. All of such *N*
identical oscillators are coupled to a single main oscillator of mass
*M* and constant *K* (see Figure 1 of [^{6}]). The absence of multipartite entanglement of
the semiclassical model is reflected in the fact that each oscillator is moving
independently from the resting *N* − 1 oscillators. However, it is
worth observing that the *N* different oscillators are coupled
through the driver oscillator of mass *M* and constant
*K*. In absence of entanglement both quantities
*M* and *K* do not depend on *N*
(the total number of oscillators). The Lagrangian in the center-of-mass coordinates
is

In such an approach it is considered exclusively the dynamics of the tapped oscillator by
assuming that the resting oscillators are coupled to this one through
*R*. Furthermore, in such a work the *N* − 2
resting oscillators are neglected in an effective way to assume that they decouple
from the relevant modes {*X*, *R*,
*x _{t}*}. The effective Lagrangian which does not
consider entanglement is then [

^{6}]

where , and
*y _{t}* =

*x*−

_{t}*R*

^{1}. Due to the decoupling to the relevant modes, in the above equation the modes different to the tapped one have been neglected.

The eigenvalues ω+, ω−, and ωt associated to 𝓛* ^{ne}* satisfy

From the above equation it can be concluded that in the limit of *N* large
enough, the time of execution of Grover’s algorithm is approximately

The above result contradicts the expected values of the time of execution of Grover’s algorithm which are [^{2}]. The contradiction in Eq. (9) consists in that this one predicts that for N small the times of execution would be very large and that for N large the time of execution would be very small.

By the way, Eq. (9) is a consequence of the absence of entanglement in the semiclassical wave version of the database search of Ref. [^{6}]. In the present paper by considering multipartite entanglement in the wave version of the unsorted database search algorithm, we eliminate the above contradiction through a two different strategies, one of the them is qualitative and the other quantitative.

3 Consideration of Multipartite Entanglement in the Wave Version of the Unsorted Database Search Algorithm Based on a Coupled Simple Harmonic Oscillators

In this section we restore the correct values of the times of execution of Grover’s algorithm within the semiclassical wave version of the unsorted database search algorithm by employing a two different approaches, one of them qualitative and the other quantitative.

3.1 Qualitative Approach

As we commented lines above, Grover’s algorithm starts with the entangled state of Eq. (5).
In other words, the uniform superposition of the different states plays a
fundamental role for the efficient execution of the algorithm. In spite of the
above, in the typical version of the wave model of the unsorted data base search
it is considered exclusively one single mode (*x _{t}*)
assuming that the

*N*− 1 remaining decouples from the relevant modes {

*X*,

*R*,

*x*}. It is worth mentioning at this stage that the target mode can be any of the

_{t}*N*modes with equal probability of occurrence.

Furthermore, multipartite entanglement demands that the modes {*X*,
*R*, *x*_{1},
*x*_{2}, ..., *x _{N}*} does
not disentangle each other. In certain sense they are mutually coupled and all
of the modes {

*x*

_{1},

*x*

_{2}, ...,

*x*} participate on the same basis each of them in the evolution of the algorithm. In such a sense it is necessary a factor of N in Eq. (9). Thus, to multiply Eq. (9) by

_{N}*N*we obtain that to consider entanglement in the wave version of the unsorted data base search, the predicted time of execution of Grover’s algorithm is

which is consistent with Grover’s algorithm [^{1}], [^{2}]. In the above we have used the fact that Grover’s algorithm is of a probabilistic nature. Likewise, we have assumed that all of the N different modes are entangled. Hence each of the different modes necessarily participate as tapped and do it with the same probability of occurrence.

3.2 Quantitative Approach

Within the wave version of the unsorted data base search algorithm all of the
*N* different oscillators of mass m each are coupled to the
mass *M*. We identify the constant *K* of the
oscillator of mass *M* as the driver of the coordination of the
movement of the *N* different oscillators. Therefore, we can
think of the constant *K*/*M* as an indirect
measurement of the multipartite “entanglement” of the different
*N* oscillators of mass m. If we observe Eqs. (8), we can see
that there is an effective time for the execution of the database search
algorithm in its semiclassical wave approach. Namely,

where ω± are given by Eq. (8). In the above, the times of execution depend on the
significative driver parameter *K*/*M* which can
be considered as an indirect measurement of the multipartite “entanglement”
associated to the different *N* oscillators of mass
*m*.

The other quantity appearing in Eq. (11) being *K*/*M* is
related to the size of the database. By varying the driver parameter
*K*/*M* in Eq. (11), we can find those range
of values for which the times of Eq. (11) are consistent with the time of
execution of Grover’s algorithm, that is

In Figure 1 it is plotted the quantity
*T _{tot}* of Eq. (11) as a function of the two
following parameters

*K*/

*M*and

*N*/

*M*. As we can appreciate from Figure 1, still there is a range of values of the parameter

*K*/

*M*for which it is satisfied Eq. (12). Due that in the present work the parameter

*K*/

*M*is identified as an indirect measure of the degree of entanglement of the

*N*different oscillators, we can conclude that to incorporate entanglement in an effective way into the wave version of the unsorted database search algorithm it is restored the value of the time of execution of Grover’s algorithm.

4 Conclusions

Along the present work we have assumed that the mass and the constants of all of the harmonic
oscillators have the same values, that is *m* = 1 and
*k* = 1. On the other hand, multipartite entanglement between the
*N* harmonic oscillators and the control oscillator of mass
*M* requires that the constant *K* must be large
enough. The later imply that any change in the state of any of the N harmonic
oscillator would modify the states of both the control oscillator and the other
*N* −1 oscillators. Due that there are in total
*N* + 1 oscillators, it is assumed that *K* ∼
(*N* + 1).

On the other hand, it is also assumed that all the *N* +1 oscillators have the
same mass, that is, *M* = *m* = 1. With such an
assumptions we found that in the limit of *N* large enough, the
quantity in Eq. (9)
is which makes that
the wave version of the unsorted data base search algorithm contradicts Grover’s
algorithm as it was already commented lines below to such equation. Such a fail of
the wave version of the unsorted database search algorithm is eliminated if one
observes that Grover’s algorithm has an intrinsic probabilistic structure. Hence,
the target mode which decouples from the rest of the harmonic oscillators can be any
of the *N* different modes with the same probability. Therefore, it
is necessary to include a factor of *N* in the Eq. (9). With that
factor, it is restored the correct time of execution of Grover’s algorithm which is
At this stage we
note that there is an overhead in terms of the number of oscillators with respect to
the quantum case. The above reflects a problem of memory space in the wave model
approach.

The approach considered in the present paper concludes that entanglement is a necessary
ingredient for the simulation of Grover’s algorithm through the wave version of the
unsorted data base search algorithm based on a coupled classical simple harmonic
oscillators. The above follows from our identification of the parameter
*K*/*M* as and indirect measure of entanglement of
the *N* harmonic oscillators. Thus, from Figure 1 we can appreciate that for certain values of such a
parameter, Eq. (12) is satisfied, with which the time of execution of Grover’s
algorithm is restored from the wave version of the unsorted data base search
algorithm.

In the present work, it is suggested that the our approach can be experimentally implemented
through the use of a system of *N* vibrating quantum dots with a
harmonic oscillator confinement potential for each of the dots. The
*N* different vibrating quantum dots must be coupled to a main
vibrating quantum dot. The value of the time of execution of the wave version of the
unsorted database search algorithm is conciliated with the value of the time of
execution of Grover’s algorithm if the coupling constants of the vibrating quantum
dots are adjusted in the way indicated in Section 2 of the present work. With the
above, the wave version of the unsorted database search algorithm becomes both a
legitimate and an efficient quantum algorithm.