# Multi-Qubit Dynamical Quantum Search Algorithm with Dissipation

A. H. Homid1,3,111E-mail<EMAIL_ADDRESS>M. Abdel-Aty1 and A.-S. F.
Obada2
1 University of Science and Technology, Zewail City for Science and
Technology, Giza, Egypt
2 Faculty of Science, Al-Azhar University, Nasr city 11884, Cairo, Egypt
3Faculty of Science, Al-Azhar University, Assiut 71524, Egypt

###### Abstract

We invoke an efficient search algorithms as a key challenge in multi-qubit
quantum systems. An original algorithm called dynamical quantum search
algorithm from which Grover algorithm is obtained at a specified time is
presented. This algorithm is distinguished by accuracy in obtaining high
probability of finding any marked state in a shorter time than Grover
algorithm time. The algorithm performance can be improved with respect to the
different values of the controlled phase. A new technique is used to generate
the dynamical quantum gates in the presence of dissipation effect that helps
in implementing the current algorithm.

##  1 Introduction

In the early 1996s, Grover search algorithm (GA) has attracted much interest,
since it could work in quantum computer. The quantum search, from
computational point of view, is proved to get approximately $O(\sqrt{N})$
operations (in comparison with the $O(N)$ classical operations), which
indicates a quadratic speedup [1, 2]. Many efforts have been devoted to
achievement of this algorithm theoretically and experimentally by using
superconducting qubits [3], trapped ions [4], atom cavity QED systems [5, 6]
and nuclear magnetic resonance [7]. Also, most previous studies for improving
the quantum GA were general and do not depend on a physical system. Further,
all previous implementations of quantum GA have been developed depending on
Grover visualizations, where the control phase parameter for the controlled
phase gates or oracle is switched to $\pi$ or $\frac{\pi}{3}$ [2, 8]. However,
an interesting work by Long et al [9] has replaced the Grover’s arbitrary
phase by $\frac{\pi}{2}$. The combination of strong resonant interactions with
tight subwavelength traps gives an excellent opportunity to implement much
faster quantum gates. However, there is no way to achieve single site
addressability in those systems. Existing proposals circumventing this problem
[2, 8] do not consider an arbitrary controlled phase parameter or dispassion
effect, as the interaction need not to be restricted by certain values of the
phase control.
In this article, we propose to add new factors allowing for significant
speedup. By combining a multi-qubit system with dissipation taking into
account an arbitrary phase control, we can achieve a fast search in a
realistic setting. We assume that the control phase is time-dependent to
realize the dynamical quantum controlled phase gates. It is shown that the
involved time is the crucial factor for controlling and searching of any
marked state regardless the number of qubits. Consequently, we define a new
algorithm called a dynamical quantum search algorithm (DQSA). The proposed
algorithm is a generalization of Grover algorithm and (i) gives a higher
probability of finding any marked state in shorter time, (ii) there is no
restriction on the controlled phase parameter and (iii) the presence of qubits
dissipation is allowed.

##  2 Quantum Search Algorithm with Dissipation

The effect of a thermal bath has been used to improve the performance of a
quantum adiabatic search algorithm [10]. Vega et al [11] compared the effects
of the field dissipation on the algorithm performance with those of a
structured environment similar to the one encountered in systems coupled to an
electromagnetic field that exists within a photonic crystal. They show that,
the algorithm performance has been improved by tuning the environment
parameters to certain regimes [12]. We consider the problem with a different
perspective, where the algorithm may be implemented within a system taking
into account the qubit dissipation.

Figure 1: Schematic diagram of quantum dots ensemble. A single surface plasmon
injected from outside is coherently scattered by the dots. Our interest is to
consider all the possible interactions between any arbitrary pair or more
inside the sample. The dissipation effect resulted from the direct contact
between the qubits and the plasmone surface.

Here, we consider a dense multi-qubit system sample interacting with a single
surface plasmon injected from outside. In this case, we consider all the
possible interactions between any arbitrary pair or more inside the sample
(see Fig.1). The direct contact between the qubits and the plasmone surface
yields dissipation effects. As a consequence, we assume that the dipole-dipole
interaction potential is nonidentical for all qubits. Other appropriate
systems could be superconducting qubits [13, 14] or double quantum dot
excitonic systems [15]. Such a model can be written as follows

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle-\lambda\Big{\\{}\sum_{s_{1}=1}^{N}J_{s_{1}}\,\hat{\sigma}^{s_{1}}_{z}+\sum_{s_{1},s_{2}=1}^{N}J_{s_{1}s_{2}}\,\hat{\sigma}^{s_{1}}_{z}\hat{\sigma}^{s_{2}}_{z}+\sum_{s_{1},s_{2},s_{3}=1}^{N}J_{s_{1}s_{2}s_{3}}\,\hat{\sigma}^{s_{1}}_{z}\hat{\sigma}^{s_{2}}_{z}\hat{\sigma}^{s_{3}}_{z}$
(1)
$\displaystyle+...+\sum_{s_{1},s_{2},s_{3},...,s_{n}=1}^{N}J_{s_{1}s_{2}s_{3}...s_{n}}\,\hat{\sigma}^{s_{1}}_{z}\hat{\sigma}^{s_{2}}_{z}\hat{\sigma}^{s_{3}}_{z}...\hat{\sigma}^{s_{n}}_{z}\Big{\\}}-i\sum_{v=1}^{N}\frac{\gamma_{v}}{2}\,\hat{\sigma}_{+}^{v}\hat{\sigma}_{-}^{v},$

where $S=\\{s_{1},s_{2},...,s_{n}\\}$ refer to the labels of the qubits, $N$
is the number of qubits, $\gamma_{u}$ are the dissipation rates of different
qubits, $J_{s_{1}}$,…, $J_{s_{n}}$ are the energies biases, $\lambda$ is the
coupling constant and $J_{s_{1}s_{2}}$, $J_{s_{1}s_{2}s_{3}}$,
$J_{s_{1}s_{2}s_{3}s_{4}}$, …, $J_{s_{1}...s_{n}}$ are the interaction
energies between two qubits or more. In the proposed model, the interaction
energy between qubits is set to provide the maximum ferromagnetic coupling.
Here, we consider the number of qubits is $N=9$ and
$S=\\{s_{1},s_{2},...,s_{9}\\}$. To realize the dynamics of the controlled
phase gates for different qubits, we write, for simplicity, that the sub-
indexes $s_{1}\equiv s$, $s_{2}\equiv j$, $s_{3}\equiv k$ and $s_{4}\equiv l$,
$s_{5}\equiv m$, $s_{6}\equiv n$ and so on.
First, we use the case of $N=2$ and $S=\\{s_{1},s_{2}\\}$. Hence, the time
evolution of system (1) in these cases is given by $\hat{U}_{2}(t)$.
Therefore, we can realize a two-qubit dynamical controlled phase gates in the
presence of qubits dissipation in the following context.

  * •

If $J_{1}=J_{2}=\pm(J_{11}+J_{22})$ and $J_{12}=J_{21}=\pm\frac{1}{2}J_{1}$,
we get $\hat{U}_{2}(t)\rightarrow\hat{P}_{ee}(t;\gamma_{1},\gamma_{2})$ and
$\hat{U}_{2}(t)\rightarrow\hat{P}_{gg}(t;\gamma_{1},\gamma_{2})$.

  * •

If $J_{1}=\pm(J_{11}+J_{22})$, $J_{2}=-J_{1}$ and
$J_{12}=J_{21}=\pm\frac{1}{2}J_{2}$, we obtain
$\hat{U}_{2}(t)\rightarrow\hat{P}_{eg}(t;\gamma_{1},\gamma_{2})$ and
$\hat{U}_{2}(t)\rightarrow\hat{P}_{ge}(t;\gamma_{1},\gamma_{2})$.

Second, we use the case of $N=3$ and $S=\\{s_{1},s_{2},s_{3}\\}$. In this
case, it is assumed that (i) $J_{kkj}=-J_{kjk}$, $J_{jkk}=-\frac{1}{2}J_{jjj}$
and $J_{kj}=J_{jk}$, $\,\forall\,j\neq k$; (ii) $F_{sjk}=J_{ss}+J_{jj}+J_{kk}$
and $J_{sjk}=w_{1}$, $\,\forall\,s\neq j\neq k$. Hence, after applying these
steps, the time evolution of system (1) is given by $\hat{U}_{3}(t)$. So, we
can realize three-qubit dynamical controlled phase gates in the presence of
qubits dissipation as follows.

  * •

If $J_{s}=\pm F_{sjk}$, $J_{jk}=\pm\frac{1}{2}J_{s}$ $(\forall\,s\neq j\neq
k)$ and $w_{1}=\pm\frac{1}{6}F_{sjk}$, we get
$\hat{U}_{3}(t)\rightarrow\hat{P}_{eee}(t;\gamma_{1},\gamma_{2},\gamma_{3})$
and
$\hat{U}_{3}(t)\rightarrow\hat{P}_{ggg}(t;\gamma_{1},\gamma_{2},\gamma_{3})$.

  * •

If $J_{1}=J_{2}=\pm F_{sjk}$, $J_{3}=\mp F_{sjk}$,
$J_{12}=\pm\frac{1}{2}J_{1}$, $J_{13}=J_{23}=\mp\frac{1}{2}J_{2}$ and
$w_{1}=\mp\frac{1}{6}F_{sjk}$ we obtain
$\hat{U}_{3}(t)\rightarrow\hat{P}_{eeg}(t;\gamma_{1},\gamma_{2},\gamma_{3})$
and
$\hat{U}_{3}(t)\rightarrow\hat{P}_{gge}(t;\gamma_{1},\gamma_{2},\gamma_{3})$.

  * •

If $J_{1}=J_{3}=\pm F_{sjk}$, $J_{2}=\mp F_{sjk}$,
$J_{12}=J_{23}=\pm\frac{1}{2}J_{2}$, $J_{13}=\pm\frac{1}{2}J_{3}$ and
$w_{1}=\mp\frac{1}{6}F_{sjk}$, then
$\hat{U}_{3}(t)\rightarrow\hat{P}_{ege}(t;\gamma_{1},\gamma_{2},\gamma_{3})$
and
$\hat{U}_{3}(t)\rightarrow\hat{P}_{geg}(t;\gamma_{1},\gamma_{2},\gamma_{3})$.

  * •

If $J_{1}=\pm F_{sjk}$, $J_{2}=J_{3}=\mp F_{sjk}$,
$J_{12}=J_{13}=\pm\frac{1}{2}J_{3}$, $J_{23}=\pm\frac{1}{2}J_{1}$ and
$w_{1}=\pm\frac{1}{6}F_{sjk}$, one obtains
$\hat{U}_{3}(t)\rightarrow\hat{P}_{egg}(t;\gamma_{1},\gamma_{2},\gamma_{3})$
and
$\hat{U}_{3}(t)\rightarrow\hat{P}_{gee}(t;\gamma_{1},\gamma_{2},\gamma_{3})$.

Third, when $N=4$ and $S=\\{s_{1},s_{2},s_{3},s_{4}\\}$, and assume that (i)
$J_{jsks}=-J_{sjsk}$ and $J_{sjks}=-J_{jssk}$, $\forall\,s\neq j\neq k$; (ii)
$J_{kss}=-\frac{1}{3}J_{kkk}$, $J_{ssk}=-J_{sks}$ and $J_{sksk}=-J_{skks}$,
$\forall\,s\neq k$; (iii) $J_{jjkk}=0$, $J_{jk}=J_{kj}$ and
$J_{ssjk}=-J_{jkss}$, $\forall\,j\neq k$, the time evolution of system (1) is
given by $\hat{U}_{4}(t)$. After that, we write (i)
$E_{sjkl}=\big{(}J_{ss}+J_{ssss}+J_{jj}+J_{jjjj}+J_{kk}+J_{kkkk}+J_{ll}+J_{llll}\big{)}$,
$\forall\,s\neq j\neq k\neq l$; (ii) $w_{1}=J_{sjk}\,(s,j,k=1,2,3)$,
$w_{2}=J_{sjk}\,(s,j,k=1,2,4)$, $w_{3}=J_{sjk}\,(s,j,k=1,3,4)$ and
$w_{4}=J_{sjk}\,(s,j,k=2,3,4)$, $\,\forall\,s\neq j\neq k$; (iii)
$\frac{1}{6}E_{sjk4}=a_{1}$, $\frac{1}{6}E_{sjk3}=a_{2}$,
$\frac{1}{6}E_{sjk2}=a_{3}$ and $\frac{1}{6}E_{sjk1}=a_{4}$. Now, one can
realize the four-qubit dynamical controlled phase gates in following.

  * •

If $J_{sjkl}=\frac{1}{24}E_{sjkl}$, $J_{s}=\pm E_{sjkl}$,
$J_{jk}=\pm\frac{1}{2}J_{s}$ $(\forall\,s\neq j\neq k)$, $w_{1}=\pm a_{1}$,
$w_{2}=\pm a_{2}$, $w_{3}=\pm a_{3}$ and $w_{4}=\pm a_{4}$, one obtains
$\hat{U}_{4}(t)\rightarrow\hat{P}_{eeee}(t;\gamma_{1},...,\gamma_{4})$ and
$\hat{U}_{4}(t)\rightarrow\hat{P}_{gggg}(t;\gamma_{1},...,\gamma_{4})$.

  * •

If $J_{sjkl}=-\frac{1}{24}E_{sjkl}$, $J_{1}=J_{2}=J_{3}=\pm E_{sjkl}$,
$J_{4}=\mp E_{sjkl}$, $J_{12}=J_{13}=J_{23}=\pm\frac{1}{2}J_{2}$,
$J_{14}=J_{24}=J_{34}=\mp\frac{1}{2}J_{3}$, $w_{1}=\pm a_{1}$, $w_{2}=\mp
a_{2}$, $w_{3}=\mp a_{3}$ and $w_{4}=\mp a_{4}$, then
$\hat{U}_{4}(t)\rightarrow\hat{P}_{eeeg}(t;\gamma_{1},...,\gamma_{4})$ and
$\hat{U}_{4}(t)\rightarrow\hat{P}_{ggge}(t;\gamma_{1},...,\gamma_{4})$.

  * •

If $J_{sjkl}=-\frac{1}{24}E_{sjkl}$, $J_{1}=J_{2}=J_{4}=\pm E_{sjkl}$,
$J_{3}=\mp E_{sjkl}$, $J_{12}=J_{14}=J_{24}=\mp\frac{1}{2}J_{3}$,
$J_{13}=J_{23}=J_{34}=\pm\frac{1}{2}J_{3}$, $w_{1}=\mp a_{1}$, $w_{2}=\pm
a_{2}$, $w_{3}=\mp a_{3}$ and $w_{4}=\mp a_{4}$, obtain
$\hat{U}_{4}(t)\rightarrow\hat{P}_{eege}(t;\gamma_{1},...,\gamma_{4})$ and
$\hat{U}_{4}(t)\rightarrow\hat{P}_{ggeg}(t;\gamma_{1},...,\gamma_{4})$.

  * •

If $J_{sjkl}=\frac{1}{24}E_{sjkl}$, $J_{1}=J_{2}=\pm E_{sjkl}$,
$J_{3}=J_{4}=\mp E_{sjkl}$, $J_{12}=J_{34}=\pm\frac{1}{2}J_{1}$,
$J_{13}=J_{14}=J_{23}=J_{24}=\pm\frac{1}{2}J_{4}$, $w_{1}=\mp a_{1}$,
$w_{2}=\mp a_{2}$, $w_{3}=\pm a_{3}$ and $w_{4}=\pm a_{4}$, then
$\hat{U}_{4}(t)\rightarrow\hat{P}_{eegg}(t;\gamma_{1},...,\gamma_{4})$ and
$\hat{U}_{4}(t)\rightarrow\hat{P}_{ggee}(t;\gamma_{1},...,\gamma_{4})$.

  * •

If $J_{sjkl}=-\frac{1}{24}E_{sjkl}$, $J_{1}=J_{3}=J_{4}=\pm E_{sjkl}$,
$J_{2}=\mp E_{sjkl}$, $J_{12}=J_{23}=J_{24}=\mp\frac{1}{2}J_{4}$,
$J_{13}=J_{14}=J_{34}=\pm\frac{1}{2}J_{1}$, $w_{1}=\mp a_{1}$, $w_{2}=\mp
a_{2}$, $w_{3}=\pm a_{3}$ and $w_{4}=\mp a_{4}$, one obtains
$\hat{U}_{4}(t)\rightarrow\hat{P}_{egee}(t;\gamma_{1},...,\gamma_{4})$ and
$\hat{U}_{4}(t)\rightarrow\hat{P}_{gegg}(t;\gamma_{1},...,\gamma_{4})$.

  * •

If $J_{sjkl}=\frac{1}{24}E_{sjkl}$, $J_{1}=J_{3}=\pm E_{sjkl}$,
$J_{2}=J_{4}=\mp E_{sjkl}$, $J_{12}=J_{14}=J_{23}=J_{34}=\pm\frac{1}{2}J_{4}$,
$J_{13}=J_{24}=\mp\frac{1}{2}J_{4}$, $w_{1}=\mp a_{1}$, $w_{2}=\pm a_{2}$,
$w_{3}=\mp a_{3}$ and $w_{4}=\pm a_{4}$, we get
$\hat{U}_{4}(t)\rightarrow\hat{P}_{egeg}(t;\gamma_{1},...,\gamma_{4})$ and
$\hat{U}_{4}(t)\rightarrow\hat{P}_{gege}(t;\gamma_{1},...,\gamma_{4})$.

  * •

If $J_{sjkl}=\frac{1}{24}E_{sjkl}$, $J_{1}=J_{4}=\pm E_{sjkl}$,
$J_{2}=J_{3}=\mp E_{sjkl}$, $J_{12}=J_{13}=J_{24}=J_{34}=\pm\frac{1}{2}J_{2}$,
$J_{14}=J_{23}=\pm\frac{1}{2}J_{4}$, $w_{1}=\pm a_{1}$, $w_{2}=\mp a_{2}$,
$w_{3}=\mp a_{3}$ and $w_{4}=\pm a_{4}$, then
$\hat{U}_{4}(t)\rightarrow\hat{P}_{egge}(t;\gamma_{1},...,\gamma_{4})$ and
$\hat{U}_{4}(t)\rightarrow\hat{P}_{geeg}(t;\gamma_{1},...,\gamma_{4})$.

  * •

If $J_{sjkl}=-\frac{1}{24}E_{sjkl}$, $J_{1}=\pm E_{sjkl}$,
$J_{2}=J_{3}=J_{4}=\mp E_{sjkl}$, $J_{12}=J_{13}=J_{14}=\mp\frac{1}{2}J_{1}$,
$J_{23}=J_{24}=J_{34}=\mp\frac{1}{2}J_{2}$, $w_{1}=\pm a_{1}$, $w_{2}=\pm
a_{2}$, $w_{3}=\pm a_{3}$ and $w_{4}=\mp a_{4}$, one obtains
$\hat{U}_{4}(t)\rightarrow\hat{P}_{eggg}(t;\gamma_{1},...,\gamma_{4})$ and
$\hat{U}_{4}(t)\rightarrow\hat{P}_{geee}(t;\gamma_{1},...,\gamma_{4})$.

In the above notations (First, Second and Third) the top sign is used to
realize the first part while the lower sign is used to realize the second
part.
Using the same technique, one realizes the dynamical controlled phase gates in
the presence of the dissipation for the case of $N=5,6,7,8$ and $9$. The
calculations of the dynamics gates when $N=5,6,7,8$ and $9$ lengthy and are
not presented here. But we present some results of application of these
dynamic controlled phase gates for different qubits in calculating the
probabilities of marked and un-marked states in the presence of dissipation,
see appendix. If $\gamma_{i}=0,i=1,2,...,9$, one obtains the ideal dynamical
controlled phase gates. The dynamical conditional phase gates for $N$ qubits
is governed by a dynamical control parameter phase

$\beta_{N}(t)=\frac{2^{N}\theta_{N}}{\hbar}\lambda t.$ (2)

The parameter $\theta_{N}$ for different qubits is given by:

$\displaystyle\begin{array}[]{ll}\theta_{2}=\sum_{r=1}^{2}J_{rr},\,\theta_{3}=\theta_{2}+J_{33},\,\theta_{4}=\sum_{r=1}^{4}(J_{rr}+J_{rrrr}),\\\
\theta_{5}=\theta_{4}+J_{55},\,\theta_{6}=\sum_{r=1}^{6}(J_{rr}+J_{rrrr}+J_{rrrrrr}),\,\theta_{7}=\theta_{6}+J_{77},\\\
\theta_{8}=\sum_{r=1}^{8}(J_{rr}+J_{rrrr}+J_{rrrrrr}+J_{rrrrrrrr})\,\,\textrm{and}\,\,\theta_{9}=\theta_{8}+J_{99}.\end{array}$

For an individual qubit and based on the available experimental data as given
in [16, 17], we assume that the value of $\lambda J_{s}=\theta_{N}$
$(s=1,2,3,...,N)$ to realize one qubit gate in the presence of dissipation.
Now, one can realize a one-qubit gate $\hat{W}(\gamma_{s})$ in the following
context. The time evolution of system under $\hat{H}_{s}=-\lambda
J_{s}\hat{\sigma}_{z}^{s}$ is given by $\hat{V}_{1}^{s}(t)$, and the time
evolution of system under
$\hat{H}^{s}=\exp(i\pi\hat{\sigma}_{y}^{s}/4)\hat{H}_{s}\exp(-i\pi\hat{\sigma}_{y}^{s}/4)-\frac{i\gamma_{s}}{2}\,\hat{\sigma}_{+}^{s}\hat{\sigma}_{-}^{s},$
is given by $\hat{V}_{2}^{s}(t)$. Then, it is concluded that

$\displaystyle\hat{W}(\gamma_{s})$ $\displaystyle=$ $\displaystyle
e^{\frac{i\pi}{2}}\hat{V}_{1}^{s}\Big{(}\frac{\pi\hbar}{4\theta_{N}}\Big{)}\hat{V}_{2}^{s}\Big{(}\frac{\pi\hbar}{4\xi_{s}}\Big{)}\hat{V}_{1}^{s}\Big{(}\frac{\pi\hbar}{4\theta_{N}}\Big{)}=\frac{1}{\sqrt{2}}\exp(\frac{-\pi\gamma_{s}}{16\xi_{s}})\Big{\\{}(1+\frac{\gamma_{s}}{4\xi_{s}})|g_{s}\rangle\langle
g_{s}|$ (4) $\displaystyle+\frac{\theta_{N}}{\xi_{s}}|g_{s}\rangle\langle
e_{s}|+\frac{\theta_{N}}{\xi_{s}}|e_{s}\rangle\langle
g_{s}|-(1-\frac{\gamma_{s}}{4\xi_{s}})|e_{s}\rangle\langle e_{s}|\Big{\\}},$

with $4\xi_{s}=\sqrt{16\theta_{N}^{2}-\gamma_{s}^{2}}$.
The current algorithm has two registers, the $N$ qubits in the first register
and one qubit in the second register. Therefore, the proposed system contains
$N+1$ qubits. Initialize an $N+1$ qubits system to the state
$|g_{1},g_{2},...,g_{N}\rangle$ $|e_{N+1}\rangle$, where the state in the
second register refers to the excited state of an auxiliary working qubit.
Now, we implement the dynamical quantum search algorithm in the presence of
dissipation as follows: Performing $\hat{W}^{\otimes
N}(\gamma_{1},\gamma_{2},...,\gamma_{N})$ on the first register and performing
$\hat{W}(\gamma_{N+1})$ on the second register. Then, we apply the dynamical
controlled phase gates $\hat{P}_{x}(t;\gamma_{1},...,\gamma_{N})$ on the first
register ( where $x\in\Lambda$ and $\Lambda=\\{g_{1}g_{2}...g_{N-1}g_{N}$,
$g_{1}g_{2}...g_{N-1}e_{N}$, $g_{1}g_{2}...g_{N-2}e_{N-1}g_{N}$, …,
$e_{1}e_{2}...e_{N-1}g_{N}$, $e_{1}e_{2}...e_{N-1}e_{N}\\}$) and apply
$\hat{W}^{-1}(\gamma_{N+1})$ on the second register. Consequently, it is to
observe the state of $\Lambda$ which has been stored in the data qubits before
applying the dynamical controlled phase gates or oracles. Also, we note that
after applying this step the dynamical controlled phase gates have effects
only on the states to be searched in the first register and the state of the
second register does not change. So, we can omit the auxiliary working qubit
or one can discard the state of the second register at this point via
normalization of this state. Finally, we apply a new dynamical diffusion
transform matrix,
$\hat{D}(t;\gamma_{1},...,\gamma_{N})=\exp(i\beta_{N}(t))\hat{W}^{\otimes
N}(\gamma_{1},...,\gamma_{N})\hat{P}_{g_{1}g_{2}...g_{N}}(t;\gamma_{1},...,\gamma_{N})\hat{W}^{\otimes
N}(\gamma_{1},...,\gamma_{N})$, on the first register and measure the
resulting state. Consequently, the final formula of the dissipation DQSA is
given by:

$\mathop{\mathrm{DQSA}}\nolimits=\prod_{i=1}^{N-1}\Big{\\{}\hat{D}(t;\gamma_{1},...,\gamma_{N})\hat{P}_{x}(t;\gamma_{1},...,\gamma_{N})\Big{\\}}^{N-i}\hat{W}^{\otimes
N}(\gamma_{1},...,\gamma_{N})|g_{1}g_{2}...g_{N-1}g_{N}\rangle,$ (5)

where $N-1$ is the number of iterations. In the absence of dissipation, the
matrix elements of $\hat{D},$ at any time, are $D_{ij}=D_{ji}$ for $i\neq j$
$(i,j=1,2,...,2^{N})$ while all diagonal elements $D_{ii}$ are equal. Also, in
the presence of qubits dissipation rates, whether similar or different values,
the matrix elements at any time are $D_{ij}=D_{ji}$ for $i\neq j$ while all
diagonal elements $D_{ii}$ are often close to each others.
Now, we give the probabilities definitions of finding any marked state
$\rho_{r_{1}r_{2}...r_{N}}$ and finding unmarked states (or remaining states)
$\varrho_{j_{1}j_{2}...j_{N}}$. The probabilities are exactly calculated in
the presence of different values of dissipation rates and different $N$, where
both of $r_{1}r_{2}...r_{N}$ and $j_{1}j_{2}...j_{N}$ are one subset of the
set $\Lambda$. The probabilities $\rho_{r_{1}r_{2}...r_{N}}$ and
$\varrho_{j_{1}j_{2}...j_{N}}$ are give by:

$\displaystyle\rho_{r_{1}r_{2}...r_{N}}=\rho_{r_{1}r_{2}...r_{N}}(t;\gamma_{1},...,\gamma_{N})=\Big{|}\langle
r_{1}r_{2}...r_{N}|I_{N-1}(t;\gamma_{1},...,\gamma_{N})\hat{W}^{\otimes
N}(\gamma_{1},...,\gamma_{N})|g_{1}g_{2}...g_{N}\rangle\Big{|}^{2}.$ (6)
$\displaystyle\varrho_{j_{1}j_{2}...j_{N}}=\varrho_{j_{1}j_{2}...j_{N}}(t;\gamma_{1},...,\gamma_{N})=\Big{|}\langle
j_{1}j_{2}...j_{N}|I_{N-1}(t;\gamma_{1},...,\gamma_{N})\hat{W}^{\otimes
N}(\gamma_{1},...,\gamma_{N})|g_{1}g_{2}...g_{N}\rangle\Big{|}^{2},$ (7)

where
$I_{N-1}(t;\gamma_{1},...,\gamma_{N})=\displaystyle\prod_{i=1}^{N-1}\Big{\\{}\hat{D}(t;\gamma_{1},...,\gamma_{N})\hat{P}_{r_{1}r_{2}...r_{N}}(t;\gamma_{1},...,\gamma_{N})\Big{\\}}^{N-i}$.
From Eq.7, in calculating the probabilities of unmarked states, it should be
noted that at least one of $j_{1}j_{2}...j_{N}$ is not equal to one of
$r_{1}r_{2}...r_{N}$ for the same qubit. Here, we write, for simplicity, that
the sub-indexes in the marked state $\rho_{e_{1}e_{2}}\equiv\rho_{ee}$ and un-
marked states ($\varrho_{e_{1}g_{2}}$, $\varrho_{g_{1}e_{2}}$,
$\varrho_{g_{1}g_{2}}$ $\equiv$ Pr. r. s.); marked state
$\rho_{e_{1}g_{2}e_{3}}\equiv\rho_{ege}$ and unmarked states
($\varrho_{e_{1}e_{2}e_{3}}$, $\varrho_{e_{1}e_{2}g_{3}}$,
$\varrho_{e_{1}g_{2}g_{3}}$, $\varrho_{g_{1}e_{2}e_{3}}$,
$\varrho_{g_{1}e_{2}g_{3}}$, $\varrho_{g_{1}g_{2}e_{3}}$,
$\varrho_{g_{1}g_{2}g_{3}}$ $\equiv$ Pr. r. s.); and so on for all states of
the $N-$qubit. Also, $\rho_{g_{1}e_{2}g_{3}e_{4}}\equiv\rho_{gege}$ and so on.

##  3 Discussion

No. | $\lambda t_{p}/\hbar$ | Present Pr. | $\lambda t_{G}/\hbar$ | Grover Pr.
---|---|---|---|---
2 | $(0.9425\pi)/4\theta_{2}$ | 1 | $\pi/4\theta_{2}$ | 1
3 | $(0.6723\pi)/8\theta_{3}$ | 1 | $\pi/8\theta_{3}$ | 0.9453
4 | $(0.6933\pi)/16\theta_{4}$ | 1 | $\pi/16\theta_{4}$ | 0.9613
5 | $(0.8661\pi)/32\theta_{5}$ | 1 | $\pi/32\theta_{5}$ | 0.9992
6 | $(0.9899\pi)/64\theta_{6}$ | 0.9635 | $\pi/64\theta_{6}$ | 0.9635
7 | $(0.9906\pi)/128\theta_{7}$ | 0.8335 | $\pi/128\theta_{7}$ | 0.8335
8 | $(0.9906\pi)/256\theta_{8}$ | 0.6503 | $\pi/256\theta_{8}$ | 0.6503
9 | $(0.995\pi)/512\theta_{9}$ | 0.4662 | $\pi/512\theta_{9}$ | 0.4662
Table 1: A comparison of the time ($t_{p}$) of the present algorithm and
Grover time ($t_{G}$) for finding higher probability (Pr.) of any marked state
of different qubits in the case of $\gamma_{i}=0,\,i=1,2,...,9$.

In table (1), we compare our proposed algorithm (OA) results with Grover
algorithm’s results. It is shown that the probability reach $100\%$ for the
two-qubit case, in both: our case and GA case, but the time needed in our
case, $t_{p},$ is shorter than the Grover time, $t_{G},$ (say,
$t_{p}=0.9425\pi\hbar/4\lambda\theta_{2}$ and
$t_{G}=\pi\hbar/4\lambda\theta_{2}$). This means that the efficiency of our
algorithm is higher than the efficiency of GA. Once we increase the number of
involved qubits, we see that a significant difference between OA and GA, is
seen. The probability of finding any marked state using 3 qubits, 4 qubits and
5 qubits in OA is $100\%$ while the corresponding case using GA does not
exceed $94.53\%$ for 3 qubits, $96.13\%$ for 4 qubits and $99.92\%$ for 5
qubits. Moreover, the times needed of finding any marked state are shorter
than Grover times. This is an interesting advantage of the proposed algorithm
since the sensitivity is much higher than the Grover sensitivity to the number
of qubits. Also, we can say that using OA, exact observation is clearly
obtained even whether the number of involved qubits gets higher (say 5) and
one still have $100\%$ probability of finding any marked states. When we
consider the number of qubits is greater than or equal 6, it is shown that the
same probability of finding any marked state is obtained using OA and GA, but
the time needed to obtain such probability is shorter in our case, which is
still gives OA an advantage. It is interesting to observe here that as the
number of qubits is increased the probability find any desired state still
coincide using OA while in Grover case there does not coincide, where from
$1-6$ qubits the results do not coincide with the fact that increasing the
number of qubits leads to decreasing the probability. Moreover, as the number
is increased further, GA shows the logic behavior. In these calculations, we
expect that our results (times and probabilities) are almost the same as GA
for the large number of qubits (say, $N>12$). The reason of such agreement
between our results and Grover results is that when $N$ is increased the time
for finding the highest probability of any marked state is increased to reach
almost $\frac{\pi\hbar}{2^{13}\lambda\theta_{13}}$ for finding the probability
of any marked state when $N=13$. After that, one obtains the highest
probability for finding any desired states for the different qubits at times
$\frac{\pi\hbar}{2^{r}\lambda\theta_{r}}\,(r=13,14,...,N)$ , this case is
compatible with Grover case. This observation shows that using the dynamical
quantum search algorithm has some advantage, where GA is obtained as a special
case from the proposed general algorithm.
In Fig.2 we plot the probabilities of finding any marked or unmarked states as
a function of dynamical phase $\beta_{N}(t)/\pi,\,N=2,3,...,9$, in the absence
of dissipation. It is shown that for 2, 3, 4 and 5 qubits, see Fig.2a and
Fig.2b, the probabilities of marked states reach one at several points rather
than Grover observation, where GA shows the maximum probability at
$\beta_{N}(t)=\pi$ only. In 3 qubits and 4 qubits cases, we find that the
behavior of curves peaks are between the two maxima there is a local minimum.
While 2 and 5 qubits cases the maximum stays longer as a function of the
phase, which means that the long lived maximum probability is obtained for 2
and 5 qubits compared with 3 and 4 qubits cases where the maximum stays for a
short period. This observation is no longer existing once the number of qubits
is increased (say $\geq 6$), where the maxima probability always less than
one, see Fig.2c and Fig.2d. Also, the unmarked states are observed for the low
number of qubits and reach the maximum of $25\%$ for 2 qubits case, while this
value is decreased once the number of qubits is increased. This phenomenon
becomes more pronounced once the number of qubits exceeds $6$ after this
number of qubits the probabilities are very close to zero. From our further
calculations (which are not presented here), the probabilities of unmarked
states vanish completely when the number of qubits is greater than 20 qubits.

Figure 2: Dynamical quantum search algorithm for the multi qubits in the
absence of dissipation for all marked states (solid lines for $2-,4-,6-$ and
$8-$qubit; dashed lines for $3-,5-,7-$ and $9-$qubit) and all un-marked states
(dashed lines for $2-,4-,6-$ and $8-$qubit; solid lines for $3-,5-,7-$ and
$9-$qubit), where $N=2,3$ in (a), $N=4,5$ in (b), $N=6,7$ in (c) and $N=8,9$
in (d).

Figure 3: Dynamical quantum search algorithm for the probabilities of marked
states $\rho_{egee}$ and $\rho_{geege}$ in the presence of dissipation, where
$\gamma_{i}=\gamma\,(i=1,...,4)$ in (a); $\gamma_{j}=\gamma\,(j=1,...,5)$ in
(b); $\gamma_{1}=\gamma_{2}=\gamma_{4}$ and
$t_{p}=\frac{0.02813\pi\hbar}{\lambda\theta_{4}}$ in (c);
$\gamma_{1}=\gamma_{3}=\gamma_{5}$, $\gamma_{2}=\gamma_{4}$ and
$t_{p}=\frac{0.027065\pi\hbar}{\lambda\theta_{5}}$ in (d).

Figure 4: Dynamical quantum search algorithm for the probabilities of marked
states $\rho_{eg}$, $\rho_{geg}$, $\rho_{gegg}$ and $\rho_{gegege}$ in the
presence of similar dissipation rates.

Figure 5: Dynamical quantum search algorithm with dissipation for 5 qubits to
compare the values of $\rho_{eeeee}$ and $\rho_{ggggg}$.

In Fig.3, we display the behavior of the dynamical search of the probabilities
of marked states $\rho_{egee}$ and $\rho_{geege}$ with the same dissipation
rates. In the absence of dissipation, see Fig.2b and table (1), the
probabilities of the current marked states or any other marked states at times
$t_{p}=0.04333\pi\hbar/\lambda\theta_{4}$ for 4 qubits and
$t_{p}=0.027065\pi\hbar/\lambda\theta_{5}$ for 5 qubits are $100\%$ and the
probabilities of un-marked states are zero. It is shown that when times are
decreased from these values, the probabilities values of finding any marked
states are less than $100\%$ and decrease more and more according to the
decreasing times. With these time decreases, the probabilities of finding any
marked states are decreased and this decreasing is added to the un-marked
states probabilities. This means that the revivals for the probabilities of
un-marked states when the time is decreased more and more to become larger
when the time is close to zero. However, when the time reaches zero, the
probabilities of any marked or un-marked states are $6.25\%$ and $3.125\%$ for
4 and 5 qubits, respectively. This means that both marked and un-marked states
have the same chance to contribute. In the presence of weak dissipation rate,
one sees that the probabilities of current marked states in Fig.3a and Fig.3b
or any other marked states at different values of the time are gradually
decreased. The probabilities values of any desired state for any number of $N$
at a weak values of similar dissipation are often close to each others. If the
dissipation value is taken to be $\gamma_{i}=\lambda\theta_{N}$
$(i=1,2,...,N)$, the probabilities values of any desired state are sometimes
increased or decreased to each other and one can not distinguish between the
probability of any marked and un-marked states at any time. However, it is
difficult to observe the probability of any marked state for any qubit at
large dissipation rates, where small values of the probability of marked state
is shown. Which means that both marked and un-marked states have the same
chance to contribute.
Also, we display the behavior of the probabilities of current marked states in
Fig.3c and Fig.3d for different values of qubits dissipation rates with time
$t_{p}=0.02813\pi\hbar/\lambda\theta_{4}$ for 4 qubits and
$t_{p}=0.027065\pi\hbar/\lambda\theta_{5}$ for 5 qubits. In the absence of
dissipation the probabilities of any marked state for 4 qubits and 5 qubits at
these values of times start from 0.8332 and from 1, respectively. When a weak
value of qubits-dissipation is considered, one finds that the probabilities of
the current states in Fig.3c and Fig.3d and any other marked states for 4 and
5 qubits are gradually decreased. The values of any desired states for 4 and 5
qubits in weak dissipation are often close to each other. It is shown that for
strong dissipation, the probabilities of any marked states for 4 and 5 qubits
are decreased further, while one can not distinguish between different
probabilities of some un-marked states. It is interesting to mention here that
the observation of any desired state will be more difficult when dissipation
rate is equal to $\gamma_{3}=\gamma_{4}=\lambda\theta_{4}$ for 4 qubits and
$\gamma_{2}=\gamma_{5}=\lambda\theta_{5}$ for 5 qubits or more than that,
where there is some probabilities of un-marked states are larger than marked
state. This means that both marked and un-marked states have the same chance
to contribute.
However, the behavior of the probability of finding any remaining marked state
for any different number of $N$ in the dissipation presence, is similar to the
previous behavior, see Fig.4. It is shown that, the time is a main factor in
the appearance of different probabilities of marked states using different
number of qubits. Therefore, the probabilities for the remaining marked states
(3 states for 2 qubits, 7 states for 3 qubits, 14 states for 4 qubits, 29
states for 5 qubits and 63 states for 6 qubits) have the same behavior as in
the figures displayed in the above. In Fig.5, we plot DQSA for the
probabilities $\scalebox{1.0}{$\rho$}_{eeeee}$ and
$\scalebox{1.0}{$\rho$}_{ggggg}$ when the dissipation rates is considered. It
is shown that the probabilities $\scalebox{1.0}{$\rho$}_{ggggg}$ and
$\scalebox{1.0}{$\rho$}_{eeeee}$ are gradually decreased once dissipation rate
takes small values, and the values of $\scalebox{1.0}{$\rho$}_{eeeee}$ are
less than the values of $\scalebox{1.0}{$\rho$}_{ggggg}$ at any time. Also,
one observes that the probability $\scalebox{1.0}{$\rho$}_{ggggg}$ is shown as
larger values compared with the probability $\scalebox{1.0}{$\rho$}_{eeeee}$
which shows small values once dissipation rate $\gamma/\lambda\theta_{5}$ is
increased. Therefore, we can distinguish the probability
$\scalebox{1.0}{$\rho$}_{gg...g}$ for multi qubits at any time once
dissipation rate is decreased or increased (or varied).
Now, we give a short discussion to observe un-marked states in the presence of
dissipation rates either similar or different. The probabilities of un-desired
states for weak dissipation are distinguished by a very small values and non-
zero compared with the absence of dissipation. While for strong dissipation,
the probability of the desired state is becoming less and is decreased due to
the influence of strong dissipation and some of this decrease is added to the
un-desired states. On the other hand, the results of finding any marked and
un-marked states, in table (1) are completely in agreement with the results
shown in Fig.3 and Fig.4, under the same conditions.

##  4 Conclusions

We have demonstrated on-demand generation of dynamic multi-qubit quantum
search algorithm, which can be maintained for multi-qubit system. The
algorithm we have demonstrated, gives favorable performance for multi-qubit
(up to 9 qubits) and would survive even in the presence of the qubit
dissipation. We have focused our attention on the probabilities of the first 9
qubits and have obtained that their optimal values come either to
diagonalizing or to symmetrizing probabilities, that implies analytical
formula for its calculation. Comparing these results with Grover algorithm
analysis allows to give a similar result when the number of involved qubits
get much larger, and is an example of a prefect observation which is suitable
for a particular purpose, e.g., for the time dependent interaction and
dissipation processes. The long life time of this multi-qubit algorithm is due
to a high symmetry of the generated forms. We have used different values of
the controlled phase to improve the suggested algorithm performance. A new
technique is used to generate the dynamical quantum gates that helps in
implementing the current algorithm. The multi-qubit character of the
interaction used for the creation of dynamical algorithm allows for the
potential applications and extension of the present method to other models,
possibly including, superconducting materials or solid state models with long
lived spin states.

## Appendix

In this appendix, we took four optional values of times (say,
$\frac{0.331\pi\hbar}{2^{N}\lambda\theta_{N}}$,
$\frac{0.566\pi\hbar}{2^{N}\lambda\theta_{N}}$, $t_{Np}$ and
$\frac{\pi\hbar}{2^{N}\lambda\theta_{N}}$) to calculate the probabilities of
some marked and unmarked states for different number of $N$. The time $t_{Np}$
gives the highest probabilities for marked states with different values of
$N$.

Times | $t=\frac{0.331\pi\hbar}{4\lambda\theta_{2}}$ | $t=\frac{0.566\pi\hbar}{4\lambda\theta_{2}}$ | $t_{2p}=\frac{0.9425\pi\hbar}{4\lambda\theta_{2}}$ | $t=\frac{\pi\hbar}{4\lambda\theta_{2}}$
---|---|---|---|---
$\rho_{ee}$ | 0.5583 | 0.8537 | 0.9625 | 0.9618
Pr. r. s. |  5cm0.1417, 0.1411, 0.1427 |  5cm0.0407, 0.0405, 0.0397 |  5cm 1.5605e, 1.5524e, 1.7800d |  5cm7.7440d, 7.5690d, 0
$\rho_{ge}$ | 0.5634 | 0.8595 | 0.9668 | 0.9659
Pr. r. s. |  5cm0.1389, 0.1405, 0.1428 |  5cm 0.0399, 0.0390, 0.0399 |  5cm 1.5556e, 1.6690d, 2.5570d |  5cm 7.5690d, 0, 1.0000c
$\rho_{gg}$ | 0.5697 | 0.8667 | 0.9722 | 0.9712
Pr. r. s. |  5cm 0.1378, 0.1404, 0.1401 |  5cm 0.0382, 0.0388, 0.0390 |  5cm 1.6490d, 9.6400c, 2.5700d |  5cm 0, 1.0000c, 1.0000c
Table 2: Outputs of dynamical quantum search algorithm for some probabilities of marked states $\rho_{r_{1}r_{2}}$ ($r_{1},r_{2}=e,g$) and unmarked states or remaining states (Pr. r. s.) of 2-qubit for different values of time, where $\frac{\gamma_{1}}{\lambda}=\frac{\theta_{2}}{113}$ and $\frac{\gamma_{2}}{\lambda}=\frac{\theta_{2}}{90}$; $c=\times 10^{-6}$, $d=\times 10^{-5}$ and $e=\times 10^{-4}$. Times | $t=\frac{0.331\pi\hbar}{4\lambda\theta_{2}}$ | $t=\frac{0.566\pi\hbar}{4\lambda\theta_{2}}$ | $t_{2p}=\frac{0.9425\pi\hbar}{4\lambda\theta_{2}}$ | $t=\frac{\pi\hbar}{4\lambda\theta_{2}}$
---|---|---|---|---
$\rho_{ee}$ | 0.0961 | 0.1224 | 0.1142 | 0.1108
Pr. r. s. |  5cm0.0921, 0.0933, 0.1792 |  5cm0.0625, 0.0632, 0.0680 |  5cm 0.0428, 0.0434, 0.0049 |  5cm0.0426, 0.0432, 0.0043
$\rho_{ge}$ | 0.1683 | 0.1700 | 0.1148 | 0.1089
Pr. r. s. |  5cm 0.0528, 0.0699, 0.1660 |  5cm 0.0491, 0.0248, 0.0486 |  5cm 0.0333, 0.0041, 5.7600e |  5cm 0.0329, 0.0040, 3.1329e
$\rho_{gg}$ | 0.3244 | 0.2640 | 0.1098 | 0.0997
Pr. r. s. |  5cm0.0308, 0.0713, 0.0712 |  5cm0.0127, 0.0251, 0.0239 |  5cm 0.0021, 3.2962e, 3.4306e |  5cm 0.0021, 1.7640d, 9.2160d
Table 3: The same as Table (2) but $\frac{\gamma_{1}}{\lambda}=\frac{4\theta_{2}}{5}$ and $\frac{\gamma_{2}}{\lambda}=\frac{7\theta_{2}}{9}$. Times | $t=\frac{0.331\pi\hbar}{8\lambda\theta_{3}}$ | $t=\frac{0.566\pi\hbar}{8\lambda\theta_{3}}$ | $t_{3p}=\frac{0.6723\pi\hbar}{8\lambda\theta_{3}}$ | $t=\frac{\pi\hbar}{8\lambda\theta_{3}}$
---|---|---|---|---
$\rho_{eee}$ | 0.5989 | 0.9085 | 0.9348 | 0.8776
Pr. r. s. |  5cm 0.0520, 0.0524, 0.0516, 0.0522, |  |  |
0.0516, 0.0517, 0.0518 |  5cm0.0052, 0.0053, 0.0046, 0.0052, |  |  |
0.0047, 0.0046, 0.0044 |  5cm 1.0345e, 1.1197e, 4.1220d, 1.0625e, |  |  |
4.5460d, 3.7480d, 2.5210d |  5cm 0.0064, 0.0063, 0.0064, 0.0064, |  |  |
0.0065, 0.0064, 0.0063 |  |  |  |
$\rho_{eeg}$ | 0.6051 | 0.9170 | 0.9435 | 0.8855
Pr. r. s. |  5cm 0.0515, 0.0512, 0.0522, 0.0511, |  |  |
0.0520, 0.0513, 0.0515 |  5cm0.0052, 0.0046, 0.0049, 0.0047, |  |  |
0.0048, 0.0044, 0.0042 |  5cm 9.5290d, 3.6000d, 5.9300d, 4.1780d, |  |  |
5.6250d, 2.1130d, 1.3220d |  5cm 0.0068, 0.0068, 0.0065, 0.0069, |  |  |
0.0066, 0.0067, 0.0065 |  |  |  |
$\rho_{ege}$ | 0.6085 | 0.9218 | 0.9483 | 0.8898
Pr. r. s. |  5cm 0.0517, 0.0509, 0.0517, 0.0509, |  |  |
0.0511, 0.0519, 0.0513 |  5cm 0.0053, 0.0046, 0.0046, 0.0046, |  |  |
0.0044, 0.0046, 0.0041 |  5cm 9.8960d, 3.4340d, 3.4490d, 3.0920d, |  |  |
1.9240d, 3.6170d, 8.8400c |  5cm 0.0069, 0.0070, 0.0067, 0.0070, |  |  |
0.0069, 0.0067, 0.0067 |  |  |  |
$\rho_{gee}$ | 0.6065 | 0.9191 | 0.9454 | 0.8874
Pr. r. s. |  5cm 0.0516, 0.0510, 0.0511, 0.0512, |  |  |
0.0518, 0.0522, 0.0514 |  5cm 0.0052, 0.0047, 0.0046, 0.0044, |  |  |
0.0047, 0.0048, 0.0041 |  5cm9.6260d, 4.0000d, 3.2500d, 2.0500d, |  |  |
4.4680d, 5.0500d, 1.1250d |  5cm 0.0069, 0.0069, 0.0069, 0.0068, |  |  |
0.0067, 0.0065, 0.0066 |  |  |  |
$\rho_{gge}$ | 0.6162 | 0.9325 | 0.9590 | 0.8998
Pr. r. s. |  5cm0.0504, 0.0505, 0.0513, 0.0507, |  |  |
0.0515, 0.0507, 0.0515 |  5cm 0.0046, 0.0044, 0.0046, 0.0041, |  |  |
0.0048, 0.0041, 0.0041 |  5cm 2.4820d, 1.5170d, 3.0440d, 6.6100c, |  |  |
4.1810d, 7.7600c, 9.9700c |  5cm0.0075, 0.0074, 0.0071, 0.0072, |  |  |
0.0071, 0.0072, 0.0069 |  |  |  |
Table 4: Outputs of dynamical quantum search algorithm for some probabilities of marked states $\rho_{r_{1}r_{2}r_{3}}$ $(r_{1},r_{2},r_{3}=e,g)$ and unmarked states of 3-qubit for different values of time, where $\frac{\gamma_{1}}{\lambda}=\frac{\theta_{3}}{113}$, $\frac{\gamma_{2}}{\lambda}=\frac{\theta_{3}}{90}$ and $\frac{\gamma_{3}}{\lambda}=\frac{\theta_{3}}{140}$. Times | $t=\frac{0.331\pi\hbar}{8\lambda\theta_{3}}$ | $t=\frac{0.566\pi\hbar}{8\lambda\theta_{3}}$ | $t_{3p}=\frac{0.6723\pi\hbar}{8\lambda\theta_{3}}$ | $t=\frac{\pi\hbar}{8\lambda\theta_{3}}$
---|---|---|---|---
$\rho_{eee}$ | 0.0217 | 0.0234 | 0.0201 | 0.0113
Pr. r. s. |  5cm 0.0183, 0.0201, 0.0257, 0.0204, |  |  |
0.0261, 0.0289, 0.0596 |  5cm0.0129, 0.0140, 0.0088, 0.0142, |  |  |
0.0089, 0.0094, 0.0044 |  5cm0.0093, 0.0100, 0.0054, 0.0101, |  |  |
0.0054, 0.0055, 4.8420e |  5cm 0.0033, 0.0034, 0.0026, 0.0034, |  |  |
0.0026, 0.0023, 0.0051 |  |  |  |
$\rho_{eeg}$ | 0.0317 | 0.0297 | 0.0247 | 0.0135
Pr. r. s. |  5cm 0.0145, 0.0160, 0.0256, 0.0162, |  |  |
0.0261, 0.0238, 0.0539 |  5cm 0.0109, 0.0085, 0.0085, 0.0086, |  |  |
0.0087, 0.0050, 0.0027 |  5cm 0.0072, 0.0051, 0.0051, 0.0051, |  |  |
0.0052, 0.0024, 2.6240d |  5cm 0.0018, 7.9524e, 0.0023, 7.7284e, |  |  |
0.0023, 9.2416e, 0.0030 |  |  |  |
$\rho_{ege}$ | 0.0356 | 0.0327 | 0.0268 | 0.0140
Pr. r. s. |  5cm 0.0152, 0.0146, 0.0241, 0.0161, |  |  |
0.0207, 0.0282, 0.0520 |  5cm 0.0111, 0.0075, 0.0071, 0.0081, |  |  |
0.0042, 0.0084, 0.0022 |  5cm0.0072, 0.0043, 0.0040, 0.0045, |  |  |
0.0019, 0.0047, 2.3050d |  5cm0.0017, 5.8564e, 0.0016, 4.7089e, |  |  |
7.2900e, 0.0016, 0.0026 |  |  |  |
$\rho_{gee}$ | 0.0362 | 0.0332 | 0.0271 | 0.0141
Pr. r. s. |  5cm 0.0153, 0.0145, 0.0159, 0.0203, |  |  |
0.0243, 0.0279, 0.0518 |  5cm 0.0112, 0.0074,0.0079, 0.0040, |  |  |
0.0070, 0.0082, 0.0022 |  5cm 0.0073, 0.0042, 0.0044, 0.0018, |  |  |
0.0039, 0.0045, 3.1760d |  5cm 0.0016, 5.4289e, 4.4521e, 7.0225e, |  |  |
0.0015, 0.0015, 0.0025 |  |  |  |
$\rho_{gge}$ | 0.0737 | 0.0571 | 0.0448 | 0.0213
Pr. r. s. |  5cm 0.0105, 0.0097, 0.0193, 0.0152, |  |  |
0.0192, 0.0153, 0.0420 |  5cm0.0045, 0.0026, 0.0087, 0.0010, |  |  |
0.0085, 9.8881e, 1.7730e |  5cm0.0017, 7.9322e, 0.0041, 2.4912e, |  |  |
0.0039, 2.3850e, 9.4625e |  5cm2.7040d, 5.1840d, 1.4161e, |  |  |
4.8400c, 1.2100e, 5.7600c, 0.0014 |  |  |  |
Table 5: The same as Table (4) but $\frac{\gamma_{1}}{\lambda}=\frac{4\theta_{3}}{5}$, $\frac{\gamma_{2}}{\lambda}=\frac{7\theta_{3}}{9}$ and $\frac{\gamma_{3}}{\lambda}=\frac{5\theta_{3}}{8}$. Times | $t=\frac{0.331\pi\hbar}{16\lambda\theta_{4}}$ | $t=\frac{0.566\pi\hbar}{16\lambda\theta_{4}}$ | $t_{4p}=\frac{0.6933\pi\hbar}{16\lambda\theta_{4}}$ | $t=\frac{\pi\hbar}{16\lambda\theta_{4}}$
---|---|---|---|---
$\rho_{eeee}$ | 0.5433 | 0.8570 | 0.8900 | 0.8499
Pr. r. s. |  5cm 0.0269, 0.0267, 0.0259, 0.0269, |  |  |
0.0258, 0.0259, 0.0256, 0.0268, |  |  |  |
0.0259, 0.0259, 0.0256, 0.0258, |  |  |  |
0.0255, 0.0256, 0.0253 |  5cm 0.0032, 0.0032, 0.0030, 0.0033, |  |  |
0.0030, 0.0030, 0.0030, 0.0032, |  |  |  |
0.0030, 0.0030, 0.0030, 0.0030, |  |  |  |
0.0030, 0.0030, 0.0029 |  5cm 6.5530d, 5.7800d, 5.3210d, 6.9200d, |  |  |
5.7250d, 5.4500d, 6.3610d, 6.3370d, |  |  |  |
5.5210d, 5.2520d, 6.1450d, 5.5210d, |  |  |  |
6.5810d, 6.2820d, 7.2650d |  5cm 0.0016, 0.0016, 0.0018, 0.0016, |  |  |
0.0018, 0.0018, 0.0019, 0.0016, |  |  |  |
0.0018, 0.0018, 0.0019, 0.0018, |  |  |  |
0.0019, 0.0019, 0.0020 |  |  |  |
$\rho_{eegg}$ | 0.5611 | 0.8840 | 0.9178 | 0.8767
Pr. r. s. |  5cm 0.0260, 0.0262, 0.0265, 0.0257, |  |  |
0.0254, 0.0255, 0.0260, 0.0257, |  |  |  |
0.0254, 0.0255, 0.0259, 0.0254, |  |  |  |
0.0251, 0.0252, 0.0248 |  5cm 0.0028, 0.0029, 0.0030, 0.0028, |  |  |
0.0027, 0.0028, 0.0029, 0.0028, |  |  |  |
0.0027, 0.0028, 0.0029, 0.0027, |  |  |  |
0.0027, 0.0027, 0.0027 |  5cm 1.3220d, 2.5780d, 2.8090d, 1.8010d, |  |  |
2.3840d, 2.1730d, 4.4690d, 1.6850d, |  |  |  |
2.3050d, 2.0450d, 3.9290d, 2.3050d, |  |  |  |
3.0250d, 2.7250d, 3.5050d |  5cm 0.0020, 0.0019, 0.0018, 0.0020, |  |  |
0.0021, 0.0021, 0.0020, 0.0020, |  |  |  |
0.0021, 0.0021, 0.0020, 0.0021, |  |  |  |
0.0022, 0.0022, 0.0023 |  |  |  |
$\rho_{eggg}$ | 0.5729 | 0.9019 | 0.9364 | 0.8947
Pr. r. s. |  5cm 0.0257, 0.0254, 0.0255, 0.0260, |  |  |
0.0254, 0.0256, 0.0259, 0.0254, |  |  |  |
0.0251, 0.0252, 0.0249, 0.0251, |  |  |  |
0.0248, 0.0249, 0.0253 |  5cm 0.0027, 0.0026, 0.0026, 0.0028, |  |  |
0.0026, 0.0027, 0.0028, 0.0026, |  |  |  |
0.0026, 0.0026, 0.0026, 0.0026, |  |  |  |
0.0026, 0.0026, 0.0027 |  5cm 3.9200c, 6.4800c, 5.4500c, 1.8180d, |  |  |
6.8500c, 1.6290d, 1.8180d, 6.1300c, |  |  |  |
9.2500c, 8.4100c, 1.2010d, 1.0130d, |  |  |  |
1.4050d, 1.3010d, 2.8250d |  5cm0.0022, 0.0022, 0.0022, 0.0021, |  |  |
0.0023, 0.0022, 0.0021, 0.0022, |  |  |  |
0.0023, 0.0023, 0.0024, 0.0023, |  |  |  |
0.0024, 0.0024, 0.0023 |  |  |  |
$\rho_{geeg}$ | 0.5629 | 0.8867 | 0.9206 | 0.8795
Pr. r. s. |  5cm 0.0259, 0.0263, 0.0257, 0.0254, |  |  |
0.0256, 0.0253, 0.0254, 0.0251, |  |  |  |
0.0265, 0.0254, 0.0257, 0.0254, |  |  |  |
0.0259, 0.0251, 0.0248 |  5cm 0.0028, 0.0029, 0.0028, 0.0027, |  |  |
0.0028, 0.0027, 0.0027, 0.0027, |  |  |  |
0.0030, 0.0027, 0.0028, 0.0027, |  |  |  |
0.0029, 0.0027, 0.0027 |  5cm 1.1450d, 2.5480d, 1.4450d, 1.9010d, |  |  |
1.6180d, 2.1530d, 1.9720d, 2.5610d, |  |  |  |
2.6330d, 1.8500d, 3.3650d, 2.0250d, |  |  |  |
4.2530d, 2.5000d, 3.1610d |  5cm 0.0020, 0.0019, 0.0020, 0.0021, |  |  |
0.0021, 0.0022, 0.0021, 0.0022, |  |  |  |
0.0019, 0.0021, 0.0021, 0.0022, |  |  |  |
0.0021, 0.0022, 0.0023 |  |  |  |
Table 6: Outputs of dynamical quantum search algorithm for some probabilities of marked states $\rho_{r_{1}r_{2}r_{3}r_{4}}$ ($r_{1},r_{2},r_{3},r_{4}=e,g$) and unmarked states of 4-qubit for different values of time, where $\frac{\gamma_{1}}{\lambda}=\frac{\theta_{4}}{113}$, $\frac{\gamma_{2}}{\lambda}=\frac{\theta_{4}}{90}$, $\frac{\gamma_{3}}{\lambda}=\frac{\theta_{4}}{140}$ and $\frac{\gamma_{4}}{\lambda}=\frac{\theta_{4}}{100}$. Times | $t=\frac{0.331\pi\hbar}{16\lambda\theta_{4}}$ | $t=\frac{0.566\pi\hbar}{16\lambda\theta_{4}}$ | $t_{4p}=\frac{0.6933\pi\hbar}{16\lambda\theta_{4}}$ | $t=\frac{\pi\hbar}{16\lambda\theta_{4}}$
---|---|---|---|---
$\rho_{eeee}$ | 0.0034 | 0.0025 | 0.0018 | 0.0017
Pr. r. s. |  5cm 0.0038, 0.0035, 0.0041, 0.0038, |  |  |
0.0045, 0.0042, 0.0055, 0.0038, |  |  |  |
0.0046, 0.0042, 0.0056, 0.0047, |  |  |  |
0.0064, 0.0057, 0.0153 |  5cm 0.0017, 0.0016, 0.0015, 0.0017, |  |  |
0.0016, 0.0015, 0.0011, 0.0018, |  |  |  |
0.0016, 0.0015, 0.0011, 0.0016, |  |  |  |
0.0010, 0.0011, 4.9361e |  5cm 9.1801e, 8.9317e, 8.3357e, 9.2909e, |  |  |
8.0002e, 8.2450e, 0.0011, 9.3466e, |  |  |  |
7.9141e, 8.2273e, 0.0011, 7.8829e, |  |  |  |
9.7501e, 0.0010, 0.0025 |  5cm 0.0010, 9.6100e, 7.7284e, 0.0010, |  |  |
8.0089e, 7.7841e, 4.2025e, 0.0010, |  |  |  |
8.0089e, 7.7841e, 4.2436e, 8.0656e, |  |  |  |
4.4944e, 4.3264e, 1.0000$\times 10^{-8}$ |  |  |  |
$\rho_{eegg}$ | 0.0071 | 0.0048 | 0.0037 | 0.0026
Pr. r. s. |  5cm0.0024, 0.0035, 0.0035, 0.0029, |  |  |
0.0037, 0.0035, 0.0049, 0.0030, |  |  |  |
0.0037, 0.0035, 0.0050, 0.0034, |  |  |  |
0.0047, 0.0043, 0.0127 |  5cm 7.7860e, 0.0012, 0.0012, 8.0725e, |  |  |
8.3394e, 8.4361e, 0.0010, 8.0770e, |  |  |  |
8.3449e, 8.4817e, 0.0010, 7.5393e, |  |  |  |
4.4753e, 4.8322e, 6.1274e |  5cm 3.1058e, 4.9130e, 5.4666e, 2.7961e, |  |  |
3.2020e, 3.4613e, 9.2273e, 2.7505e, |  |  |  |
3.1720e, 3.4289e, 9.1705e, 3.0266e, |  |  |  |
4.9689e, 5.3170e, 0.0018 |  5cm4.0000e, 5.6169e, 5.8081e, 4.0000e, |  |  |
3.4225e, 3.3489e, 2.5281e, 4.0000e, |  |  |  |
3.4596e, 3.3856e, 2.6244e, 3.2761e, |  |  |  |
1.1881e, 1.1449e, 8.4640d |  |  |  |
$\rho_{eggg}$ | 0.0148 | 0.0100 | 0.0079 | 0.0044
Pr. r. s. |  5cm 0.0019, 0.0026, 0.0025, 0.0045, |  |  |
0.0026, 0.0043, 0.0044, 0.0023, |  |  |  |
0.0025, 0.0024, 0.0033, 0.0025, |  |  |  |
0.0036, 0.0034, 0.0095 |  5cm 4.1785e, 4.9360e, 5.0265e, 0.0011, |  |  |
4.9034e, 9.2489e, 0.0010, 4.3810e, |  |  |  |
3.8849e, 3.9641e, 3.4597e, 3.8628e, |  |  |  |
3.0980e, 3.3857e, 0.0012 |  5cm 1.3136e, 1.2770e, 1.3549e, 4.2365e, |  |  |
1.2456e,3.3808e, 4.0637e, 1.3130e, |  |  |  |
1.4257e, 1.4788e, 3.0025e, 1.3925e, |  |  |  |
2.9314e, 2.9786e, 0.0019 |  5cm1.2100e, 1.2996e, 1.2996e, 2.9584e, |  |  |
1.2996e, 2.4649e, 2.8900e, 1.0816e, |  |  |  |
6.5610d, 6.5610d, 4.0000c, 6.5610d, |  |  |  |
2.8900c, 4.0000c, 4.9000d |  |  |  |
$\rho_{geeg}$ | 0.0084 | 0.0054 | 0.0041 | 0.0029
Pr. r. s. |  5cm0.0024, 0.0037, 0.0026, 0.0032, |  |  |
0.0029, 0.0037, 0.0029, 0.0038, |  |  |  |
0.0038, 0.0033, 0.0046, 0.0038, |  |  |  |
0.0056, 0.0040, 0.0122 |  5cm 7.0922e, 0.0012, 6.8965e, 7.2320e, |  |  |
7.0381e, 7.4378e, 6.4129e, 4.2530e, |  |  |  |
0.0012, 7.2041e, 8.5426e, 7.4321e, |  |  |  |
9.0625e, 4.1474e, 7.2585e |  5cm 2.6185e, 4.9985e, 2.3620e, 2.8793e, |  |  |
2.1853e, 2.5348e, 2.7625e, 4.9369e, |  |  |  |
4.7890e, 2.7716e, 7.7681e, 2.4500e, |  |  |  |
7.5024e, 4.7545e, 0.0018 |  5cm 3.4596e, 5.5696e, 3.2761e, 2.7225e, |  |  |
3.4225e, 2.9584e, 2.5921e, 7.5690d, |  |  |  |
5.4756e, 2.7556e, 1.8496e, 2.9929e, |  |  |  |
2.4025e, 7.9210d, 1.2769e |  |  |  |
$\rho_{ggeg}$ | 0.0195 | 0.0125 | 0.0095 | 0.0054
Pr. r. s. |  5cm 0.0018, 0.0026, 0.0019, 0.0020, |  |  |
0.0027, 0.0051, 0.0021, 0.0027, |  |  |  |
0.0027, 0.0051, 0.0021, 0.0027, |  |  |  |
0.0050, 0.0028, 0.0083 |  5cm 3.2786e, 3.9785e, 3.4632e, 3.2057e, |  |  |
3.9469e, 0.0010, 3.2050e, 3.3730e, |  |  |  |
3.9469e, 0.0010, 3.1688e, 3.3377e, |  |  |  |
9.7604e, 3.2677e, 0.0014 |  5cm 9.8410d, 8.8650d, 1.1365e, 1.2874e, |  |  |
8.6210d, 3.5257e, 1.2789e, 2.8346e, |  |  |  |
8.5520d, 3.3962e, 1.2580e, 2.8157e, |  |  |  |
3.2185e, 2.7693e, 0.0019 |  5cm 7.9210d, 9.0250d, 6.2410d, 2.9160d, |  |  |
9.0250d, 2.7556e, 2.9160d, 1.4400c, |  |  |  |
9.0250d, 2.6569e, 2.9160d, 1.4400c, |  |  |  |
2.5921e, 1.4400c, 1.5625e |  |  |  |
Table 7: The same as Table (6) but $\frac{\gamma_{1}}{\lambda}=\frac{4\theta_{4}}{5}$, $\frac{\gamma_{2}}{\lambda}=\frac{7\theta_{4}}{9}$, $\frac{\gamma_{3}}{\lambda}=\frac{5\theta_{4}}{8}$ and $\frac{\gamma_{4}}{\lambda}=\frac{3\theta_{4}}{4}$. Times | $t=\frac{0.331\pi\hbar}{32\lambda\theta_{5}}$ | $t=\frac{0.566\pi\hbar}{32\lambda\theta_{5}}$ | $t_{5p}=\frac{0.8661\pi\hbar}{32\lambda\theta_{5}}$ | $t=\frac{\pi\hbar}{32\lambda\theta_{5}}$
---|---|---|---|---
$\rho_{eeeeg}$ | 0.4478 | 0.7707 | 0.8562 | 0.8536
Pr. r. s. |  5cm0.0159, 0.0153, 0.0156, 0.0153, |  |  |
0.0155, 0.0151, 0.0149, 0.0152, |  |  |  |
0.0156, 0.0150, 0.0148, 0.0151, |  |  |  |
0.0149, 0.0148, 0.0146, 0.0153, |  |  |  |
0.0156, 0.0150, 0.0149, 0.0151, |  |  |  |
0.0149, 0.0149, 0.0147, 0.0150, |  |  |  |
0.0148, 0.0148, 0.0146, 0.0149, |  |  |  |
0.0146, 0.0146, 0.0144 |  5cm0.0035, 0.0035, 0.0037, 0.0035, |  |  |
0.0036, 0.0035, 0.0035, 0.0035, |  |  |  |
0.0037, 0.0035, 0.0036, 0.0035, |  |  |  |
0.0036, 0.0036, 0.0036, 0.0035, |  |  |  |
0.0036, 0.0035, 0.0036, 0.0035, |  |  |  |
0.0035, 0.0036, 0.0036, 0.0035, |  |  |  |
0.0036, 0.0036, 0.0036, 0.0036, |  |  |  |
0.0036, 0.0036, 0.0037 |  5cm{5.9130, 3.9610, 5.5840, 4.1210, |  |  |
5.1610, 3.7060, 3.4850, 3.9240, |  |  |  |
5.8760, 3.6500, 3.4640, 3.7060, |  |  |  |
3.4850, 3.4640, 3.2850, 3.9610, |  |  |  |
5.4410, 3.7060, 3.4850, 3.7370, |  |  |  |
3.5080, 3.4850, 3.2980, 3.6770, |  |  |  |
3.4640, 3.4450, 3.2740, 3.4640, |  |  |  |
3.2850, 3.2740, 3.1370}d |  5cm 1.0000a, 1.6000b, 8.1000b, 2.5000b, |  |  |
2.5000b, 4.0000a, 1.0000a, 1.6000b, |  |  |  |
1.2100c, 0, 9.0000a, 1.0000a, |  |  |  |
4.0000a, 9.0000a, 3.6000b, 2.5000b, |  |  |  |
6.4000b, 1.0000a, 4.0000a, 4.0000a, |  |  |  |
1.0000a, 4.0000a, 2.5000b, 1.0000a, |  |  |  |
4.0000a, 9.0000a, 4.9000b, 4.0000a, |  |  |  |
2.5000b, 3.6000b, 1.0000c |  |  |  |
$\rho_{eegeg}$ | 0.4553 | 0.7835 | 0.8707 | 0.8683
Pr. r. s. |  5cm0.0154, 0.0156, 0.0152, 0.0150, |  |  |
0.0157, 0.0150, 0.0153, 0.0152, |  |  |  |
0.0150, 0.0149, 0.0147, 0.0150, |  |  |  |
0.0154, 0.0147, 0.0145,0.0152, |  |  |  |
0.0150, 0.0150, 0.0148, 0.0150, |  |  |  |
0.0153, 0.0148, 0.0146, 0.0149, |  |  |  |
0.0147, 0.0147, 0.0145, 0.0148, |  |  |  |
0.0146, 0.0145, 0.0143 |  5cm0.0033, 0.0035, 0.0034, 0.0034, |  |  |
0.0035, 0.0034, 0.0036, 0.0034, |  |  |  |
0.0034, 0.0035, 0.0035, 0.0034, |  |  |  |
0.0036, 0.0035, 0.0035, 0.0034, |  |  |  |
0.0034, 0.0034, 0.0035, 0.0034, |  |  |  |
0.0036, 0.0035, 0.0035, 0.0034, |  |  |  |
0.0035, 0.0035, 0.0035, 0.0035, |  |  |  |
0.0035, 0.0035, 0.0036 |  5cm{ 3.1040, 4.1210, 2.857, 2.6440, |  |  |
4.2850, 2.6690, 4.0400, 2.8260, |  |  |  |
2.6440, 2.5220, 2.3530, 2.6440, |  |  |  |
4.1650, 2.3530, 2.2180, 2.8570, |  |  |  |
2.6690, 2.6440, 2.4650, 2.6690, |  |  |  |
3.9170, 2.4650, 2.3200, 2.6210, |  |  |  |
2.4500, 2.3400, 2.2130, 2.4650, |  |  |  |
2.3200, 2.2130, 2.1200}d |  5cm 3.2400c, 4.9000b, 1.9600c, 1.4400c, |  |  |
4.9000b, 1.4400c, 4.0000a, 1.9600c, |  |  |  |
1.2100c, 1.0000c, 4.9000b, 1.2100c, |  |  |  |
1.6000b, 4.9000b, 1.6000b, 2.2500c, |  |  |  |
1.4400c, 1.2100c, 6.4000b, 1.4400c, |  |  |  |
0, 6.4000b, 2.5000b, 1.2100c, |  |  |  |
6.4000b, 4.9000b, 9.0000a, 6.4000b, |  |  |  |
2.5000b, 1.6000b, 0 |  |  |  |
$\rho_{egeee}$ | 0.4510 | 0.7761 | 0.8623 | 0.8597
Pr. r. s. |  5cm 0.0161, 0.0153, 0.0152, 0.0150, |  |  |
0.0153, 0.0151, 0.0150, 0.0149, |  |  |  |
0.0154, 0.0155, 0.0148, 0.0154, |  |  |  |
0.0148, 0.0148, 0.0146, 0.0153, |  |  |  |
0.0151, 0.0150, 0.0148, 0.0151, |  |  |  |
0.0149, 0.0148, 0.0146, 0.0155, |  |  |  |
0.0148, 0.0147, 0.0145, 0.0148, |  |  |  |
0.0146, 0.0145,0.0143, |  5cm 0.0036, 0.0034, 0.0034, 0.0035, |  |  |
0.0034, 0.0035, 0.0035, 0.0035, |  |  |  |
0.0036, 0.0036, 0.0035, 0.0036, |  |  |  |
0.0035, 0.0035, 0.0036, 0.0034, |  |  |  |
0.0035, 0.0035, 0.0035, 0.0035, |  |  |  |
0.0035, 0.0035, 0.0036, 0.0036, |  |  |  |
0.0035, 0.0035, 0.0036, 0.0035, |  |  |  |
0.0036, 0.0036, 0.0036 |  5cm{ 5.9680, 3.5730, 3.4600, 3.2210, |  |  |
3.6100, 3.3610, 3.2210, 3.0160, |  |  |  |
4.6450, 4.9130, 2.9800, 4.5140, |  |  |  |
3.0160, 2.9970, 2.8340, 3.4600, |  |  |  |
3.2210, 3.1940, 2.9970, 3.3610, |  |  |  |
3.0370, 3.0160, 2.8450, 4.7780, |  |  |  |
2.9970, 2.9800, 2.8250, 2.9970, |  |  |  |
2.8340, 2.8250, 2.7040}$d$ |  5cm 0, 8.1000b, 8.1000b, 3.6000b, |  |  |
1.0000c, 4.9000b, 3.6000b, 9.0000a, |  |  |  |
9.0000a, 3.6000b, 1.0000a, 4.0000a, |  |  |  |
4.0000a, 4.0000a, 4.0000a, 8.1000b, |  |  |  |
3.6000b, 2.5000b, 4.0000a, 3.6000b, |  |  |  |
9.0000a, 9.0000a, 1.0000a, 2.5000b, |  |  |  |
4.0000a, 1.0000a, 4.0000a, 4.0000a, |  |  |  |
1.0000a, 4.0000a, 3.6000b |  |  |  |
$\rho_{egegg}$ | 0.4704 | 0.8088 | 0.8987 | 0.8962
Pr. r. s. |  5cm 0.0152, 0.0151, 0.0150, 0.0154, |  |  |
0.0151, 0.0149, 0.0148, 0.0146, |  |  |  |
0.0150, 0.0153, 0.0152, 0.0148, |  |  |  |
0.0146, 0.0145, 0.0147, 0.0150, |  |  |  |
0.0148, 0.0148, 0.0146, 0.0149, |  |  |  |
0.0147, 0.0146, 0.0144, 0.0148, |  |  |  |
0.0146, 0.0145, 0.0148, 0.0146, |  |  |  |
0.0144, 0.0143, 0.0141 |  5cm 0.0032, 0.0032, 0.0032, 0.0034, |  |  |
0.0032, 0.0033, 0.0033, 0.0033, |  |  |  |
0.0032, 0.0034, 0.0034, 0.0033, |  |  |  |
0.0033, 0.0033, 0.0035, 0.0032, |  |  |  |
0.0033, 0.0033, 0.0033, 0.0033, |  |  |  |
0.0033, 0.0033, 0.0034, 0.0033, |  |  |  |
0.0033, 0.0033, 0.0035, 0.0033, |  |  |  |
0.0034, 0.0034, 0.0034 |  5cm 1.3810d, 1.2100d, 1.2100d, 2.3780d, |  |  |
1.3000d, 1.1530d, 1.0730d, 9.7000c, |  |  |  |
1.1890d, 2.1460d, 1.8450d, 1.0730d, |  |  |  |
9.7000c, 9.6500c, 1.5220d, 1.2100d, |  |  |  |
1.0730d, 1.0600d, 9.6500c, 1.0880d, |  |  |  |
9.7700c, 9.7000c, 9.0100c, 1.0600d, |  |  |  |
9.6500c, 9.6200c, 1.6820d, 9.6500c, |  |  |  |
9.0400c, 9.0900c, 8.3300c |  5cm 1.3690d, 1.1560d, 1.0890d, 3.2400c, |  |  |
1.1560d, 9.6100c, 9.6100c, 7.8400c, |  |  |  |
1.0890d, 3.6100c, 4.0000c, 9.0000c, |  |  |  |
7.2900c, 6.7600c, 2.2500c, 1.1560d, |  |  |  |
9.6100c, 9.0000c, 7.2900c, 9.6100c, |  |  |  |
7.8400c, 7.2900c, 5.7600c, 9.0000c, |  |  |  |
6.7600c, 6.7600c, 1.6900c, 7.2900c, |  |  |  |
5.7600c, 5.2900c, 3.6100c |  |  |  |
$\rho_{geeee}$ | 0.4486 | 0.7721 | 0.8578 | 0.8553
Pr. r. s. |  5cm 0.0160, 0.0153, 0.0153, 0.0151, |  |  |
0.0153, 0.0151, 0.0151, 0.0149, |  |  |  |
0.0152, 0.0150, 0.0150, 0.0148, |  |  |  |
0.0151, 0.0149, 0.0148, 0.0146, |  |  |  |
0.0155, 0.0156, 0.0149, 0.0155, |  |  |  |
0.0149, 0.0149, 0.0147, 0.0156, |  |  |  |
0.0148, 0.0148, 0.0146, 0.0148, |  |  |  |
0.0146, 0.0146, 0.0144 |  5cm 0.0035, 0.0034, 0.0035, 0.0035, |  |  |
0.0034,0.0035, 0.0035, 0.0035, |  |  |  |
0.0035, 0.0035, 0.0035, 0.0036, |  |  |  |
0.0035, 0.0035, 0.0036, 0.0036, |  |  |  |
0.0036, 0.0037, 0.0035, 0.0036, |  |  |  |
0.0035, 0.0035, 0.0036, 0.0037, |  |  |  |
0.0036, 0.0036, 0.0036, 0.0035, |  |  |  |
0.0036, 0.0036, 0.0037 |  5cm{ 5.9130, 3.8420, 3.8050, 3.5890, |  |  |
3.8810, 3.6200, 3.5890, 3.3700, |  |  |  |
3.8050, 3.5600, 3.5330, 3.3300, |  |  |  |
3.5600, 3.3700, 3.3490, 3.1610, |  |  |  |
5.1220, 5.4020, 3.3490, 4.9850, |  |  |  |
3.3930, 3.3700, 3.1850, 5.5450, |  |  |  |
3.3490, 3.3300, 3.0500, 3.3490, |  |  |  |
3.1720, 3.1610, 3.0260 }d |  5cm1.0000a, 3.6000b, 3.6000b, 9.0000a, |  |  |
4.9000b, 1.6000b, 9.0000a, 0, |  |  |  |
2.5000b, 4.0000a, 1.0000a, 4.0000a, |  |  |  |
4.0000a, 1.0000a, 1.0000a, 2.5000b, |  |  |  |
3.6000b, 8.1000b, 1.0000a, 1.6000b, |  |  |  |
0, 0, 1.6000b, 1.0000c, |  |  |  |
1.0000a, 4.0000a, 2.5000b, 1.0000a, |  |  |  |
1.6000b, 2.5000b, 8.1000b |  |  |  |
$\rho_{gegeg}$ | 0.4647 | 0.7993 | 0.8881 | 0.8857
Pr. r. s. |  5cm 0.0153, 0.0151, 0.0151, 0.0149, |  |  |
0.0151, 0.0154, 0.0149, 0.0147, |  |  |  |
0.0150, 0.0148, 0.0148, 0.0146, |  |  |  |
0.0149, 0.0147, 0.0146, 0.0144, |  |  |  |
0.0151, 0.0153, 0.0149, 0.0147, |  |  |  |
0.0154, 0.0147, 0.0150, 0.0148, |  |  |  |
0.0146, 0.0146, 0.0144, 0.0146, |  |  |  |
0.0150, 0.0144, 0.0142 |  5cm 0.0033, 0.0033, 0.0033, 0.0033, |  |  |
0.0033, 0.0035, 0.0033, 0.0034, |  |  |  |
0.0033, 0.0034, 0.0033, 0.0034, |  |  |  |
0.0033, 0.0034, 0.0034, 0.0034, |  |  |  |
0.0033, 0.0034, 0.0033, 0.0034, |  |  |  |
0.0034, 0.0034, 0.0036, 0.0034, |  |  |  |
0.0034, 0.0034, 0.0035, 0.0034, |  |  |  |
0.0036, 0.0035, 0.0035 |  5cm{ 1.9700, 1.7690, 1.7440, 1.6020, |  |  |
1.7960, 2.8260, 1.6020, 1.4690, |  |  |  |
1.7440, 1.5850, 1.4930, 1.3780, |  |  |  |
1.6020, 1.4690, 1.3850, 1.2970, |  |  |  |
1.7690, 2.4730, 1.5850, 1.4600, |  |  |  |
2.6960, 1.4690, 2.5360, 1.5080, |  |  |  |
1.3850, 1.3780, 1.2970, 1.4600, |  |  |  |
2.6370, 1.2970, 1.2500}d |  5cm 9.0000c, 7.2900c, 6.7600c, 5.2900c, |  |  |
7.2900c, 1.9600c, 5.7600c, 4.4100c, |  |  |  |
6.7600c, 5.2900c, 4.8400c, 3.6100c, |  |  |  |
5.2900c, 4.0000c, 3.6100c, 2.5600c, |  |  |  |
7.2900c, 2.5600c, 5.2900c, 4.0000c, |  |  |  |
2.2500c, 4.0000c, 3.6000b, 5.2900c, |  |  |  |
3.6100c, 3.6100c, 2.2500c, 4.0000c, |  |  |  |
1.6000b, 2.5600c, 1.4400c |  |  |  |
$\rho_{ggege}$ | 0.4712 | 0.8104 | 0.9004 | 0.8979
Pr. r. s. |  5cm 0.0152, 0.0150, 0.0150, 0.0148, |  |  |
0.0151, 0.0149, 0.0148, 0.0146, |  |  |  |
0.0150, 0.0148, 0.0152, 0.0145, |  |  |  |
0.0148, 0.0146, 0.0145, 0.0143, |  |  |  |
0.0150, 0.0148, 0.0154, 0.0146, |  |  |  |
0.0149, 0.0147, 0.0146, 0.0144, |  |  |  |
0.0153, 0.0145, 0.0147, 0.0146, |  |  |  |
0.0144, 0.0147, 0.0141 |  5cm 0.0032, 0.0032, 0.0032, 0.0033, |  |  |
0.0032, 0.0033, 0.0033, 0.0033, |  |  |  |
0.0032, 0.0033, 0.0034, 0.0033, |  |  |  |
0.0033, 0.0033, 0.0033, 0.0034, |  |  |  |
0.0032, 0.0033, 0.0034, 0.0033, |  |  |  |
0.0033, 0.0033, 0.0033, 0.0034, |  |  |  |
0.0034, 0.0033, 0.0035, 0.0033, |  |  |  |
0.0034, 0.0035, 0.0034 |  5cm 1.3140d, 1.1450d, 1.1240d, 9.9700c, |  |  |
1.1450d, 1.0250d, 1.0100d, 9.0900c, |  |  |  |
1.1240d, 9.9700c, 1.9490d, 9.0100c, |  |  |  |
9.9700c, 9.0400c, 9.0100c, 8.5000c, |  |  |  |
1.1450d, 1.0100d, 2.2600d, 9.0400c, |  |  |  |
1.0100d, 9.0900c, 9.0400c, 8.4500c, |  |  |  |
2.0570d, 9.0100c, 1.5220d, 9.0400c, |  |  |  |
8.4500c, 1.4450d, 7.9300c |  5cm 1.4440d, 1.2250d, 1.2250d, 1.0240d, |  |  |
1.2250d, 1.0890d, 1.0240d, 8.4100c, |  |  |  |
1.1560d, 9.6100c, 4.4100c, 7.2900c, |  |  |  |
9.6100c, 7.8400c, 7.8400c, 5.7600c, |  |  |  |
1.2250d, 1.0240d, 3.6100c, 7.8400c, |  |  |  |
1.0240d, 8.4100c, 8.4100c, 6.2500c, |  |  |  |
4.0000c, 7.8400c, 2.2500c, 7.8400c, |  |  |  |
6.2500c, 2.5600c, 4.0000c |  |  |  |
Table 8: Outputs of dynamical quantum search algorithm for some probabilities of marked states $\rho_{r_{1}r_{2}r_{3}r_{4}r_{5}}$ ($r_{1},r_{2},r_{3},r_{4},r_{5}=e,g$) and unmarked states of 5-qubit for different values of time, where $\frac{\gamma_{1}}{\lambda}=\frac{\theta_{5}}{113}$, $\frac{\gamma_{2}}{\lambda}=\frac{\theta_{5}}{90}$, $\frac{\gamma_{3}}{\lambda}=\frac{\theta_{5}}{140}$, $\frac{\gamma_{4}}{\lambda}=\frac{\theta_{5}}{100}$ and $\frac{\gamma_{5}}{\lambda}=\frac{\theta_{5}}{125}$; $a=\times 10^{-8}$ and $b=\times 10^{-7}$. Times | $t=\frac{0.331\pi\hbar}{32\lambda\theta_{5}}$ | $t=\frac{0.566\pi\hbar}{32\lambda\theta_{5}}$ | $t_{5p}=\frac{0.8661\pi\hbar}{32\lambda\theta_{5}}$ | $t=\frac{\pi\hbar}{32\lambda\theta_{5}}$
---|---|---|---|---
$\rho_{eeeeg}$ | 4.9268e | 1.9396e | 1.9925e | 1.4400e
Pr. r. s. |  5cm 3.0069e, 3.9569e, 5.4196e, 3.6756e, |  |  |
4.9844e, 4.5000e, 5.1885e, 3.9946e, |  |  |  |
5.5514e, 4.9860e, 6.0265e, 4.5785e, |  |  |  |
5.3434e, 4.7248e, 6.1245e, 4.0706e, |  |  |  |
5.6450e, 5.0705e, 6.1956e, 4.6580e, |  |  |  |
5.4445e, 4.8296e, 6.3040e, 5.1560e, |  |  |  |
6.3673e, 5.5953e, 8.0965e, 4.9360e, |  |  |  |
6.6002e, 5.5714e, 0.0032 |  5cm 8.4500d, 7.7450d, 1.0345e, 7.7450d, |  |  |
1.0100e, 8.4020d, 1.1545e, 7.7450d, |  |  |  |
1.0345e, 8.0900d, 1.0642e, 8.4020d, |  |  |  |
1.1338e, 1.1601e, 2.2978e, 7.7450d, |  |  |  |
1.0345e, 8.1170d, 1.0445e, 8.4250d, |  |  |  |
1.1133e, 1.1392e, 2.2394e, 7.9400d, |  |  |  |
1.0309e, 1.0280e, 1.9325e, 1.1234e, |  |  |  |
2.1818e, 2.4097e, 8.9730e |  5cm 8.8820d, 8.4250d, 1.0413e, 8.3520d, |  |  |
9.8810d, 8.4100d, 7.5560d, 8.6690d, |  |  |  |
1.0498e, 8.6530d, 7.8440d, 8.4100d, |  |  |  |
7.5560d, 7.2650d, 4.5050d, 8.6690d, |  |  |  |
1.0498e, 8.6530d, 7.7620d, 8.4890d, |  |  |  |
7.6250d, 7.2650d, 4.4980d, 8.7380d, |  |  |  |
7.7620d, 7.6100d, 4.2410d, 7.3300d, |  |  |  |
4.4930d, 4.6600d, 1.7557e |  5cm 5.3290d, 4.2250d, 5.4760d, 4.3560d, |  |  |
5.4760d, 4.3560d, 5.4760d, 4.2250d, |  |  |  |
5.4760d, 4.0960d, 4.9000d, 4.3560d, |  |  |  |
5.3290d, 5.6250d, 1.1236e, 4.2250d, |  |  |  |
5.4760d, 4.0960d, 4.9000d, 4.3560d, |  |  |  |
5.3290d, 5.4760d, 1.1025e, 4.0960d, |  |  |  |
4.7610d, 4.9000d, 9.6040d, 5.4760d, |  |  |  |
1.0816e, 1.1881e, 4.5369e |  |  |  |
$\rho_{eegeg}$ | 6.3684e | 2.8265e | 2.5492e | 1.9321e
Pr. r. s. |  5cm 2.7266e, 4.1876e, 3.6496e, 5.0013e, |  |  |
3.8408e, 4.3661e, 5.0960e, 3.7220e, |  |  |  |
5.1301e, 4.6045e, 5.4920e, 4.4420e, |  |  |  |
5.2370e, 4.3874e, 5.6180e, 3.7458e, |  |  |  |
5.2154e, 4.6472e, 5.5953e, 4.5256e, |  |  |  |
5.3800e, 4.4874e, 5.8154e, 4.7714e, |  |  |  |
5.7717e, 5.0338e, 7.2125e, 4.6309e, |  |  |  |
6.0842e, 4.9865e, 0.0031 |  5cm 5.9450d, 7.9570d, 6.1000d, 6.8050d, |  |  |
8.9360d, 7.6050d, 1.3093e, 6.1000d, |  |  |  |
6.8050d, 6.5970d, 8.3210d, 7.4320d, |  |  |  |
1.3093e, 9.8930d, 2.1697e, 6.1000d, |  |  |  |
6.8240d, 6.5970d, 8.3210d, 7.4450d, |  |  |  |
1.3093e, 9.7330d, 2.1125e, 6.6100d, |  |  |  |
7.9690d, 8.7880d, 1.7890e, 9.5400d, |  |  |  |
2.0561e, 2.2045e, 8.7986e |  5cm 6.2600d, 8.4500d, 6.9320d, 7.4500d, |  |  |
8.5930d, 7.1540d, 7.9560d, 6.9320d, |  |  |  |
7.5290d, 7.2200d, 6.1970d, 7.2250d, |  |  |  |
8.1250d, 5.9680d, 3.7850d, 7.0010d, |  |  |  |
7.5290d, 7.2200d, 6.1970d, 7.2250d, |  |  |  |
8.1860d, 5.9680d, 3.7700d, 7.2970d, |  |  |  |
6.2660d, 6.0650d, 3.3680d, 5.8820d, |  |  |  |
3.7570d, 3.7210d, 1.8386e |  5cm 3.2490d, 4.2250d, 3.1360d, 3.1360d, |  |  |
4.7610d, 3.6000d, 6.2410d, 3.0250d, |  |  |  |
3.0250d, 3.1360d, 3.6000d, 3.4810d, |  |  |  |
6.0840d, 4.3560d, 9.8010d, 3.0250d, |  |  |  |
3.0250d, 3.0250d, 3.6000d, 3.4810d, |  |  |  |
6.0840d, 4.3560d, 9.6040d, 3.0250d, |  |  |  |
3.4810d, 3.8440d, 8.2810d, 4.2250d, |  |  |  |
9.4090d, 1.0201e, 4.2849e |  |  |  |
$\rho_{egeee}$ | 4.7081e | 1.9066e | 1.9652e | 1.3924e
Pr. r. s. |  5cm 3.0197e, 4.1732e, 3.9569e, 5.2925e, |  |  |
3.6756e, 4.8690e, 4.5000e, 5.1209e, |  |  |  |
5.6052e, 5.1994e, 6.0740e, 4.7317e, |  |  |  |
5.3434e, 4.7912e, 6.1498e, 4.0706e, |  |  |  |
5.4689e, 5.0705e, 6.0930e, 4.6580e, |  |  |  |
5.3874e, 4.8296e, 6.2785e, 5.3737e, |  |  |  |
6.4160e, 5.7424e, 8.1252e, 5.0362e, |  |  |  |
6.6265e, 5.6186e, 0.0032 |  5cm 8.4250d, 7.7480d, 7.7450d, 8.1460d, |  |  |
7.7450d, 8.4500d, 8.5850d, 1.1285e, |  |  |  |
1.0548e, 1.0525e, 1.0900e, 1.0485e, |  |  |  |
1.1600e, 1.2125e, 2.3053e, 7.7480d, |  |  |  |
8.0000d, 8.2960d, 1.0000e, 8.6080d, |  |  |  |
1.0877e, 1.1650e, 2.2685e, 1.0525e, |  |  |  |
1.0565e, 1.1188e, 1.9669e, 1.1965e, |  |  |  |
2.2180e, 2.4473e, 8.9809e |  5cm 8.7050d, 8.9170d, 8.6690d, 8.9050d, |  |  |
8.3520d, 8.7370d, 8.5770d, 7.5560d, |  |  |  |
1.0874e, 1.0330e, 7.8440d, 9.9850d, |  |  |  |
7.7860d, 7.5850d, 4.6330d, 8.6690d, |  |  |  |
8.9920d, 8.8200d, 7.7620d, 8.6560d, |  |  |  |
7.6960d, 7.4890d, 4.6330d, 1.0600e, |  |  |  |
8.0000d, 7.9300d, 4.3720d, 7.6500d, |  |  |  |
4.4900d, 4.6600d, 1.7389e |  5cm 5.1840d, 4.2250d, 4.3560d, 4.0960d, |  |  |
4.3560d, 4.3560d, 4.4890d, 5.4760d, |  |  |  |
5.6250d, 5.6250d, 5.0410d, 5.6250d, |  |  |  |
5.4760d, 5.9290d, 1.1449e, 4.3560d, |  |  |  |
3.9690d, 4.2250d, 4.7610d, 4.4890d, |  |  |  |
5.1840d, 5.6250d, 1.1236e, 5.6250d, |  |  |  |
4.9000d, 5.3290d, 9.6040d, 5.7760d, |  |  |  |
1.0816e, 1.1881e, 4.5796e |  |  |  |
$\rho_{egegg}$ | 0.0011 | 5.8981e | 4.3525e | 3.5721e
Pr. r. s. |  5cm 2.3722e, 3.6452e, 3.4866e, 5.6537e, |  |  |
2.9377e, 3.8464e, 3.5989e, 4.1645e, |  |  |  |
3.5405e, 5.7330e, 5.5386e, 3.6650e, |  |  |  |
4.2577e, 3.9041e, 4.4434e, 3.2456e, |  |  |  |
4.3938e, 4.1296e, 5.2561e, 3.5620e, |  |  |  |
3.8020e, 3.3908e, 4.2761e, 4.2029e, |  |  |  |
5.3645e, 4.9045e, 6.5173e, 3.5066e, |  |  |  |
4.4629e, 3.7705e, 0.0027 |  5cm 4.3280d, 4.8200d, 4.9860d, 8.7300d, |  |  |
4.6800d, 4.9640d, 5.2650d, 7.5850d, |  |  |  |
4.9860d, 8.3450d, 9.3160d, 5.1220d, |  |  |  |
7.4210d, 7.9370d, 2.4592e, 4.5140d, |  |  |  |
4.5250d, 4.6730d, 6.2900d, 5.2840d, |  |  |  |
7.3970d, 7.9210d, 1.8596e, 4.6730d, |  |  |  |
6.1330d, 6.7600d, 2.2160e, 7.7440d, |  |  |  |
1.8056e, 1.9700e, 8.7425e |  5cm 4.0840d, 4.9370d, 4.8800d, 7.5560d, |  |  |
4.4690d, 4.5620d, 4.5730d, 3.8050d, |  |  |  |
4.8800d, 7.5560d, 7.8130d, 4.5730d, |  |  |  |
3.8480d, 3.8420d, 4.1650d, 4.5730d, |  |  |  |
4.7450d, 4.6820d, 4.0930d, 4.5730d, |  |  |  |
3.6490d, 3.5730d, 2.4770d, 4.6820d, |  |  |  |
4.1480d, 4.1060d, 4.3720d, 3.5730d, |  |  |  |
2.4400d, 2.6930d, 2.0740e |  5cm 1.7640d, 1.7640d, 1.8490d, 3.2490d, |  |  |
1.7640d, 1.7640d, 1.8490d, 2.6010d, |  |  |  |
1.8490d, 3.1360d, 3.6000d, 1.8490d, |  |  |  |
2.5000d, 2.8090d, 9.8010d, 1.6810d, |  |  |  |
1.4440d, 1.6000d, 2.0250d, 1.8490d, |  |  |  |
2.5000d, 2.7040d, 7.2250d, 1.6000d, |  |  |  |
1.9360d, 2.2090d, 8.6490d, 2.7040d, |  |  |  |
6.8890d, 7.7440d, 3.8416e |  |  |  |
$\rho_{geeee}$ | 4.7681e | 1.9125e | 1.9652e | 1.4161e
Pr. r. s. |  5cm 3.0197e, 4.1732e, 3.9569e, 5.2925e, |  |  |
3.6756e, 4.8690e, 4.5000e, 5.1209e, |  |  |  |
4.0325e, 5.3802e, 4.9860e, 5.9714e, |  |  |  |
4.5785e, 5.2865e, 4.7248e, 6.0745e, |  |  |  |
5.6521e, 5.2445e, 6.1956e, 4.8177e, |  |  |  |
5.4445e, 4.8865e, 6.3730e, 5.3737e, |  |  |  |
6.4160e, 5.7424e, 8.1252e, 5.0362e, |  |  |  |
6.6265e, 5.6186e, 0.0032 |  5cm 8.4500d, 7.7480d, 7.7450d, 8.1460d, |  |  |
7.7450d, 8.2690d, 8.5850d, 1.1080e, |  |  |  |
7.7450d, 7.9690d, 8.2690d, 1.0000e, |  |  |  |
8.6080d, 1.0877e, 1.1601e, 2.3273e, |  |  |  |
1.0573e, 1.0525e, 1.0504e, 1.0282e, |  |  |  |
1.1393e, 1.1912e, 2.2469e, 1.0525e, |  |  |  |
1.0309e, 1.0985e, 1.9325e, 1.1912e, |  |  |  |
2.1893e, 2.4170e, 8.9730e |  5cm 8.8820d, 8.7460d, 8.6690d, 8.9050d, |  |  |
8.3520d, 8.7370d, 8.5770d, 7.5560d, |  |  |  |
8.6690d, 8.9920d, 8.8200d, 7.6850d, |  |  |  |
8.5770d, 7.6250d, 7.4240d, 4.6400d, |  |  |  |
1.0685e, 1.0413e, 7.9210d, 9.9850d, |  |  |  |
7.6250d, 7.5850d, 4.4980d, 1.0413e, |  |  |  |
8.0000d, 7.7690d, 4.3720d, 7.6500d, |  |  |  |
4.4900d, 4.6600d, 1.7389e |  5cm 5.1840d, 4.2250d, 4.3560d, 4.0960d, |  |  |
4.3560d, 4.2250d, 4.4890d, 5.3290d, |  |  |  |
4.3560d, 3.9690d, 4.2250d, 4.7610d, |  |  |  |
4.4890d, 5.3290d, 5.6250d, 1.1236e, |  |  |  |
5.6250d, 5.6250d, 4.9000d, 5.6250d, |  |  |  |
5.4760d, 5.7760d, 1.1025e, 5.6250d, |  |  |  |
4.9000d, 5.1840d, 9.6040d, 5.7760d, |  |  |  |
1.0816e, 1.1881e, 4.5796e |  |  |  |
$\rho_{gegeg}$ | 9.8957e | 5.4797e | 3.9770e | 3.3489e
Pr. r. s. |  5cm 2.4250e, 3.6658e, 3.2125e, 4.2841e, |  |  |
3.3181e, 5.2317e, 3.7402e, 4.4225e, |  |  |  |
3.2456e, 4.4005e, 3.9005e, 4.3552e, |  |  |  |
3.8081e, 4.5610e, 3.4697e, 4.3733e, |  |  |  |
3.5605e, 5.5009e, 4.1296e, 5.0866e, |  |  |  |
5.1280e, 4.1993e, 4.9613e, 4.2434e, |  |  |  |
5.2385e, 4.1282e, 5.7556e, 4.3749e, |  |  |  |
5.2496e, 4.0385e, 0.0027 |  5cm 4.5370d, 5.0440d, 4.7140d, 4.9160d, |  |  |
5.2370d, 9.0890d, 5.3780d, 7.7930d, |  |  |  |
4.7140d, 4.9160d, 5.3650d, 7.0810d, |  |  |  |
5.3650d, 7.6330d, 8.1040d, 1.8938e, |  |  |  |
5.0690d, 7.7800d, 5.0660d, 6.7880d, |  |  |  |
9.4900d, 8.1490d, 2.4973e, 5.0660d, |  |  |  |
6.6420d, 7.4120d, 1.6553e, 7.9700d, |  |  |  |
2.4697e, 1.9769e, 8.6333e |  5cm 4.3730d, 5.0660d, 4.8250d, 4.9930d, |  |  |
4.9540d, 7.6500d, 4.8800d, 4.2500d, |  |  |  |
4.8800d, 4.9930d, 4.9960d, 4.0570d, |  |  |  |
4.8100d, 4.2970d, 3.9610d, 2.6280d, |  |  |  |
5.1970d, 7.3090d, 4.9300d, 4.3290d, |  |  |  |
7.9130d, 4.2050d, 4.5890d, 4.9300d, |  |  |  |
4.3840d, 4.0100d, 2.3780d, 4.2050d, |  |  |  |
4.7050d, 2.8010d, 1.9720e |  5cm 1.9360d, 1.9360d, 1.8490d, 1.7640d, |  |  |
2.1160d, 3.7210d, 2.0250d, 2.8090d, |  |  |  |
1.8490d, 1.7640d, 1.9360d, 2.4010d, |  |  |  |
2.0250d, 2.7040d, 3.0250d, 7.5690d, |  |  |  |
1.9360d, 3.0250d, 1.8490d, 2.3040d, |  |  |  |
3.9690d, 2.9160d, 1.0201e, 1.7640d, |  |  |  |
2.2090d, 2.6010d, 6.4000d, 2.8090d, |  |  |  |
1.0000e, 7.9210d, 3.8416e |  |  |  |
$\rho_{geggg}$ | 0.0022 | 0.0015 | 8.4776e | 7.6729e
Pr. r. s. |  5cm 2.1097e, 3.0185e, 2.9017e, 4.1992e, |  |  |
2.7421e, 4.0482e, 3.9266e, 7.1018e, |  |  |  |
2.7508e, 3.4154e, 3.1912e, 3.3525e, |  |  |  |
2.9905e, 3.0625e, 2.8240e, 3.5197e, |  |  |  |
2.9597e, 4.3034e, 4.1353e, 6.9269e, |  |  |  |
3.9869e, 7.0690e, 7.1045e, 3.3125e, |  |  |  |
3.5033e, 3.1700e, 4.2562e, 2.9245e, |  |  |  |
3.7120e, 3.2840e, 0.0015 |  5cm 4.1410d, 4.2050d, 4.2640d, 4.4960d, |  |  |
4.1490d, 4.7560d, 4.9300d, 1.2325e, |  |  |  |
4.5610d, 4.8010d, 4.9810d, 6.4660d, |  |  |  |
5.0320d, 6.6560d, 7.0130d, 1.6388e, |  |  |  |
4.2640d, 4.4960d, 4.6250d, 1.0169e, |  |  |  |
4.7970d, 1.1674e, 1.3252e, 4.9320d, |  |  |  |
6.2800d, 6.6250d, 1.5385e, 6.8170d, |  |  |  |
1.5885e, 1.6904e, 8.7125e |  5cm2.2370d, 2.4650d, 2.4340d, 2.7040d, |  |  |
2.4340d, 2.6650d, 2.7250d, 5.8130d, |  |  |  |
2.4050d, 2.3720d, 2.3120d, 1.7450d, |  |  |  |
2.2850d, 1.8000d, 1.7890d, 1.8320d, |  |  |  |
2.4650d, 2.6090d, 2.6650d, 4.8400d, |  |  |  |
2.6650d, 5.5250d, 6.1090d, 2.3410d, |  |  |  |
1.7450d, 1.7300d, 1.6490d, 1.7890d, |  |  |  |
1.7650d, 1.9400d, 1.5538e |  5cm 8.4100c, 7.2900c, 7.8400c, 7.2900c, |  |  |
7.8400c, 7.8400c, 8.4100c, 3.1360d, |  |  |  |
8.4100c, 7.8400c, 8.4100c, 1.1560d, |  |  |  |
9.0000c, 1.2960d, 1.4440d, 4.4890d, |  |  |  |
7.2900c, 6.7600c, 7.2900c, 2.3040d, |  |  |  |
8.4100c, 2.8090d, 3.3640d, 8.4100c, |  |  |  |
1.1560d, 1.2250d, 4.0960d, 1.3690d, |  |  |  |
4.3560d, 4.6240d, 3.3124e |  |  |  |
$\rho_{ggege}$ | 0.0010 | 5.6410e | 4.1425e | 3.4225e
Pr. r. s. |  5cm 2.3985e, 3.3800e, 3.5060e, 4.3181e, |  |  |
2.9690e, 3.7700e, 3.6360e, 3.6682e, |  |  |  |
3.5405e, 4.3938e, 5.4401e, 5.2474e, |  |  |  |
3.7025e, 3.7953e, 3.9041e, 4.2400e, |  |  |  |
3.5752e, 4.4770e, 5.4721e, 5.3554e, |  |  |  |
3.7402e, 3.8792e, 3.9538e, 4.3893e, |  |  |  |
5.4920e, 5.4650e, 6.7010e, 4.0784e, |  |  |  |
4.5621e, 4.0673e, 0.0027 |  5cm 4.4100d, 4.5810d, 5.0960d, 4.7860d, |  |  |
4.7780d, 5.3930d, 5.3930d, 7.7480d, |  |  |  |
4.9570d, 4.6490d, 8.9170d, 6.6250d, |  |  |  |
5.3930d, 7.5730d, 8.3060d, 1.8913e, |  |  |  |
4.9570d, 4.6490d, 8.7300d, 6.4640d, |  |  |  |
5.3780d, 7.4050d, 8.1250d, 1.8640e, |  |  |  |
8.5280d, 6.3050d, 2.2525e, 7.9460d, |  |  |  |
1.8369e, 2.5856e, 8.7466e |  5cm 4.2050d, 4.7530d, 5.0090d, 4.8680d, |  |  |
4.6450d, 4.7530d, 4.6980d, 3.7640d, |  |  |  |
5.0090d, 4.8680d, 7.6500d, 4.2100d, |  |  |  |
4.7530d, 3.7640d, 4.0000d, 2.5700d, |  |  |  |
5.0660d, 4.9330d, 7.7170d, 4.2650d, |  |  |  |
4.7530d, 3.8050d, 4.0000d, 2.5700d, |  |  |  |
7.5560d, 4.3220d, 4.6280d, 4.0410d, |  |  |  |
2.5330d, 4.3520d, 2.0506e |  5cm 1.8490d, 1.6810d, 1.9360d, 1.6000d, |  |  |
1.8490d, 1.8490d, 1.9360d, 2.7040d, |  |  |  |
1.9360d, 1.6000d, 3.6000d, 2.2090d, |  |  |  |
1.9360d, 2.6010d, 2.9160d, 7.3960d, |  |  |  |
1.8490d, 1.6000d, 3.4810d, 2.1160d, |  |  |  |
1.9360d, 2.6010d, 2.8090d, 7.2250d, |  |  |  |
3.3640d, 2.1160d, 9.0250d, 2.8090d, |  |  |  |
7.0560d, 1.0609e, 3.8416e |  |  |  |
$\rho_{gggeg}$ | 0.0023 | 0.0015 | 8.7754e | 7.8961e
Pr. r. s. |  5cm 2.1073e, 2.9890e, 2.6874e, 3.3256e, |  |  |
2.7325e, 4.1000e, 2.9097e, 2.9250e, |  |  |  |
2.9017e, 4.2772e, 3.1714e, 3.2786e, |  |  |  |
3.9637e, 7.3225e, 2.7592e, 3.3898e, |  |  |  |
2.9306e, 4.3165e, 3.1912e, 3.3525e, |  |  |  |
4.0010e, 7.3053e, 2.7914e, 3.4697e, |  |  |  |
4.1738e, 7.1381e, 3.0985e, 4.1093e, |  |  |  |
7.3157e, 3.1322e, 0.0015 |  5cm 4.1050d, 4.1490d, 4.5700d, 4.8500d, |  |  |
4.0970d, 4.6660d, 5.0850d, 6.6890d, |  |  |  |
4.2100d, 4.4100d, 4.9010d, 6.4970d, |  |  |  |
4.8400d, 1.2356e, 7.0480d, 1.6400e, |  |  |  |
4.2100d, 4.4100d, 4.9010d, 6.3400d, |  |  |  |
4.7090d, 1.1920e, 6.8850d, 1.6145e, |  |  |  |
4.5370d, 9.9970d, 6.6890d, 1.5392e, |  |  |  |
1.3058e, 1.6916e, 8.8290e |  5cm 2.1460d, 2.3720d, 2.3120d, 2.2500d, |  |  |
2.3120d, 2.6650d, 2.1940d, 1.7060d, |  |  |  |
2.3410d, 2.6090d, 2.2210d, 1.6490d, |  |  |  |
2.6280d, 5.7130d, 1.6970d, 1.8180d, |  |  |  |
2.3410d, 2.6090d, 2.2210d, 1.6490d, |  |  |  |
2.6280d, 5.5700d, 1.6970d, 1.8180d, |  |  |  |
2.6090d, 4.7540d, 1.7170d, 1.6370d, |  |  |  |
6.1090d, 1.9300d, 1.6164e |  5cm 8.4100c, 6.7600c, 7.8400c, 7.2900c, |  |  |
7.2900c, 7.2900c, 9.0000c, 1.2960d, |  |  |  |
7.2900c, 6.2500c, 7.8400c, 1.1560d, |  |  |  |
7.8400c, 3.0250d, 1.3690d, 4.3560d, |  |  |  |
6.7600c, 6.2500c, 7.8400c, 1.0890d, |  |  |  |
7.8400c, 2.9160d, 1.3690d, 4.3560d, |  |  |  |
6.7600c, 2.2090d, 1.2250d, 3.9690d, |  |  |  |
3.2490d, 4.6240d, 3.3124e |  |  |  |
Table 9: The same as Table (8) but $\frac{\gamma_{1}}{\lambda}=\frac{4\theta_{5}}{5}$, $\frac{\gamma_{2}}{\lambda}=\frac{7\theta_{5}}{9}$, $\frac{\gamma_{3}}{\lambda}=\frac{5\theta_{5}}{8}$, $\frac{\gamma_{4}}{\lambda}=\frac{3\theta_{5}}{4}$ and $\frac{\gamma_{5}}{\lambda}=\frac{6\theta_{5}}{7}$. Times | $t=\frac{0.331\pi\hbar}{64\lambda\theta_{6}}$ | $t=\frac{0.566\pi\hbar}{64\lambda\theta_{6}}$ | $t_{6p}=\frac{0.9899\pi\hbar}{64\lambda\theta_{6}}$ | $t=\frac{\pi\hbar}{64\lambda\theta_{6}}$
---|---|---|---|---
$\rho_{eeeeee}$ | 0.3216 | 0.6082 | 0.7566 | 0.7566
$\rho_{eeeegg}$ | 0.3365 | 0.6362 | 0.7914 | 0.7914
$\rho_{eeegge}$ | 0.3381 | 0.6389 | 0.7949 | 0.7950
$\rho_{eegeeg}$ | 0.3357 | 0.6346 | 0.7895 | 0.7894
$\rho_{eeggee}$ | 0.3372 | 0.6374 | 0.7931 | 0.7930
$\rho_{eegggg}$ | 0.3531 | 0.6670 | 0.8299 | 0.8299
$\rho_{egeeeg}$ | 0.3394 | 0.6416 | 0.7982 | 0.7982
$\rho_{egeegg}$ | 0.3471 | 0.6559 | 0.8162 | 0.8161
$\rho_{egegge}$ | 0.3487 | 0.6588 | 0.8198 | 0.8197
$\rho_{eggeeg}$ | 0.3463 | 0.6544 | 0.8141 | 0.8141
$\rho_{egggee}$ | 0.3479 | 0.6573 | 0.8177 | 0.8178
$\rho_{eggggg}$ | 0.3643 | 0.6879 | 0.8561 | 0.8562
$\rho_{geeege}$ | 0.3370 | 0.6369 | 0.7923 | 0.7923
$\rho_{geegeg}$ | 0.3469 | 0.6555 | 0.8155 | 0.8154
$\rho_{geeggg}$ | 0.3548 | 0.6702 | 0.8339 | 0.8339
$\rho_{gegegg}$ | 0.3519 | 0.6648 | 0.8272 | 0.8272
$\rho_{geggeg}$ | 0.3539 | 0.6685 | 0.8318 | 0.8319
$\rho_{gegggg}$ | 0.3619 | 0.6836 | 0.8506 | 0.8506
$\rho_{ggeeeg}$ | 0.3480 | 0.6575 | 0.8181 | 0.8179
$\rho_{ggegeg}$ | 0.3579 | 0.6760 | 0.8412 | 0.8411
$\rho_{ggeggg}$ | 0.3661 | 0.6912 | 0.8601 | 0.8603
$\rho_{gggege}$ | 0.3545 | 0.6699 | 0.8335 | 0.8336
$\rho_{ggggeg}$ | 0.3652 | 0.6896 | 0.8582 | 0.8582
$\rho_{gggggg}$ | 0.3736 | 0.7051 | 0.8777 | 0.8776
Table 10: Outputs of dynamical quantum search algorithm for some probabilities of marked states $\rho_{r_{1}r_{2}r_{3}r_{4}r_{5}r_{6}}$ ($r_{1},r_{2},r_{3},r_{4},r_{5},r_{6}=e,g$) of 6-qubit for different values of time, where $\frac{\gamma_{1}}{\lambda}=\frac{\theta_{6}}{113}$, $\frac{\gamma_{2}}{\lambda}=\frac{\theta_{6}}{90}$, $\frac{\gamma_{3}}{\lambda}=\frac{\theta_{6}}{140}$, $\frac{\gamma_{4}}{\lambda}=\frac{\theta_{6}}{100}$, $\frac{\gamma_{5}}{\lambda}=\frac{\theta_{6}}{125}$ and $\frac{\gamma_{6}}{\lambda}=\frac{\theta_{6}}{119}$. Times | $t=\frac{0.331\pi\hbar}{64\lambda\theta_{6}}$ | $t=\frac{0.566\pi\hbar}{64\lambda\theta_{6}}$ | $t_{6p}=\frac{0.9899\pi\hbar}{64\lambda\theta_{6}}$ | $t=\frac{\pi\hbar}{64\lambda\theta_{6}}$
---|---|---|---|---
$\rho_{eeegge}$ | 3.7540d | 1.6180d | 1.4170d | 1.4440d
Pr. r. s. |  5cm 1.6250d, 2.4980d, 2.6090d, 3.8210d, |  |  |
2.4660d, 3.4850d, 4.6250d, 2.1730d, |  |  |  |
3.1460d, 3.1690d, 3.9780d, 2.8730d, |  |  |  |
3.6010d, 3.4900d, 3.0880d, 2.4050d, |  |  |  |
3.5810d, 3.4850d, 4.6730d, 3.2800d, |  |  |  |
4.2500d, 4.0960d, 4.2120d, 2.9530d, |  |  |  |
3.6040d, 3.4820d, 3.2650d, 3.1360d, |  |  |  |
2.6120d, 2.4250d, 6.0020d, 2.4050d, |  |  |  |
3.5810d, 3.6020d, 4.8100d, 3.2800d, |  |  |  |
4.2500d, 4.2250d, 4.3940d, 2.9530d, |  |  |  |
3.7300d, 3.6040d, 3.3620d, 3.2490d, |  |  |  |
2.7540d, 2.4740d, 6.2290d, 3.3700d, |  |  |  |
4.3920d, 4.2500d, 4.5460d, 3.8530d, |  |  |  |
3.8260d, 3.5240d, 8.4500d, 3.3650d, |  |  |  |
2.8090d, 2.6650d, 6.4660d, 2.0500d, |  |  |  |
5.5810d, 5.3380d, 9.2909e |  5cm6.5600c, 8.1800c, 8.6500c, 9.3200c, |  |  |
8.6500c, 9.3200c, 1.1300d, 7.4000c, |  |  |  |
9.0100c, 9.0100c, 9.6200c, 8.5000c, |  |  |  |
9.4100c, 1.0210d, 1.1050d, 7.8500c, |  |  |  |
9.3200c, 9.3200c, 9.8500c, 9.0100c, |  |  |  |
9.6200c, 1.1050d, 1.1600d, 8.5000c, |  |  |  |
9.6200c, 9.4100c, 1.0980d, 9.4100c, |  |  |  |
1.1050d, 1.1140d, 3.6170d, 7.8500c, |  |  |  |
9.3200c, 9.3200c, 9.8500c, 9.0100c, |  |  |  |
9.6200c, 1.1050d, 1.1570d, 8.5000c, |  |  |  |
9.6200c, 9.4100c, 1.0980d, 9.4100c, |  |  |  |
1.1050d, 1.1140d, 3.5300d, 9.3200c, |  |  |  |
9.8500c, 9.6200c, 1.0890d, 9.6200c, |  |  |  |
1.0930d, 1.1650d, 3.2080d, 9.4100c, |  |  |  |
1.1050d, 1.1050d, 3.5280d, 1.1140d, |  |  |  |
3.7890d, 3.8800d, 2.7073e |  5cm 5.7800c, 6.7400c, 7.4000c, 7.9300c, |  |  |
7.4000c, 7.9300c, 8.4800c, 6.2500c, |  |  |  |
7.2500c, 7.2500c, 7.4000c, 6.7400c, |  |  |  |
6.7400c, 7.4000c, 3.6000c, 6.7400c, |  |  |  |
7.9300c, 7.9300c, 7.9300c, 7.4000c, |  |  |  |
7.4000c, 7.9300c, 4.4900c, 6.7400c, |  |  |  |
6.7400c, 6.7400c, 3.6000c, 6.2500c, |  |  |  |
3.2500c, 3.2500c, 7.3000b, 6.7400c, |  |  |  |
7.9300c, 7.9300c, 7.9300c, 7.4000c, |  |  |  |
7.4000c, 7.9300c, 4.4900c, 7.2500c, |  |  |  |
6.7400c, 6.7400c, 4.1000c, 6.2500c, |  |  |  |
3.2500c, 3.2500c, 7.3000b, 7.4000c, |  |  |  |
7.9300c, 7.4000c, 4.9000c, 6.8900c, |  |  |  |
4.1000c, 4.1000c, 2.5000b, 6.7400c, |  |  |  |
3.6000c, 3.2500c, 5.8000b, 2.9200c, |  |  |  |
8.5000b, 1.0400c, 6.9530d |  5cm 5.2900c, 6.7600c, 7.2900c, 7.8400c, |  |  |
7.2900c, 7.8400c, 8.4100c, 6.2500c, |  |  |  |
7.2900c, 7.2900c, 7.2900c, 6.7600c, |  |  |  |
6.7600c, 7.2900c, 3.6100c, 6.7600c, |  |  |  |
7.8400c, 7.8400c, 7.8400c, 7.2900c, |  |  |  |
7.2900c, 7.8400c, 4.4100c, 7.2900c, |  |  |  |
6.7600c, 6.7600c, 3.6100c, 6.2500c, |  |  |  |
3.2400c, 3.2400c, 4.9000b, 6.7600c, |  |  |  |
7.8400c, 7.8400c, 7.8400c, 7.2900c, |  |  |  |
7.2900c, 8.4100c, 4.4100c, 7.2900c, |  |  |  |
6.7600c, 6.7600c, 4.0000c, 6.2500c, |  |  |  |
3.2400c, 3.2400c, 4.9000b, 7.2900c, |  |  |  |
7.8400c, 7.2900c, 4.8400c, 7.2900c, |  |  |  |
4.0000c, 4.0000c, 9.0000a, 6.7600c, |  |  |  |
3.6100c, 3.2400c, 3.6000b, 2.8900c, |  |  |  |
6.4000b, 8.1000b, 6.8890d |  |  |  |
$\rho_{eggeeg}$ | 4.5730d | 2.1760d | 1.7690d | 1.7640d
Pr. r. s. |  5cm 1.5170d, 2.4400d, 2.3140d, 3.5810d, |  |  |
2.1380d, 3.2570d, 3.1460d, 4.2500d, |  |  |  |
2.1250d, 3.3140d, 2.9530d, 3.8450d, |  |  |  |
2.7700d, 3.4810d, 3.2490d, 2.9000d, |  |  |  |
2.2600d, 3.5890d, 3.2570d, 4.5140d, |  |  |  |
3.0600d, 3.9780d, 3.7300d, 3.8260d, |  |  |  |
3.1330d, 3.4820d, 3.0420d, 3.1400d, |  |  |  |
2.4400d, 2.2100d, 5.3890d, 2.2250d, |  |  |  |
3.4850d, 3.2570d, 4.3920d, 3.0370d, |  |  |  |
3.9850d, 3.7370d, 3.8970d, 2.8730d, |  |  |  |
3.7220d, 3.3650d, 3.2040d, 3.0260d, |  |  |  |
2.4740d, 2.1760d, 5.9240d, 3.1460d, |  |  |  |
4.2410d, 3.9850d, 4.2850d, 3.4850d, |  |  |  |
3.4900d, 3.0770d, 7.9010d, 3.2500d, |  |  |  |
2.7620d, 2.4740d, 5.9130d, 1.8850d, |  |  |  |
5.0580d, 5.1210d, 9.0817e |  5cm 6.1700c, 7.7300c, 7.7300c, 9.1400c, |  |  |
7.3000c, 8.8100c, 8.8100c, 9.5300c, |  |  |  |
7.7300c, 1.0180d, 8.3200c, 9.2800c, |  |  |  |
8.3200c, 9.0500c, 8.5000c, 1.0280d, |  |  |  |
7.7300c, 1.0530d, 8.8100c, 9.5300c, |  |  |  |
8.3200c, 9.2800c, 9.2800c, 1.0240d, |  |  |  |
1.0180d, 9.0500c, 1.2260d, 9.0500c, |  |  |  |
1.2290d, 1.0400d, 3.5300d, 7.7300c, |  |  |  |
9.1400c, 9.1400c, 9.5300c, 8.3200c, |  |  |  |
9.5300c, 9.2800c, 1.0240d, 8.3200c, |  |  |  |
9.2800c, 9.2800c, 1.0280d, 8.5000c, |  |  |  |
1.0330d, 1.0330d, 3.4450d, 8.8100c, |  |  |  |
9.5300c, 9.5300c, 1.0250d, 9.2800c, |  |  |  |
1.0250d, 1.0250d, 3.1250d, 9.0500c, |  |  |  |
1.2290d, 1.0330d, 3.3620d, 1.0400d, |  |  |  |
3.6170d, 3.7060d, 2.6645e |  5cm 4.7700c, 6.2500c, 6.2500c, 7.2500c, |  |  |
5.7800c, 6.7400c, 6.7400c, 6.7400c, |  |  |  |
5.7800c, 7.9300c, 6.7400c, 6.7400c, |  |  |  |
6.2500c, 6.2500c, 6.2500c, 3.2500c, |  |  |  |
6.2500c, 7.9300c, 6.7400c, 7.2500c, |  |  |  |
6.7400c, 6.7400c, 6.7400c, 3.9700c, |  |  |  |
7.9300c, 6.2500c, 4.4900c, 5.7800c, |  |  |  |
3.7300c, 2.6100c, 8.5000b, 6.2500c, |  |  |  |
7.2500c, 7.2500c, 7.2500c, 6.7400c, |  |  |  |
6.7400c, 6.7400c, 3.9700c, 6.2500c, |  |  |  |
6.2500c, 6.2500c, 3.2500c, 5.7800c, |  |  |  |
2.9200c, 2.9200c, 8.5000b, 6.7400c, |  |  |  |
6.7400c, 6.7400c, 3.9700c, 6.2500c, |  |  |  |
3.6000c, 3.6000c, 4.5000b, 5.7800c, |  |  |  |
4.1000c, 2.9200c, 6.8000b, 2.6100c, |  |  |  |
1.0400c, 1.2500c, 7.1370d |  5cm 4.8400c, 6.2500c, 6.2500c, 7.2900c, |  |  |
5.7600c, 6.7600c, 6.7600c, 6.7600c, |  |  |  |
5.7600c, 7.8400c, 6.7600c, 6.7600c, |  |  |  |
6.2500c, 6.2500c, 6.2500c, 3.2400c, |  |  |  |
6.2500c, 7.8400c, 6.7600c, 7.2900c, |  |  |  |
6.7600c, 6.7600c, 6.7600c, 4.0000c, |  |  |  |
7.8400c, 6.2500c, 4.4100c, 5.7600c, |  |  |  |
3.6100c, 2.5600c, 6.4000b, 6.2500c, |  |  |  |
7.2900c, 7.2900c, 7.2900c, 6.7600c, |  |  |  |
6.7600c, 6.7600c, 4.0000c, 6.2500c, |  |  |  |
6.2500c, 6.2500c, 3.2400c, 5.7600c, |  |  |  |
2.8900c, 2.8900c, 6.4000b, 6.7600c, |  |  |  |
6.7600c, 6.7600c, 4.0000c, 6.2500c, |  |  |  |
3.6100c, 3.2400c, 2.5000b, 6.2500c, |  |  |  |
4.0000c, 2.8900c, 4.9000b, 2.5600c, |  |  |  |
1.0000c, 1.2100c, 7.0560d |  |  |  |
$\rho_{geeggg}$ | 8.6440d | 4.8050d | 3.0980d | 3.1360d
Pr. r. s. |  5cm 1.3000d, 2.1250d, 2.1250d, 3.4450d, |  |  |
1.9610d, 3.2500d, 3.1410d, 5.2480d, |  |  |  |
1.7690d, 2.6260d, 2.5290d, 3.2530d, |  |  |  |
2.3410d, 2.9250d, 2.8250d, 2.4660d, |  |  |  |
1.8820d, 2.8730d, 2.7970d, 3.8480d, |  |  |  |
2.5970d, 3.4810d, 3.3640d, 3.7570d, |  |  |  |
2.3720d, 2.7200d, 2.6170d, 2.0410d, |  |  |  |
2.3400d, 1.5850d, 1.4120d, 4.5890d, |  |  |  |
2.0420d, 3.3610d, 3.2500d, 5.3780d, |  |  |  |
3.0650d, 5.0000d, 4.8820d, 2.4340d, |  |  |  |
3.0340d, 2.9250d, 2.6930d, 2.6260d, |  |  |  |
2.2100d, 2.0530d, 3.1120d, 2.6960d, |  |  |  |
3.6010d, 3.4810d, 3.9860d, 3.1370d, |  |  |  |
3.3300d, 3.1250d, 4.7610d, 2.5250d, |  |  |  |
1.8050d, 1.6200d, 4.9330d, 1.2200d, |  |  |  |
4.3600d, 4.2010d, 8.1577e |  5cm 6.1300c, 7.6100c, 7.6100c, 8.9000c, |  |  |
7.2200c, 8.9000c, 8.5300c, 1.2850d, |  |  |  |
6.8500c, 8.0800c, 8.0800c, 8.7200c, |  |  |  |
8.0800c, 8.2100c, 7.9400c, 9.0000c, |  |  |  |
7.2200c, 8.4500c, 8.4500c, 9.3200c, |  |  |  |
8.0800c, 9.0100c, 9.0100c, 9.7000c, |  |  |  |
7.7300c, 8.4500c, 8.4500c, 9.0000c, |  |  |  |
7.9400c, 9.0100c, 9.0400c, 3.2020d, |  |  |  |
7.2200c, 8.9000c, 8.9000c, 1.2850d, |  |  |  |
8.5300c, 1.2850d, 1.2850d, 8.0800c, |  |  |  |
8.2100c, 8.2100c, 9.6200c, 7.9400c, |  |  |  |
9.0000c, 9.0100c, 3.2180d, 8.0800c, |  |  |  |
9.0100c, 9.0100c, 9.7700c, 8.7200c, |  |  |  |
9.6500c, 9.6500c, 2.9860d, 7.9400c, |  |  |  |
9.0100c, 9.0100c, 3.0420d, 9.0400c, |  |  |  |
3.2850d, 3.3700d, 2.5801e |  5cm 3.8600c, 5.2000c, 5.2000c, 6.2500c, |  |  |
4.7700c, 5.6500c, 5.6500c, 8.4800c, |  |  |  |
4.7700c, 5.2000c, 5.2000c, 5.2000c, |  |  |  |
5.2000c, 4.7700c, 4.7700c, 2.2100c, |  |  |  |
4.7700c, 5.6500c, 5.6500c, 5.6500c, |  |  |  |
5.2000c, 5.2000c, 5.2000c, 3.2500c, |  |  |  |
5.2000c, 4.7700c, 4.7700c, 2.2100c, |  |  |  |
4.3600c, 1.9400c, 1.6900c, 1.7000c, |  |  |  |
4.7700c, 6.1200c, 6.1200c, 8.4800c, |  |  |  |
5.6500c, 7.9300c, 7.9300c, 5.2000c, |  |  |  |
4.7700c, 4.7700c, 2.5000c, 4.7700c, |  |  |  |
2.2100c, 2.2100c, 5.8000b, 5.6500c, |  |  |  |
5.6500c, 5.6500c, 3.6000c, 5.2000c, |  |  |  |
2.8100c, 2.8100c, 1.7000b, 4.3600c, |  |  |  |
1.9400c, 1.9400c, 1.4500c, 1.6900c, |  |  |  |
1.9700c, 1.9700c, 7.6690d |  5cm 4.0000c, 4.8400c, 4.8400c, 6.2500c, |  |  |
4.8400c, 5.7600c, 5.7600c, 8.4100c, |  |  |  |
4.4100c, 5.2900c, 5.2900c, 5.2900c, |  |  |  |
5.2900c, 4.8400c, 4.8400c, 2.2500c, |  |  |  |
4.8400c, 5.7600c, 5.7600c, 5.7600c, |  |  |  |
5.2900c, 5.2900c, 5.2900c, 3.2400c, |  |  |  |
5.2900c, 4.8400c, 4.8400c, 2.2500c, |  |  |  |
4.4100c, 1.9600c, 1.6900c, 1.4400c, |  |  |  |
4.8400c, 6.2500c, 5.7600c, 8.4100c, |  |  |  |
5.7600c, 8.4100c, 7.8400c, 5.2900c, |  |  |  |
4.8400c, 4.8400c, 2.5600c, 4.4100c, |  |  |  |
2.2500c, 1.9600c, 3.6000b, 5.7600c, |  |  |  |
5.7600c, 5.2900c, 3.2400c, 5.2900c, |  |  |  |
2.8900c, 2.8900c, 1.0000a, 4.4100c, |  |  |  |
1.9600c, 1.9600c, 1.2100c, 1.6900c, |  |  |  |
1.6900c, 1.9600c, 7.5690d |  |  |  |
$\rho_{gegggg}$ | 1.5541e | 1.1444e | 5.7600d | 5.6250d
Pr. r. s. |  5cm 1.3010d, 1.9720d, 1.9720d, 2.9840d, |  |  |
1.8320d, 2.7880d, 2.7380d, 4.1960d, |  |  |  |
1.7650d, 2.6450d, 2.5540d, 3.8650d, |  |  |  |
2.3780d, 3.6500d, 3.5600d, 6.2450d, |  |  |  |
1.8850d, 2.6010d, 2.6010d, 3.1690d, |  |  |  |
2.4250d, 2.8730d, 2.6960d, 2.2850d, |  |  |  |
2.1760d, 2.5290d, 2.4340d, 1.8640d, |  |  |  |
2.2810d, 1.5700d, 1.4050d, 2.8250d, |  |  |  |
1.9010d, 2.8850d, 2.7880d, 4.3250d, |  |  |  |
2.6450d, 3.9650d, 3.8420d, 6.4810d, |  |  |  |
2.4650d, 3.7690d, 3.6500d, 6.2570d, |  |  |  |
3.4450d, 6.0840d, 6.0880d, 2.4250d, |  |  |  |
2.9780d, 2.8730d, 2.5290d, 2.5970d, |  |  |  |
2.0800d, 1.8850d, 4.0180d, 2.3410d, |  |  |  |
1.6640d, 1.5700d, 3.1060d, 1.2500d, |  |  |  |
2.6690d, 2.5000d, 4.2250e |  5cm 5.4900c, 6.5000c, 6.5000c, 7.6100c, |  |  |
6.5000c, 7.6100c, 7.6100c, 8.6500c, |  |  |  |
6.5000c, 7.6100c, 7.6100c, 8.3200c, |  |  |  |
7.6100c, 8.3200c, 8.3200c, 1.5130d, |  |  |  |
6.5000c, 7.6100c, 7.2400c, 8.0100c, |  |  |  |
7.2400c, 8.0100c, 7.7200c, 7.9300c, |  |  |  |
7.2400c, 7.2500c, 7.2500c, 7.8800c, |  |  |  |
7.2500c, 7.8500c, 7.8500c, 2.4580d, |  |  |  |
6.5000c, 7.6100c, 7.6100c, 8.6500c, |  |  |  |
7.6100c, 8.3200c, 8.3200c, 1.4170d, |  |  |  |
7.6100c, 8.3200c, 8.3200c, 1.5130d, |  |  |  |
8.3200c, 1.5880d, 1.5880d, 7.2400c, |  |  |  |
8.0100c, 8.0100c, 7.9300c, 7.2500c, |  |  |  |
7.8800c, 7.8800c, 2.3300d, 7.2500c, |  |  |  |
7.8800c, 7.8500c, 2.3930d, 7.8500c, |  |  |  |
2.4580d, 2.5250d, 1.8178e |  5cm2.7200c, 3.4900c, 3.4900c, 3.8600c, |  |  |
3.4900c, 3.8600c, 3.8600c, 4.2500c, |  |  |  |
3.1400c, 3.8600c, 3.8600c, 3.8600c, |  |  |  |
3.4900c, 3.8600c, 3.8600c, 5.7800c, |  |  |  |
3.1400c, 3.8600c, 3.4900c, 3.4900c, |  |  |  |
3.4900c, 3.1400c, 3.1400c, 1.3700c, |  |  |  |
3.4900c, 3.1400c, 3.1400c, 1.1600c, |  |  |  |
2.8100c, 1.1600c, 9.7000b, 1.2200c, |  |  |  |
3.4900c, 3.8600c, 3.8600c, 4.2500c, |  |  |  |
3.8600c, 4.2500c, 3.8600c, 5.3300c, |  |  |  |
3.4900c, 3.8600c, 3.8600c, 5.7800c, |  |  |  |
3.8600c, 5.7800c, 5.7800c, 3.4900c, |  |  |  |
3.1400c, 3.1400c, 1.3700c, 3.1400c, |  |  |  |
1.3700c, 1.1600c, 1.0100c, 2.8100c, |  |  |  |
1.1600c, 1.1600c, 1.2200c, 9.7000b, |  |  |  |
1.4500c, 1.4500c, 3.9940d |  5cm2.8900c, 3.6100c, 3.2400c, 4.0000c, |  |  |
3.2400c, 4.0000c, 4.0000c, 4.0000c, |  |  |  |
3.2400c, 3.6100c, 3.6100c, 4.0000c, |  |  |  |
3.6100c, 4.0000c, 4.0000c, 5.7600c, |  |  |  |
3.2400c, 3.6100c, 3.6100c, 3.2400c, |  |  |  |
3.6100c, 3.2400c, 3.2400c, 1.4400c, |  |  |  |
3.2400c, 2.8900c, 2.8900c, 1.2100c, |  |  |  |
2.8900c, 1.0000c, 1.0000c, 1.2100c, |  |  |  |
3.2400c, 4.0000c, 4.0000c, 4.4100c, |  |  |  |
3.6100c, 4.0000c, 4.0000c, 5.2900c, |  |  |  |
3.6100c, 4.0000c, 4.0000c, 5.7600c, |  |  |  |
4.0000c, 5.7600c, 5.7600c, 3.6100c, |  |  |  |
3.2400c, 3.2400c, 1.4400c, 2.8900c, |  |  |  |
1.2100c, 1.2100c, 8.1000b, 2.8900c, |  |  |  |
1.0000c, 1.0000c, 1.0000c, 8.1000b, |  |  |  |
1.2100c, 1.4400c, 3.8440d |  |  |  |
$\rho_{ggggeg}$ | 1.1466e | 1.0372e | 5.0180d | 4.9000d
Pr. r. s. |  5cm1.3520d, 2.0450d, 2.0250d, 2.8330d, |  |  |
1.9010d, 2.7880d, 2.6010d, 3.1690d, |  |  |  |
1.8320d, 2.6450d, 2.4250d, 2.8730d, |  |  |  |
2.4650d, 3.6500d, 2.5000d, 1.9700d, |  |  |  |
1.9010d, 2.8850d, 2.6450d, 3.2570d, |  |  |  |
2.6000d, 3.8420d, 2.8730d, 2.5290d, |  |  |  |
2.4650d, 3.6250d, 2.5970d, 2.0570d, |  |  |  |
3.3320d, 5.3330d, 1.5850d, 3.2170d, |  |  |  |
1.9720d, 2.8850d, 2.6450d, 3.3700d, |  |  |  |
2.6930d, 3.9650d, 2.9530d, 2.6570d, |  |  |  |
2.5540d, 3.7440d, 2.5970d, 2.1690d, |  |  |  |
3.4450d, 5.3380d, 1.6810d, 3.3130d, |  |  |  |
2.6930d, 4.0690d, 2.9530d, 2.7560d, |  |  |  |
3.6250d, 5.5570d, 2.1050d, 4.5140d, |  |  |  |
3.4180d, 5.4920d, 1.6810d, 3.5300d, |  |  |  |
5.1850d, 2.7450d, 4.2850e |  5cm5.4900c, 6.8500c, 6.8500c, 7.6500c, |  |  |
6.5000c, 7.2400c, 7.2400c, 7.7200c, |  |  |  |
6.1300c, 7.2400c, 7.2400c, 7.7200c, |  |  |  |
7.2400c, 8.0100c, 6.9800c, 7.8500c, |  |  |  |
6.5000c, 7.2400c, 7.2400c, 7.7200c, |  |  |  |
7.2400c, 8.0100c, 7.7200c, 7.8800c, |  |  |  |
7.2400c, 8.0100c, 7.2500c, 7.8800c, |  |  |  |
8.0100c, 1.4690d, 7.8500c, 2.3930d, |  |  |  |
6.5000c, 8.0200c, 7.6500c, 7.7200c, |  |  |  |
7.2400c, 8.3200c, 7.7200c, 7.8800c, |  |  |  |
7.2400c, 8.0100c, 7.2500c, 7.8800c, |  |  |  |
8.0100c, 1.4690d, 7.8500c, 2.3930d, |  |  |  |
7.2400c, 8.3200c, 7.7200c, 7.9300c, |  |  |  |
8.0100c, 1.3250d, 7.8800c, 2.3300d, |  |  |  |
8.0100c, 1.3960d, 7.8500c, 2.3300d, |  |  |  |
1.5440d, 2.4580d, 1.7620e |  5cm 2.8100c, 3.4900c, 3.4900c, 3.8600c, |  |  |
3.4900c, 3.8600c, 3.8600c, 3.4900c, |  |  |  |
3.1400c, 3.8600c, 3.4900c, 3.1400c, |  |  |  |
3.4900c, 3.8600c, 3.1400c, 1.3700c, |  |  |  |
3.4900c, 3.8600c, 3.8600c, 3.4900c, |  |  |  |
3.8600c, 3.8600c, 3.1400c, 1.6000c, |  |  |  |
3.8600c, 3.8600c, 3.1400c, 1.3700c, |  |  |  |
3.8600c, 5.3300c, 1.1600c, 1.0100c, |  |  |  |
3.4900c, 3.8600c, 3.8600c, 3.4900c, |  |  |  |
3.8600c, 4.2500c, 3.4900c, 1.6000c, |  |  |  |
3.8600c, 3.8600c, 3.1400c, 1.3700c, |  |  |  |
3.8600c, 5.3300c, 1.1600c, 1.0100c, |  |  |  |
3.8600c, 4.2500c, 3.4900c, 1.6000c, |  |  |  |
3.8600c, 4.9000c, 1.3700c, 8.2000b, |  |  |  |
3.8600c, 5.3300c, 1.1600c, 8.2000b, |  |  |  |
5.3300c, 1.2200c, 3.5060d |  5cm 2.8900c, 3.6100c, 3.6100c, 4.0000c, |  |  |
3.2400c, 4.0000c, 3.6100c, 3.6100c, |  |  |  |
3.2400c, 4.0000c, 3.6100c, 3.2400c, |  |  |  |
3.6100c, 4.0000c, 2.8900c, 1.2100c, |  |  |  |
3.2400c, 4.0000c, 3.6100c, 3.6100c, |  |  |  |
4.0000c, 4.0000c, 3.2400c, 1.4400c, |  |  |  |
3.6100c, 4.0000c, 3.2400c, 1.2100c, |  |  |  |
3.6100c, 5.2900c, 1.0000c, 8.1000b, |  |  |  |
3.6100c, 4.0000c, 3.6100c, 3.6100c, |  |  |  |
4.0000c, 4.0000c, 3.2400c, 1.4400c, |  |  |  |
3.6100c, 4.0000c, 3.2400c, 1.4400c, |  |  |  |
4.0000c, 5.2900c, 1.2100c, 8.1000b, |  |  |  |
4.0000c, 4.0000c, 3.2400c, 1.6900c, |  |  |  |
4.0000c, 4.8400c, 1.2100c, 6.4000b, |  |  |  |
4.0000c, 5.2900c, 1.2100c, 8.1000b, |  |  |  |
5.2900c, 1.0000c, 3.4810d |  |  |  |
Table 11: The same as Table (10) but
$\frac{\gamma_{1}}{\lambda}=\frac{4\theta_{6}}{5}$,
$\frac{\gamma_{2}}{\lambda}=\frac{7\theta_{6}}{9}$,
$\frac{\gamma_{3}}{\lambda}=\frac{5\theta_{6}}{8}$,
$\frac{\gamma_{4}}{\lambda}=\frac{3\theta_{6}}{4}$,
$\frac{\gamma_{5}}{\lambda}=\frac{6\theta_{6}}{7}$ and
$\frac{\gamma_{6}}{\lambda}=\frac{9\theta_{6}}{10}$.

First, it is easy to see that the sum of all probabilities for all marked and
unmarked states at any time in the absence of dissipation rates is equal to
one, see Fig.2. Also, from Figs.3, Fig.4, Fig.5 and the above tables, one can
note that the sum of all probabilities in the presence of dissipation at
different times are gradually decreased than one, and reaches almost zero at
high values of dissipation rates.
Second, from Tables (2-11), one observes and distinguishes between different
values of times for the probabilities of each marked state at weak values of
the dissipation rates. Moreover, we observe for times $t_{Np}$, and a weak
dissipation, that the probabilities of each marked states have higher
probabilities than any marked state probabilities through other times. Further
increasing of $\gamma_{i}$ and for different values of times, one can not
distinguish the probabilities of some marked states for the different number
of qubits, where some of un-marked states due to the influence of large
dissipation exchange their values either increase or decrease the
probabilities of marked states. Moreover, if dissipation rates are taken to be
$\gamma_{i}=\lambda\theta_{N}$, $i=1,2,...,N$, one determines and observes at
different times the probabilities of some marked states for different values
of the number of $N$. However, when $\gamma_{i}>\lambda\theta_{N}$, we can not
distinguish any marked states where very small values for the probability of
marked state is shown. Which means that both marked and un-marked states have
the same chance to contribute. One concludes that at
$\gamma_{i}>\lambda\theta_{N}$, the system is destroyed completely as well as
it can not search for any desired state whatever the number of qubits. It is
interesting to discuss the influence of large values of dissipation on the
probabilities of marked states at these optional times in the following
scenario:
Through the optional times $\frac{0.331\pi\hbar}{2^{N}\lambda\theta_{N}}$ and
the number of qubits up to $N=5$, we can not distinguish and observe the
probability of any marked state for any number of qubits except the
probabilities of marked states of the kind $q_{2}=\\{gg,eg,ge\\}$,
$q_{3}=\\{ggg,egg,geg,gge\\}$, $q_{4}=\\{gggg,eggg,gegg,ggeg,ggge\\}$ and
$q_{5}=\\{ggggg,egggg,geggg,ggegg,gggeg,gggge\\}$. As for the case of $N\geq
6$, we can not distinguish the probabilities of any marked states, where the
probabilities of un-marked states show larger values compared with the
probability of marked state which shows a small values.
At times $\frac{0.566\pi\hbar}{2^{N}\lambda\theta_{N}}$, one distinguishes the
probability of marked state for $2-$, $3-$ and $4-$qubit but for 5-qubit one
can not distinguish the probabilities of any marked states except the marked
states which are of the kind $q_{5}$ one can distinguish the probabilities for
it. While through this time when $N\geq 6$, we can not distinguish and
determine the probability of any marked state, where both marked and un-marked
states have the same chance to contribute in the search.
Also, at times $t_{Np}$, we distinguish the probability of any marked state
for different number of qubits until $N=5$ except the probabilities of the
marked states $|eeee\rangle$ and $|eeeee\rangle$. For $N=6$, one can not
observe and determine the probability of any marked state except the marked
states of the kind $q_{6}=\\{gggggg,eggggg,gegggg,ggeggg,gggegg,$
$ggggeg,ggggge\\}$, where the value of any marked state probability for this
kind is distinguished and shows small value and larger than the probabilities
of the remaining states. However, we expect through these times for $N>6$ that
one can not distinguish and determine the probability of any marked state.
Finally, at times $\frac{\pi\hbar}{2^{N}\lambda\theta_{N}}$, it is shown that
the probability of any marked state for the different number of qubits until
$N=4$ can be distinguished. As for 5-qubit and 6-qubit, we can not distinguish
the probability of some marked state, see Tables (9,11), except the marked
states of the kind $q_{5}$ and $q_{6}$. After that, through these times, one
can not observe the probabilities of any marked states at $N=7$ and more.
Consequently, through the large dissipation values, most marked states are
distinguished through the optional times $t_{Np}$ for the different number of
qubits. As a result, one can distinguish and observe, at large values of
dissipation, the probabilities of many different marked states for multi
qubits at different values of times. These observations give an advantage of
using the dynamical quantum search algorithm with the dissipation are other
quantum search algorithms.

## References

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