Experimental NMR Realization of A Generalized Quantum Search Algorithm
G. L. Long$^{1,2,3,4}$
H. Y. Yan$^{1,2}$
Y. S. Li$^{1,2}$
C. C. Tu$^{1,2}$
J.X.Tao$^{1,2}$
H.M.Chen$^{1}$
M. L. Liu$^5$
X. Zhang$^5$
J. Luo$^5$
L. Xiao$^{1,2,5}$
X. Z. Zeng$^5$
A generalized quantum search algorithm, where phase inversions for the marked state and %%@ the prepared state are replaced by $\pi/2$ phase rotations, is realized in a 2-qubit NMR %%@ heteronuclear system. The quantum algorithm searches a marked state with a smaller step %%@ compared to standard Grover algorithm. Phase matching requirement in quantum searching is %%@ demonstrated by comparing it with another generalized algorithm where the two phase %%@ rotations are $\pi/2$ and $3\pi/2$ respectively. Pulse sequences which include non 90 %%@ degree pulses are given.
nmr2.TEX
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\title{Experimental NMR Realization of A Generalized  Quantum Search Algorithm}
\author{G. L. Long$^{1,2,3,4}$,H. Y. Yan$^{1,2}$,Y. S. Li$^{1,2}$, C. C. Tu$^{1,2}$,	
J.X.Tao$^{1,2}$,H.M.Chen$^{1}$,\\ M. L. Liu$^5$, X. Zhang$^5$, J. Luo$^5$,
 L. Xiao$^{1,2,5}$, X. Z. Zeng$^5$}
\address{$^1$Department of Physics, Tsinghua University, Beijing 100084, P.R.China\\
$^2$Key Laboratory for Quantum Information and Measurements, Ministry of Education, %%@
Beijing 100084, 
P.R. China\\
$^3$Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100080, P. R. %%@
China\\
$^4$Center of Atomic, Molecular and Nanosciences, Tsinghua University, Beijing 100084, P. %%@
R. China\\
$^5$ Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
\\ Wuhan Institute of Physics and Mathematics,\\  Chinese Academy of Sciences, Wuhan %%@
430071, P. R. China}
\date{\today}
\maketitle

\begin{abstract}

A generalized quantum search algorithm, where phase inversions  for the marked state and %%@
the prepared state are replaced by $\pi/2$ phase rotations,  is realized in a 2-qubit NMR %%@
heteronuclear system. The quantum algorithm searches a marked state with a smaller step %%@
compared to standard Grover algorithm. Phase matching requirement in quantum searching is %%@
demonstrated by comparing it with another generalized algorithm where the two phase %%@
rotations are $\pi/2$ and $3\pi/2$ respectively.  Pulse sequences which include non 90 %%@
degree pulses are given.
\end{abstract}

\pacs{PACS numbers: 03.67.Lx, 89.70.+c, 89.80.+h}

Grover's quantum search algorithm is one of the most important development in quantum  
computation\cite{r1}. It achieves quadratic speedup in searching a marked state in an %%@
unordered list over classical searching algorithms.  It has many potential applications in %%@
various fields of interests, for instance in deciphering the DES encryption %%@
code\cite{science} and algorithms that need searching. Typical examples of the algorithms %%@
that need searching are the Simon problem\cite{simon}, the Hamilton's circuit %%@
problem\cite{hamilton}, the hiddens shift problem\cite{twawley} and quantum %%@
counting\cite{counting}. There are several generalizations of the Grover algorithm. A %%@
modification of the algorithm can search for a ``chain" of $m$ marked items in %%@
$O(\sqrt{N/m})$ iterations\cite{r2}. In the standard Grover algorithm, the quantum %%@
database is built on an evenly distributed quantum superposition, and a generalization is %%@
made to allow the algorithm to work on a biased database where the amplitudes of the items %%@
in a database are not even\cite{arbi}. Searching in an arbitrary entangled superposition  %%@
is given in Ref.\cite{carlini}

In some cases, one needs a quantum searching engine that searches an item with a smaller %%@
step. For instance, in the Simon algorithm\cite{simon} where ${\pi\over 2}$-phase %%@
rotations rather than phase inversions are used and in the case where the number of marked %%@
states is more than $N/4$\cite{kim}, standard Grover algorithm can not be used. In %%@
addition, for small $N$, state of the quantum computer may not be exactly the marked state %%@
during the search process, and  there is  small probability that the algorithm may fail. %%@
In problems that certainty of success is vital, this should be avoided. A generalized %%@
quantum search algorithm\cite{r3,r4} suits this purpose, where the searching step can be %%@
anything between that of Grover algorithm and zero. This is done by replacing the two %%@
phase inversions in Grover's algorithm with smaller phase rotations($\phi$ for phase %%@
rotation of the marked state, and $\theta$ for phase rotation of the prepared state %%@
$|0...0\ket$ ). It has been found that with an evenly distributed database, arbitrary %%@
phase rotation is not applicable\cite{r3}, and only when the two phase rotations satisfy %%@
the phase matching requirement $\theta=\phi$ that an efficient quantum searching algorithm %%@
can be constructed\cite{r4}. An approximate expression $2\sin{\theta \over 2}\;\sqrt{1 %%@
\over {N}}$ was given for the search step\cite{r4}, where $\theta$ is the rotation angle %%@
of the marked state\cite{r4}. Exact expressions are given in a $SO(3) $geometric %%@
picture\cite{r4p}.

NMR implementation of Grover's algorithm has been realized in 2-qubit and 3-qubit %%@
systems\cite{r5,r6,r7}. Quantum counting has also been realized in NMR %%@
system\cite{rcounting}. Experimental studies  are important in
demonstrating quantum algorithms, investigating  effects of gate imperfection and  %%@
decoherence,  and in identifying problems in building a practical quantum computer. In %%@
this Letter, we report the experimental realization of this phase matching quantum search %%@
algorithm where the phase inversions are replaced by rotations through $\pi/2$ in a 2 %%@
qubit heteronuclear system using NMR technique. To demonstrate the effect of phase %%@
matching,  another  experiment where   $\theta=\pi/2$, $\phi=3\pi/2$ is also performed.  %%@
Different from standard Grover algorithm, the pulse sequences used here contain non 90 %%@
degree pulses, and the delay pulses need not be multiples of ${1\over 4J}$. 

Grover algorithm consists of  four steps in an iteration\cite{r2}: 
1)a phase inversion of the marked state $I_{\tau}=I-2|\tau\rangle\langle\tau|$;
2) the Walsh-Hadamard transformation W; 
3) a phase inversion of the prepared state $|0\rangle$,
$I_{0}=I-2|0\rangle\langle 0|$; and 
4) the Walsh-Hadamard transformation. The operator for one Grover iteration is $Q=-W %%@
I_{0}WI_{\tau}$. The steps 2-4 are combined to give the inversion about average $D$ which %%@
has the following matrix
\begin{eqnarray}
D_{ij}=W\;I_\tau \;W=\left\{\begin{array}{ll}{2\over N} & {\rm i\ne j}\\
                               {2\over N}-1 & {\rm i=j}\end{array}\right.
							   \end{eqnarray} 


In the generalized quantum search algorithm, the phase inversions are replaced by %%@
arbitrary phase rotations. The corresponding operators, indicated by a ``g" in the %%@
superscript, are
\begin{eqnarray}
Q^g=D^g\;I_\tau^g=W\;I^g_0\;W\;I^g_\tau,
\end{eqnarray}
where
\begin{eqnarray}
I_{\tau}^{g}&=&I-(1-e^{i\phi}|\tau\rangle\langle\tau|),\nonumber\\
I_{0}^{g}&=&I-(1-e^{i\theta}|0\rangle\langle 0|),\nonumber\\
D^g&=&W\;I^g_{0}\; W.
\end{eqnarray}
When $\theta=\phi=\pi$ the Grover algorithm is recovered. It is helpful to give the %%@
detailed expressions for a 2-qubit system. We assume that the marked state is $\tau=3$. %%@
The operators are,
\begin{eqnarray}
I_{\tau=3}^{g}=\left[\begin{array}{rrrr}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&e^{i\phi}
           \end{array}
    \right],
\end{eqnarray}
and
\begin{eqnarray}
Q^{g}= W\; I_{0}^{g}\; W
 =\left[
      \begin{array}{rrrr}1&1&1&1\\ 
                         1&-1&1&-1\\
		     			 1&1&-1&-1\\
						 1&-1&-1&1\\
				         \end{array}
						   \right]
                           \left[\begin{array}{rrrr}
						   e^{i\theta}&0&0&0\\ 
						   0&1&0&0          \\ 
						   0&0&1&0 
						   \\ 0&0&0&1
						          \end{array}
						   \right]
						   \left[\begin{array}{rrrr}1&1&1&1\\ 
                                                     1&-1&1&-1\\
													 1&1&-1&-1\\
													 1&-1&-1&1\\
					              \end{array}
						   \right].
\end{eqnarray}

For demonstration in this experiment, we choose $\theta=\phi=\pi /2$. In table \ref{t1}, %%@
we give the state vector for the quantum computer in each step during a quantum searching %%@
process where $|c\ket=1/\sqrt{N}\sum_{i\ne \tau} |i\ket$. When phase matching requirement %%@
is not satisfied, the quantum algorithm will not work. To demonstrate this, we also give %%@
the state vectors of the quantum computer for the case with $\theta=\pi/2$ and %%@
$\phi=3\pi/2$. Both algorithms are performed for 10 searching iterations. The probability %%@
for finding the marked state is the square of the coefficient of $|\tau\ket$.  These state %%@
vectors are converted to density matrices for comparisons with experiment. 


In the experiment, the working media is $H_{2}PO_{3}$.  The 2 qubits are the nuclear spins %%@
of the H-atom and the P-atom. The observed J-coupling between $^{1}H$
and $^{31}P$ is {\bf 647.451 Hz}. The experiment was performed on a Bruker 500 ARX NMR %%@
spectrometer. The frequencies for $ ^{1}H$(qubit A) and $^{31}$P (qubit B) are   500MHz
and 220MHz respectively.  The Hamiltonian for this system can be modelled 
as a two-spin system with a weak coupling interaction,
\begin{eqnarray}
H=\omega_{A}\;I_{AZ}+\omega_{B}\;I_{BZ}+2\pi J_{AB}\;I_{AZ}\;I_{BZ},
\end{eqnarray}
where $I_{AZ}=\frac {1}{2}\sigma _{ZA}$ is the angular momentum operator in the %%@
$\widehat{{\bf e}_{z}}$
direction for spin A. $\omega _{A}$ and $\omega _{B}$ describe the free precession %%@
frequencies of the nuclear spins $A$ and $B$ respectively. The magnetic field is in %%@
$-$$\widehat{{\bf e}_{z}}$ direction. The quantum gate operations needed in quantum %%@
computation can be constructed by a combination of some hard pulses and the delay pulses. %%@
Compared to the pulse sequences in the Grover algorithm for 2-qubit system, one needs only %%@
a small modification in the  pulse sequences. First let's denote the following operators
\begin{eqnarray}
X^{\theta}&=&exp[-i\theta I_{x}],\nonumber\\
\overline{{X}^{\theta}}&=&  exp[i\theta I_{x}], \nonumber\\
Y^{\theta}&=& exp[-i\theta I_{y}],\nonumber\\ 
\overline{{Y}^{\theta}}&=& exp[i\theta I_{y}],
\end{eqnarray} 
which are radio frequency(rf) pulses for rotations about $\hat{x}$-axis through $\theta$, %%@
$-\theta$, rotations about $\hat{y}$-axis through $\theta$ and $-\theta$ respectively.  In %%@
addition to these hard pulses, we also have
\begin{eqnarray}
\tau ^{t}=exp[-2\pi i\;J_{AB}\;I_{ZA}\;I_{ZB}\;t],\nonumber
\end{eqnarray}
which is a delay pulse  where the system undergoes an evolution during period $t$ in the %%@
doubly rotating frame. We denote $|\uparrow\rangle=|0\rangle$ and %%@
$|\downarrow\rangle=|1\rangle$.

We used temporal averaging\cite{r8} to produce the effective pure state $|00\rangle$. The %%@
pulse sequences for the temporal averaging and the Hadmard-Walsh transformation  are %%@
standard\cite{r5,fuliping}:
\begin{eqnarray}
P_{0}: &&I(none);\nonumber\\
P1: &&Y_{B}^{\frac {\pi}{2}}\tau ^{\frac {1}{2J}}
	X_{B}^{\frac {\pi}{2}}Y_{A}^{\frac 	{\pi}{2}}\tau ^{\frac {1}{2J}}X_{A}^{\frac %%@
{\pi}{2}},\nonumber\\
P2:&&Y_{A}^{\frac {\pi}{2}}\tau ^{\frac {1}{2J}}
	X_{B}^{\frac {\pi}{2}}X_{A}^{\frac 	{\pi}{2}}\tau ^{\frac {1}{2J}}X_{B}^{\frac %%@
{\pi}{2}},\nonumber \\
W=&&(X_{A}^{{\frac {\pi}{2}}})^{2}\overline{Y_{A}}^{\frac {\pi}{2}}(X_{B}^{{\frac %%@
{\pi}{2}}})^{2}\overline{Y_{B}}^{\frac {\pi}{2}}.\nonumber
\end{eqnarray}
The pulse sequences for  the generalized quantum search algorithm are obtained by %%@
modifying the pulse sequences used for Grover's algorithm in  Ref.\cite{r5}:
\begin{eqnarray}
I^{g}&=&Y_{A}^{\frac {\pi}{2}}\overline{X_{A}}^{\frac {\phi}{2}}\overline{Y_{A}}^{\frac %%@
{\pi}{2}}Y_{B}^{\frac{\pi}{2}}\overline{X_{B}}^{\frac{\phi}{2}}\overline{Y_{B}}^{\frac{\pi%%@
%%@
}{2}}\tau ^{\frac {2\pi-\phi}{2J\pi}},\nonumber\\
D^g&=&Y_{A}^{\frac {\pi}{2}}X_{A}^{\frac {\theta}{2}}\overline{Y_{A}}^{\frac %%@
{\pi}{2}}Y_{B}^{\frac{\pi}{2}}X_{B}^{\frac {\theta}{2}}\overline{Y_{B}}^{\frac %%@
{\pi}{2}}\tau ^{\frac{2\pi-\theta}{2J\pi}}.
\end{eqnarray}
The pulse sequence for a complete generalized search  iteration $Q^g=D^g\;I^g$ is
\begin{eqnarray}
Q^g=\overline{X_{A}}^{\frac {\theta}{2}}Y_{A}^{\frac {\pi}{2}}X_{B}^{\frac %%@
{\theta}{2}}Y_{B}^{\frac {\pi}{2}}\tau ^{\frac{2\pi-\theta}{2J\pi}}X_{A}^{\frac %%@
{\phi}{2}}\overline{Y_{A}}^{\frac {\pi}{2}}X_{B}^{\frac {\phi}{2}}\overline{Y_{B}}^{\frac %%@
{\pi}{2}}\tau ^{\frac {2\pi-\phi}{2J\pi}}.
\end{eqnarray}

It is interesting to study the length of time in a given quantum search algorithm with  %%@
arbitrary phases. A hard pulse takes microsecond to complete while a delay pulse takes %%@
about a millisecond to complete. An algorithm with large $\theta$ or $\phi$ takes less %%@
time to complete. Therefore in practice, it is better to use large phase rotations to make %%@
the computation time short if one is given the freedom. It also worth pointing that in %%@
general, the hard pulses may not be the multiples of ${\pi\over 2}$ as in other NMR %%@
quantum computations, and the delay pulses may also takes noninteger multiples of ${1\over %%@
4J}$. This is different from the pulse sequences used in standard Grover algorithm, for %%@
instance in \cite{r5,r7}.


Two sets of experiments: phase matched searching  with $\theta=\phi={\frac{\pi}{2}}$ and %%@
phase mismatched searching with
$\theta={\frac{\pi}{2}},\phi={\frac{3\pi}{2}}$, have been performed.
State tomography is used  to obtain the density matrices\cite{r9}. Density matrices are %%@
experimentally constructed for all the 10 iterations in the two sets of experiments. It %%@
took quite some time and labor to get them. 
  In table \ref{t2}, we have given the relative error defined as %%@
$\delta\rho=\frac{||\rho_{th}-\rho_{exp}||_2}{||\rho||_2}$. However, this  error is not %%@
solely the ''genuine" errors\cite{r10,r10p} from gate imperfection and decoherence that %%@
occurs during the quantum computation process. Part of the  error is caused by doing the %%@
integration of areas of the spectrum by hand during the density matrix construction.  It %%@
is interesting to note that the relative errors at later stages are sometimes even smaller %%@
than the early stages. For instance at step 7 in the phase-matched case, the relative %%@
error is only 15\% , the smallest among the 10 steps.  Similar result is observed in other %%@
NMR quantum computing experiment, for instance in \cite{r5}. This is perhaps because %%@
imperfect gate errors cancel out each other. 
For the economy of the paper length, we give only the density matrix for the 6th iteration %%@
where the success rate is the largest in Fig.1 for phase matched searching. In Fig.2 the %%@
same is plotted for the phase mismatched searching. For clarity, we plot separately the %%@
real and imaginary parts.
From these figures, it is seen that the agreement between theoretical prediction and %%@
experimental data is good. In particular, when phase matching is satisfied, the algorithm %%@
can find the marked state with high probability. However, when the phase matching %%@
requirement is seriously violated,   the probability of finding the marked state is  very %%@
low. Phase mismatching leads reduction in success rate, and it should be avoided in %%@
practice. However, Grover's algorithm has some intrinsic robustness against errors, a %%@
small amount of phase mismatching, like those errors from NMR pulse manipulations,  will %%@
not cause a big loss in the probability for finding the marked state.


In summary, a generalized quantum search algorithm has been demonstrated in a 2-qubit NMR %%@
system. Pulse sequences are given. Non 90 degree pulses are used and tested to give good %%@
performance. It also demonstrate that phase matching in quantum searching is  important.

This work is support in part by China National Science Foundation under Grant No.60073009, %%@
the excellent university teacher's fund of China Education Ministry, Fok Ying-Tung %%@
education foundation. 
\begin{table}
\begin{center}
\caption{Theoretical State Vector of the Register in Each Search Iteration}
\label{t1}
\begin{tabular}{|c|c|c|}\hline
      &phase matching&phase mismatching\\ 
      &$\theta=\phi=\pi/2$&$\theta=\pi/2$,$\phi=3\pi/2$\\ \hline     
step1:&$|\psi_{1}\rangle=0.90|\tau\rangle+0.43|c\rangle$&
$|\psi_{1}\rangle=0.25|\tau\rangle+0.97|c\rangle$\\ \hline
step2:&$|\psi_{2}\rangle=0.97|\tau\rangle+0.22|c\rangle$&
$|\psi_{2}\rangle=0.62|\tau\rangle+0.78|c\rangle$\\ \hline
step3:&$|\psi_{3}\rangle=0.65|\tau\rangle+0.76|c\rangle$&
$|\psi_{3}\rangle=0.06|\tau\rangle+1.00|c\rangle$\\ \hline
step4:&$|\psi_{4}\rangle=0.39|\tau\rangle+0.92|c\rangle$&
$|\psi_{4}\rangle=0.59|\tau\rangle+0.80|c\rangle$\\ \hline
step5:&$|\psi_{5}\rangle=0.78|\tau\rangle+0.62|c\rangle$&
$|\psi_{5}\rangle=0.36|\tau\rangle+0.93|c\rangle$\\ \hline
step6:&$|\psi_{6}\rangle=1.00|\tau\rangle+0.01|c\rangle$&
$|\psi_{6}\rangle=0.41|\tau\rangle+0.91|c\rangle$\\ \hline
step7:&$|\psi_{7}\rangle=0.80|\tau\rangle+0.60|c\rangle$&
$|\psi_{7}\rangle=0.57|\tau\rangle+0.82|c\rangle$\\ \hline
step8:&$|\psi_{8}\rangle=0.40|\tau\rangle+0.92|c\rangle$&
$|\psi_{8}\rangle=0.13|\tau\rangle+0.99|c\rangle$\\ \hline
step9:&$|\psi_{9}\rangle=0.63|\tau\rangle+0.77|c\rangle$&
$|\psi_{9}\rangle=0.63|\tau\rangle+0.78|c\rangle$\\ \hline
step10:&$|\psi_{10}\rangle=0.97|\tau\rangle+0.24|c\rangle$&
$|\psi_{10}\rangle=0.19|\tau\rangle+0.98|c\rangle$\\ \hline
\end{tabular}
\end{center}
\end{table}
	  
\begin{table}
\begin{center}
\caption{The errors of the experiments}
\label{t2}
\begin{tabular}{ccc}
       &$\theta=\phi={\frac{\pi}{2}}$&$\theta={\frac{\pi}{2}},\phi={\frac{3\pi}{2}}$\\
  Step1&$\% 18$&$\% 17$\\
  Step2&$\% 21$&$\% 28$\\
  Step3&$\% 20$&$\% 27$\\
  Step4&$\% 21$&$\% 21$\\
  Step5&$\% 27$&$\% 20$\\
  Step6&$\% 20$&$\% 20$\\
  Step7&$\% 15$&$\% 16$\\
  Step8&$\% 24$&$\% 33$\\
  Step9&$\% 22$&$\% 30$\\
  Step10&$\% 22$&$\% 33$\\
 \end{tabular}
\end{center}
\end{table}
\begin{thebibliography}{99}
\bibitem{r1} Grover L. K.,{\it Phys. Rev. Lett.}, {\bf 79} 325 (1997)
\bibitem{science} Brassard G., {\it Science} {\bf 275}, 627 - 628 (1997).
\bibitem{simon}Simon D. R., {\it Proceedings of 35th annual Symposium on the foundations %%@
of computer Sciences}, 116-123; Brassard G. and Peter H\"{o}yer. {\it Proc. of Fifth %%@
Israeli Symposium on Theory of Computing and Systems (ISTCS'97)}, pp. 12-23, 1997; 
\bibitem{twawley}J. Twamley, {\it J. Phys.} {\bf A33} 8973, (2000). 
\bibitem{hamilton} H. Guo et al, A quantum algorithm for finding a Hamiltonian circuit,   %%@
{\it Commun. Theor. Phys.} {\bf 35} (2001) 385-388.
\bibitem{counting} Brassard G. et al, {\it Proc. of 25th International Colloquium on %%@
Automata, Languages, and Programming (ICALP'98)}, {\bf  Vol.1443} of Lecture Notes in %%@
Computer Science, pp. 820-831, 1998. 
\bibitem{r2} Grover L. K.,{\it Phys. Rev. Lett.},{\bf 80} 4329 (1998).	
\bibitem{arbi} Biron, D., Biham, O., Biham, E., Grassl, M., Lidar, D.A.
 {\it Lecture Notes in Computer Science} vol. 1509, 140 - 147
(Springer, 1998).		
\bibitem{carlini} A. Carlini and A. Hosoya, {\it Phys. Lett.} {\bf A 280}, 114, (2001). %%@
Also available at quant-ph/9909089.
\bibitem{kim}Chi D. and Kim J., {\it Chaos, Solitons, and Fractals} {\bf 10} 1689 (1999).
\bibitem{r3}Long G. L.  et al,{\it Commun. Theor. Phys.}{\bf 32} 335 (1999).
\bibitem{r4}Long G. L. et al, {\it Phys. Lett.}{\bf A 262} 27 (1999). 
\bibitem{r4p} Long G. L. et al, {\it J. Phys.} {\bf A34}, 867 (2001)
\bibitem{r5}Chuang I. L. et al,{\it Phys. Rev. Lett} {\bf  80} 3408 (1998).
\bibitem{fuliping} L.P. Fu et al, {\it Chin. J. Magn. Res.}{\bf 16}, 341 (1999)
\bibitem{r6}Jones J. A. et al,{\it Nature},{\bf 393} 344 (1998).
\bibitem{r7}Vandersypen L. M. K. et al,{\it Appl. Phys. Lett.} {\bf 76} 646 (2000).
\bibitem{rcounting} J A Jones and M Mosca, {\it Phys. Rev. Lett.}{\bf 83}, 1050 (1999)
\bibitem{r8}Knill E. et al, {\it Phys. Rev} {\bf A57} 3348 (1998).
\bibitem{r9}Chuang I. L. et al,{\it Proc. R. Soc. Lond} {\bf A 454}, 447 (1998).
\bibitem{r10} Long G. L. et al, {\it Phys. Rev} {\bf A 61}, 042305 (2000). 
\bibitem{r10p} Pablo-Norman B. and Ruiz-Altaba M., {\it Phys. Rev.} {\bf A61} 012301  %%@
(2000)
\end{thebibliography}

\noindent Figure captions\\
\noindent Fig.1 Comparisons of theoretical and experimental density matrices 
with $\theta=\phi=\pi/2$ for iteration 6. (a) and (b) are the theoretical real part and %%@
imaginary part of the density matrices respectively, whereas (c) and (d) are the %%@
corresponding experimental ones. The probability of finding the marked state is the sum  %%@
of squares of the 11 component which are in the upper right corners of the figures.\\
\noindent Fig.2 Same as Fig.2 but for $\theta=\pi/2,\phi=3\pi/2$.\\
\end{document}

quant-ph0009059.extracted_bib
\begin{thebibliography}{99}
\bibitem{r1} Grover L. K.,{\it Phys. Rev. Lett.}, {\bf 79} 325 (1997)
\bibitem{science} Brassard G., {\it Science} {\bf 275}, 627 - 628 (1997).
\bibitem{simon}Simon D. R., {\it Proceedings of 35th annual Symposium on the foundations %%@
of computer Sciences}, 116-123; Brassard G. and Peter H\"{o}yer. {\it Proc. of Fifth %%@
Israeli Symposium on Theory of Computing and Systems (ISTCS'97)}, pp. 12-23, 1997; 
\bibitem{twawley}J. Twamley, {\it J. Phys.} {\bf A33} 8973, (2000). 
\bibitem{hamilton} H. Guo et al, A quantum algorithm for finding a Hamiltonian circuit,   %%@
{\it Commun. Theor. Phys.} {\bf 35} (2001) 385-388.
\bibitem{counting} Brassard G. et al, {\it Proc. of 25th International Colloquium on %%@
Automata, Languages, and Programming (ICALP'98)}, {\bf  Vol.1443} of Lecture Notes in %%@
Computer Science, pp. 820-831, 1998. 
\bibitem{r2} Grover L. K.,{\it Phys. Rev. Lett.},{\bf 80} 4329 (1998).	
\bibitem{arbi} Biron, D., Biham, O., Biham, E., Grassl, M., Lidar, D.A.
 {\it Lecture Notes in Computer Science} vol. 1509, 140 - 147
(Springer, 1998).		
\bibitem{carlini} A. Carlini and A. Hosoya, {\it Phys. Lett.} {\bf A 280}, 114, (2001). %%@
Also available at quant-ph/9909089.
\bibitem{kim}Chi D. and Kim J., {\it Chaos, Solitons, and Fractals} {\bf 10} 1689 (1999).
\bibitem{r3}Long G. L.  et al,{\it Commun. Theor. Phys.}{\bf 32} 335 (1999).
\bibitem{r4}Long G. L. et al, {\it Phys. Lett.}{\bf A 262} 27 (1999). 
\bibitem{r4p} Long G. L. et al, {\it J. Phys.} {\bf A34}, 867 (2001)
\bibitem{r5}Chuang I. L. et al,{\it Phys. Rev. Lett} {\bf  80} 3408 (1998).
\bibitem{fuliping} L.P. Fu et al, {\it Chin. J. Magn. Res.}{\bf 16}, 341 (1999)
\bibitem{r6}Jones J. A. et al,{\it Nature},{\bf 393} 344 (1998).
\bibitem{r7}Vandersypen L. M. K. et al,{\it Appl. Phys. Lett.} {\bf 76} 646 (2000).
\bibitem{rcounting} J A Jones and M Mosca, {\it Phys. Rev. Lett.}{\bf 83}, 1050 (1999)
\bibitem{r8}Knill E. et al, {\it Phys. Rev} {\bf A57} 3348 (1998).
\bibitem{r9}Chuang I. L. et al,{\it Proc. R. Soc. Lond} {\bf A 454}, 447 (1998).
\bibitem{r10} Long G. L. et al, {\it Phys. Rev} {\bf A 61}, 042305 (2000). 
\bibitem{r10p} Pablo-Norman B. and Ruiz-Altaba M., {\it Phys. Rev.} {\bf A61} 012301  %%@
(2000)
\end{thebibliography}