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Non-coherent Multi-User Detection Based on Quantum Search Sándor Imre, Member, IEEE and Ferenc Balázs, Student Member, IEEE Budapest University of Technology and Economics Department of Telecommunications Mobile Communications Laboratory H-1117 Budapest,Magyar Tudósok krt. 2., HUNGARY Abstract— 3G and 4G mobile systems are based on CDMA technology. In order to increase the efficiency of CDMA receivers large amount of effort is invested to develop suitable multi-user detector techniques. However, at this moment there are only suboptimal solutions available because of the rather high complexity of optimal detectors. One of the possible receiver technologies can be the quantum assisted computing devices which allows high level parallelism in computation. The first commercial devices are estimated by 2004, which meets the advent of 3G and 4G systems. In this paper we introduce a novel quantum computation based Quantum Multi-user detectior (QMUD), employing Grover’s search algorithm, which provides optimal solution. The proposed algorithm is robust to any kind of noise. Keywords — Multi-user detection, Grover’s Algorithm, Quantum Computing, Quantum Signal Processing I. I NTRODUCTION Subscribers of the next generation wireless systems will communicate simultaneously, sharing the same frequency band. All around the world 3G mobile systems apply Direct Sequence - Code Division Multiple Access (DS-CDMA) promising due to its high capacity and inherent resistance to interference, hence it comes into the limelight in many communication systems. Another physical layer scheme, Orthogonal Frequency Division Access (OFDM), is also often used e.g. for Wireless LANs (WLAN) or HiperLAN, where the subscriber’s signal is transmitted via a group of orthogonal frequencies, providing Inter Channel Interference (ICI) exemption. Nevertheless due to the frequency selective property of the channel, in case of CDMA communication the orthogonality between user codes at the receiver is lost, which leads to performance degradation. Single-User detectors were overtaxed and showed rather poor performance even in multi-path environment [1]. To overcome this problem, in recent years Multi-User Detection (MUD) has received considerable attention and become one of the most important signal processing task in wireless communication. Verdu [1] has proved that the optimal solution is consistent with the optimization of a quadratic function, which yields in MLSE (Maximum-Likelihood Sequence Estimation) receiver. However, to find the optimum is an NP-hard problem as the number of users grows. Many authors proposed sub-optimal linear and nonlinear solutions such as Decorrelating Detector, MMSE (Minimum Mean Square Error) detector, Multistage Detector, Hoppfield neural network or Stochastic Hoppfield neural network [1], [2], [3], [4], and the references therein. One can find a comparison of the performance of the above mentioned algorithms in [5]. Nonlinear sub-optimal solutions provide quite good performance, however, only asymptotically. Quantum computation based algorithms seem to be able to fill this long-felt gap. Beside the classical de century scription, which we recently use, researchers in the early raised the idea of quantum theory, which nowadays becomes remarkable in coding theory, information theory and for signal processing [6]. Nowadays, every scientist applies classical computation, using sequential computers. Taking into account that Moore’s law can not be held for the next ten years because silicon chip transistors reach atomic scale, therefore new technology is required. Intel, IBM, AT&T and other companies invest large amount of research to develop devices based on quantum principle. Successful experiments share up that within 3-4 years quantum computation (QC) assisted devices will be available on the market as enabling technology for 3G and 4G systems. This paper is organized as follows: in Section II. we shortly review the necessity of multi-user detection, as well as the applied quantum computation principles are shown. In Section III. Grover’s quantum search algorithm is introduced. In Section IV. we discuss the applied system model and the proposed novel multi-user detection algorithm, respectively. Finally we conclude our paper in Section V. II. T HEORETICAL BACKGROUNDS A. Multi-User Detection One of the major attributes of CDMA systems is the multiple usage of a common frequency and time slot. Despite the interference caused by the multiple access property, the users can be distinguished by their codes. Let us investigate an uplink DS-DCDMA system, where the symbol of the user is denoted by , "! . In DS-CDMA systems an information bearing bit is encoded by means of a user specific code of length of the processing gain (PG)[1]. In case of uplink communication we assume perfect power control. In the receiver side it is not required synchronization between input signals and user specific codes, however we make our decision on symbols. Applying &% BPSK modulation, the output signal user, denoted by #$ , is given as of the # &% ' )( *+$ ,$ &% - &% where *+ and ,$ are the energy associated to the user’s bit and the user continuous signature waveform, respectively /10.243 5 &% <; &% ;@BA ,$ . >=? - 6-718:9 @BA <; ; denotes, the time duration of one&% chip, is the chip of 9 the code word of subscriber and =C refers to the chip elementary waveform, which could be DFE GHJI NM % M @1O &% K"L =C . otherwise We investigate a one path uplink wideband CDMA propagation channel, however, an extension to multi-path model could be done, effortlessly. The channel distortion for user &% the &% is modelled by a simple XW$ , where SU and W impulse response function PQ R TSU"V are the path gain and the delay of the user, respectively. They @ O are assumed to be constant during a symbol period of . We consider uplink scenario because downlink can be regarded as a special, simplified case of uplink. This model contains almost all elements of a typical WCDMA channel except multipath propagation, which was omitted to simplify the explanation of the new quantum computation based multi-user detection scheme. However, based on the results of the present Y paper, multipath propagation can be included into the channel model easily. The received signal is the sum of arriving signals plus a Gaussian noise component and thus can be written as follows: 5 & % & % & % Z & % ' \[ P 4 ]^# 1 `_ a 7 .3 5 [ 7b3 &% ( *+"SU$,$ cW1:_ &% - (1) &% where is the number of users using the same band, _ is white 8 Gaussian noise with constant d spectral density. A conventional detector contains a filter bank of e fg filters signature waveforms &% which are matched to the corresponding &% , and channel impulse response P that calculates the following decision variable: h K L Z &% ,$ &% lk1mn (2) i ( *+"SU'j 8 h h The traditional ”single-user” detector (SUD) simply calculates the sign of expression (2) yielding o pQqQr tsuwvx1 C! . This method results in poor & performance, as contains not only the signal transmitted by the user but h an interference term generated by the other users: i Xz C{-yC| } signal 5 [ ~C y ~ ~ 7b3 ~7 z {-| } z$_4 {n|$ } (3) noise multiple access interference where yC ~ is defined as follows: y ~ C ( + * ( * ~ SU$S ~ j 8 K"L ," &% & % % , ~ W ~ lk system. According to the axioms of quantum mechanics, every physical system can be characterized by means of its states ´.µ 1 in the Hilbert vector space over the complex numbers ¶ , whereas2a3 physical transformation can be described as a Unitary operator · ³·i¸ , respectively. The inner product ¹º ´.µ maps the ordered pair of vectors to ¶ with the properties [6]: » Positivity: ¹º º µ-¼ for º µ , » Linearity: ¹ ´° (a º 3 µ +b º G µ )=a ¹ ´° º 3 µ +b ¹ ´° º G µ , » Skew symmetry: ¹ ´° º µ e¹ ´° º µg½ . In the classical information theory the smallest information conveying unit is the bit. The counterpart unit in quantum information is called the ”quantum bit”, the qubit. Its state can be described by G means of the state ´.µ , ´.µ ¿¾ µ XÀ µ , where ¾^G ÀÁ)¶ refers G to the complex probability amplitudes and ¾ À ¤ [8], [9], [6]. The expression ¾ denotes the probability that afterG measuring the qubit it can be found in computational base µ , and À shows the probability to be in computational base µ . In more general descrip ´.µ is set up from qubits tion an d -bit ”quantum register” (qregister) "à µ de$ computational bases, where d spanned by states can be stored in the qregisters at the same time [7]. Å 243 Å 5Å ´.µ Ä ´ µÆ´ Ƕ' (6) 7B8 Å ÁÉ ; ; d denotes G the number of states and È where , ¹ µ , µ ´ ¹ Ê , Ë , respectively. It is worth mentioning, that a transformation · on a qregister is executed parallel on all d stored states, which is called quantum parallelization. 23 To provide irreversibilmust be unitary ity of transformation, · · ³· ¸ , where the super¡Ì script refers to the Hermitian conjugate or adjoint of · . The quantum registers can be set in a general state using quantum gates [10], [11], [6], which can be represented by means of a unitary operation, described by Í a quadratic matrix. Applying Î four basic gates, twice the Hadamard and twice the phase shift gates Í (4) K L &% &% % and _ ( * S . 8 , _ lk is a zero mean white Gaussian noise due to linear transformation. The optimal Bayesian detection reduces the optimal multi-user detection scheme to the following minimization problem in vector form [1]: Z J : "gv U? 2 3u< x b3 n (5) K 23 $gv ? 24 3uw x.3 gX c ¡ where refers to the optimization variable and ¢ ¤£ ¥U ~¦ § g¨g©¨ fg«ª . Unfortunately, the search for the global optimum of (5) usually proves to be rather tiresome, which prevents real time detection (its complexity by exhaustive search is ¬ ). [ Therefore, our objective is to develop a new, powerful detection technique which paves the way toward real time MUD even in highly loaded systems. Since classical multi-user detection schemes only try to minimize the probability of error in noisy and high interference environment, they, even also optimal solutions, can commit an error. Actually, these classical approaches make compromise between computational complexity, probability of error and time barrier required for efficient operation. On the hand, QMUD provides for typical 3¡8 2other and it is able to indicate undistinguishCDMA systems ®°¯'± ²³ able decision situations for correction by higher layer protocols [7]. `Ï Ð Ñ ( Î Ï ÓÒCÔ 6nÕÖ Ñ any states can be prepared [11]. III. G ROVER ’ S Q UANTUM S EARCH A LGORITHM Keeping the fact in view that the decision is made for a single symbol, i.e. itself and its inverse form of the subscriber specific code word @1O will be detected within the time duration , multi-user detection can be traced back to find a pattern - a code word - in a suitable prepared database (DB). One will see later in Section IV-C that not a given pattern has to be found in this database essentially, rather, it should be determined only the fact, whether a given pattern is in the database or not. In this section we shortly review the optimal quantum database search algorithm later on a proper preparation of the database will be given. Consider a large unsorted database, which contains entries of d , to find >the × desired value with any classical algorithms would need at steps. Grover [12], [13], [14] has derived a quantum search least × algorithm that took precisely ( iterations to carry out the search, which is the optimal solution, as it was proved in [15]. Boyer et al [16] introduced upper bounds for the required number of evaluations for several number of identical solutions in the database. A. The Optimal Quantum Searching Algorithm Considering a database with d¿ Ø Ù B. Quantum Computation Theory Quantum theory is a mathematical model of physical systems. To describe such a model one needs to specify the representation of the 3 ' ÁÚ 1 à entries and a binary function if the entry belonging to index otherwise Say ket Û (using Dirac’s notation). Ù is in the DB (7) Ü Ý úb ï æwçéè ê Þß ë à çéá ì â í ß ã ã ä å ä áä â î |g> |g> q/2 Fig. 1. The Grover operator ð Ù dÁ . The function Ù in (7) can have the value 1 either at a single or multiple values of , depending on how many identical entries in ñ a particular database exist. Grover’s quantum searching algorithm consists of four simple operations as depicted in Figure 1. First one invokes the so called oracle ( ò ) with the following computation rule: q/2 úa |g> Ø Ù O Ù Ù µ # µ óôõ µ # ö µ (8) Ù µ where and # µ , respectively, are the wanted state and an auxiliary µ 2 µ ÷ G state. It is easy to prove that applying # µ Ï Ñ the output of the oracle will be µ 2 µ Ù Ù µ ÷ G µ if is not in the database û µ Ï Ñ Ù µ O óôùøúú Ï µ 2 µ Ñ Ù Ù ( ÷ G µ Ï if is in the database Ñ úúü or in more common compact mathematical form Ù µ ô O $ý Ôwþ Ö Ù µ (9) µ which is simply a reflection to the axis ¾ in Figure 2 [10]. After the oracle the next operation in the Grover’s algorithm applies the _ Í dimensional Hadamard ( ) gate on the input qubits, followed by a Controlled Phase Shifter, changes the sign of all computational Ù which , as well as the last operation is again a basis states except Í for Hadamard gate ( ), similar to the second one. If the input state to the algorithm is chosen to 23 5 ÿ1µ Í µ Ù µ Ä (10) B 7 8 ( d þ g where µ has a dimension of _ , µ z {-| } µ . The expression ÿ1µ in à (10) has uniform probability amplitudes, then the output can be computed as follows. The Hadamard and the controlled phase shifter inverts the state ÿ1µ about its mean: ñ Í¨Î Í Í µ Í Í Í (11) ò ¹ ò where is the _ dimensional identity operation and and so utilizing (10), (11) becomes [6] ñ ÿ1µ ÿ' ¹ Uò Í Í ¸ (12) B. Required Number of Evaluations in Grover’s Search Algorithm ¾ µ ( d À µ 5 þ ( 5 þ Ù ) states µ , that do not lead to a where (13) sums up the d solution of the search problem, denoted by and (14) does the same Ù with states µ , which leads to it, denoted by . The variables and d refer to the number of identical entries and the total number of entries in the qregister, respectively. The initial state of the search is given as Ù µ (13) Ù µ (14) 5 ÿ1µ Ù µ ( d d d 5 ( d ¾ µ Ù µ d À µ (15) The projection of ÿ1µ on the axes ¾ µ and À µ is given as E 2 s Ï Ä Ñ suwx Ï Ä E Ñ Ä (16) as showed in Figure 2, which gives rise to periodicity. The denominator of (16) corresponds to the fact that state ÿ1µ has unity length. The optimal number of iterations depends on the initial angle between ÿ1µ and ¾ µ , as well as the number of the identical entries in the database. In [13] it was showed that the rotation of state ÿ1µ to the desired state À µ after © evaluations of quantum search is ñ ÿBµ s Ï X ¾ µ Ñ suwx Ï Ñ À µ (17) whereas in [15] it was proved that the optimal number of iterations © is given as E # Í 2 The followingñ question may arise: How many times should a Grover operator applied for a proper result? For this purpose we show a geometrical description. We suppose two basis state vectors Fig. 2. Geometrical view of Grover’s quantum search algorithm ©B In case of .-Td floor !" $ $%'&& s Ä && && ~ 7 þ)( *,+ ~ 7 þ (18) orE if it exists no multiple entries in the qregister the yielding from (16), where the optimal numE 0/ Ä to © ber of evaluations is equal [15]. It is worth mentioning /21 3 8 that in case of d 5 46 , the initial angle 7 , which leads to 2 Ä only one evaluation. angle suwx IV. Q UANTUM M ULTI -U SER D ETECTOR E MPLOYING Q UANTUM S EARCH For the optimal decision it would be necessary a fully comprehensive knowledge about the symbols sent by all the subscribers in the coverage area of a base station, a realization of the delay and noise, which is typical for a particular communication channel and all the user specific codes. All this information cannot be stored in a single database. Now, we propose a way to deal with this problem. xz|y { A. Applied Databases and Qregisters Let us assume we would like to detect the user’s symbol, denoted by . To design a suitable multi-user detector all of the user specific code words 9? have to be known3 atG the base station, collected database =: £ 9 , 9 , 9 K ª . In addition we in a 0 :<; sized ´ , which contains all the possible received signal create a qregister Ù configurations µ without ´.µ 5Å Ä>,?A@ 7B8 243 ×SR,T $1UNQP k ×SV X$ (19) T X WAP<k ÏZY W\[ R Ñ ^]N (20) T where N denotes the round up operation to an integer. It is expedih ent to chose d0JKLAM as a power of 2, but [16] has extended the search algorithm to any arbitrary d , which will not be discussed here. Ù µ Furthermore, ,=4 , which turns out as Ù µ we define a function = and = according to the two possible values of as in (3) from well as it is required to set up two qregisters of type of ´.µ : ´ 3 µ and ´ 23 µ , respectively, with a length of d O , where the subscript and gives the sign of predicted value of in (3). Why not to store h itself in one ´.µ either? In this case it would be enough to use only one qregister and the length of ´.µ should be increased only ones. Howh ever, we were not able to recognize that a given may originates form e and Á , as well. Employing two qregisters it is possible to indicate that a given originates for theoretically undistinguishable symbols and in contrast to classical algorithms we do not make&% a wrong decision. After all searching iterations the received symbol Z is contained by only one qregister ´ 3 µ or ´ 23 µ in a proper situation, which indicates that either the symbol or was transmitted by user . Ù Unfortunately, the function = µ G =4 is not a reversible operation, that shows us after searching (doing a measurement) one cannot make a statement: the observed signal was disturbed by which subscriber’s signal and by which noise and delay, therefore we are able to make & decision only on the user’s symbol and not for all other interferers based on these ; two registers. Furthermore, we follow a more general notation ´ 6 µ , £ ª with attention, the operations on both ´ 6 µ qregisters can be done in a similar way. B. Quantum Search in Qregister h kmlonop q p u r s t pv w g i f Fig. 3. The proposed Grover search circuit TABLE I ´ Ù Ù µ Ù where µ are the computational base states. Since every possible noise and delay states for a given transmitted bit is involved in the prepared register, it is superfluous to send any phase information that allows non-coherent detection. The first §c$ qubit represent all the combination of transmitted by all the interferer, e.g. if the the ; symbols value of qubit in ´.µ is equal to 1, it shows that the user ; by 2 6 , other¿ has been transmitted a symbol denoted utilization wise if the qregister has an entry . For a proper [ of the qregister the next two parts in ´.µ is Mreserved O all quantized &% M @ for . For the state of the noise _ and the delay W : W sake C_ B , of simplicity we assume to apply a linear quantization _t where £w d à -id à ª and C _ B refers to the quantization step. It is remarkable, if the probability density function (pdf) of the target quantity is known, such as by the Gaussian noise, an adequate nonlinear quantization could be used, whereby most significant values can be sampled denser, which can decrease the number of required qbits for description of noise. The values of the delay W can be generated in a K L ; DW B , where ; could be ; EGF . With straitforward manner, W the above mentioned description the size of qregister ´.µ is given by dIHKJKLAMi ONQP k j _ `bcea d In the previous section we proposed to set up two qregisters for every user’s specific code word, which contains all the delayed and disturbed copy of them, corresponding to the received ”bit” is either 1 Q UANTUM M ULTI -U SER D ETECTOR DECISION RULE TABLE AFTER THE MEASUREMENT _ `a} d _ `a~} d 0 0 0 Decision no message was sent the bit ”-1” was sent the bit ”1” was sent no decision is possible 0 or -1. The qregisters have to be built up just one time at all. It is obvious to choose a suitable database searching algorithm, to see which qregister contains the received bit, if any at all. We apply the optimal ñ quantum search algorithm proposed by Grover [12] and&% depicted in Figure 1 for our purpose. We feed the received signal Z (1) to the Ø Ù oracle ò , where the function = µ G =4- Z is evaluated such that Ø 1 SJ ÁÚ if SN X otherwise (21) K L &% &% % where Z ( *+"SU 8 Z ,$ lk . Assuming, there is again solutions for the search in qregister ´ 6 µ , O d µ O À µ ´ 6 µ O (22) ¾ h d d Ù where ¾ µ consists of such configurations of µ , which does not results i Z , while À µ does. Because of the fact of tight bound, in real application less iterations would be also enough. However, we do not need to perform a complete search as QMUD because we are not interested in the search result but only the fact is interesting whether a given configuration is involved in ´ 6 µ or not. Therefore it is worth stepping forward to quantum counting based on Grover iteration. C. Quantum Counting Performing multi-user detection it is obvious not to look for a pattern in a database itself but to indicate whether or not a solution even t ó A r r õ Z¤ ¥ o¦ ¡ ¢ £ §« ¬ ¨o © ª ®C¯´ µ °Q¶0± ² ³ ·» ¼ ¸o½ ¹ º j j j j Fig. 4. The Oracle employed by the proposed quantum multi-user detection scheme á âQã ä åQæ ç èQé í îÉï|ðòñ ó ôõ ö ÷ p(2 j) øAú ù û üÉý|þòÿ p(2 j) V. C ONCLUSIONS 0 ÀÁ ¿ ¾ p(2 j)  ê ëQì ! " # ÛÝ Ü g ÃDÄÆÇÉÈ Å Ï\ÐCÒÓ Ñ Ê\ËCÍQÎ Ì Ô\Õ0×Ø Ù Ú Ö 1 p(2 In this paper we presented a quantum computation based multi-user detection algorithm, which involves the Grover’s quantum search combined with the quantum counting method. The new method utilizes one of the possible future receiver technologies of 3G and 4G mobile systems, the so called quantum assisted computing. QMUD provides optimal detection in finite time and complexity when classical methods can achieve only suboptimal solutions. The proposed algorithm has strong resistance against MAI and noise. It should be emphasized that unlikeL the original Grover search we do not need a real database , but only the index qregister of size d HKJKLAM contains the of size database.Ä 2 $ % &('* ) j) Þàg ß Fig. 5. The quantum counter circuit exists, consequently, the number of solutions have to be summed up. This quantum counting algorithm can be accomplished with combining the Grover quantum search algorithm, introduced in Section III, with the phase estimation based on the quantum Fourier transformation [10]. The quantum counting algorithm is able to estimate the eigenvalues of the Grover iterations G . ñ ÿ1µ 0/ ÿ1µ + , V (23) 1 which is in a close connection with the demanded number of solutions on the search. %In addition to the original Grover circuit (see Fig. 1) a new register µ of © new qubits should be feed to the quantum ñ counting algorithm, as in Figure 5. The original Grover block in Figure 1 becomes a controlled Grover block, where the transformation (a rotation with angle , such as in the original Grover algorithm) will be only evaluated if the control signal equals to one. The goal of quantum counting is to estimate V to 1 bits of% accuracy, with probability of success 3 2 . The state of the register µ after the controlled Grover iterations is G6587:9 G Ò 6 . µ ÿ1µ G 5 ~4 (24) 1 7B8 where © refers to the required number of qubits in the new qregister. As a last step an inverse quantum Fourier transformation is executed on (24), is to estimate V . In case of V ¼ after a measurement, the demanded pattern is contained in the qregister ´ 6 µ , if it is equal to zero, it is not included in. % µ can be calculated in funcThe required number of 2 ; qbits © in < Ò6> tion of desired accuracy of V and the probability =?S Z , ~ 8© ?_ of , where false detection. ~ This error is caused if V is not in form of @ of false @ . Hence it can be regarded as the probability detection. Fortunately, we are able the upperboud 1 . ©B ,1 XWAP<k Ï 2 'P<k Ñ ] / The accuracy of V , 1 , is given in worst case, where s6ACBBD1Ç! W d KH JmL M ] (25) ³ , (26) Therefore, we have a direct connection between the detection error % probability and the required length of register µ . Example 1. As an example let us suppose the number of the subscribers in the cell is O4FE . The noise is represented on 10 bits, which means that d à GE? , that is equal to a resolution of approximately 0.19dB on a 200dB range. The delay is represented on 10 bits, too. Considering a maximal delay of IH0J s, which is typical for an outGHz [17], the delay resolution will be door mobile channel on ¬ DW Be² I K E ns. From the settings above (20), the required length of the qregister is I d HKJKLAM2 7\4 , which leads to2413¡8 M L , according to (26). To obtain a probability of error 2 ²t for estimating V with % accuracy 1 , the number of added bits © in qregister µ is equal to 65 bits. R EFERENCES [1] S. Verdu, Multiuser Detection. Cambridge University Press, 1998. [2] G. Jeney, J. Levendovszky, “Stochastic hopfield network for multi-user detection,” in European Conf. Wireless Technology, pp. 147–150, 2000. Paris. [3] M. Varnashi, B. Aazhang, “Multistage detection for asynchronous codedivision multiple access communication,” IEEE Trans. on Communication, vol. 38, April 1990. [4] B. Aazhang, B.-P. Paris, G.C. Orsak, “Neural networks for multiuser detection in code-division multiple-access communications,” IEEE Trans. on Communications, vol. 40, pp. 1212–1222, July 1992. [5] G. Jeney, S. Imre, L. Pap, A. Engelhart, T. Dogan, W.G. Teich, “Comparison of different multiuser detectors based on recurrent neural networks,” COST 262 Workshop on Multiuser Detection in Spread Spectrum Communication, Schloss Reisensburg, Germany, pp. 61–70, January 2001. [6] J. Preskill, “Lecture notes on quantum computation.” http://www.theory.caltec.edu/preskill/ph229, 1998-. [7] S. Imre, F. Balázs, “Quantum multi-user detection,” Proc. 1st. Workshop on Wireless Services & Applications, Paris-Evry, France, pp. 147–154, July 2001. ISBN: 2-7462-0305-7. [8] P.W. Shor, “Quantum computing,” Documenta Mathematica, vol. 1-1000, 1998. Extra Volume ICM. [9] D. Deutsch, “Quantum theory of probability and decisions,” Proc. R. Soc. London, 2000. [10] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, 2000. [11] A. Ekert, P. Hayden, H. Inamori, “Basic concepts in quantum computation,” 16 January 2000. [12] L. Grover, “How fast can a quantum computer search?,” quantph/9809029v3, Bell Labs, April 1999. [13] L.K. Grover, “A fast quantum mechanical algorithm for database search,” Proceedings, 28th Annual ACM Symposium on the Theory of Computing, pp. 212–219, May 1996. also quant-ph/9605043. [14] L. Grover, “Searching with quantum computers,” quant-ph/0011118, Bell Labs, November 2000. [15] C. Zalka, “Grover’s quantum searching algorithm is optimal,” quantph/9711070v2, December 1999. [16] M. Boyer, G. Brassard, P. Hoyer, A. Tapp, “Tight bounds on quantum searching,” Proceedings 4th Workshop on Physics and Computation, pp. 36–43, 1996. quant-ph/9605034. [17] L.M. Correia (ed.), Wireless Flexible Pesonalised Communications:COST 259, European Co-operation in Radio Research. Wiley & Sons, 2001.