Download жгед symbol of the згедй из "! user is denoted by # $иж! , 5 $87!9 A

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$S ~ j 8
K"L
,"
&%
& %
%
, ~ ƒ
W ~ lk system. According to the axioms of quantum mechanics, every physical system can be characterized by means of its states š ´.µ 1 in the
Hilbert vector space over the complex numbers ¶ , whereas2‰a3 physical transformation can be described as a Unitary operator ·
³·i¸ ,
respectively. The inner product ¹º š ´.µ maps the ordered pair of vectors
to ¶ with the properties [6]:
» Positivity: ¹º š º µ-¼ for š º µ ,
» Linearity: ¹ ´°š (a š º 3 µ +b š º G µ )=a ¹ ´°š º 3 µ +b ¹ ´°š º G µ ,
» Skew symmetry: ¹ ´°š º µ e¹ ´°š º µg½ .
In the classical information theory the smallest information conveying unit is the bit. The counterpart unit in quantum information
is called the ”quantum bit”, the qubit.
Its state can be described by
G
means of the state š ´.µ , š ´.µ ¿¾ š µ XÀ š µ , where
¾^G ÀÁ)¶ refers
G
to the complex probability
amplitudes and š ¾ š š À š ¤ [8], [9],
[6]. The expression š ¾ š denotes the probability that afterG measuring
the qubit it can be found in computational base š µ , and š À š shows the
probability to be in computational base š µ . In more general descripš ´.µ is set up from qubits
tion an d -bit ”quantum
register” (qregister)
"Ã
š
µ
de$ computational bases, where dÂ
spanned by
states can be stored in the qregisters at the same time [7].
Å
243 Å
5Å
š ´.µ œÄ
´ š µÆ´ Ƕ'
(6)
7B8
Å
ÁÉ ;
;
d denotes G the number of states and È
where
, ¹ š µ ,
š µ
š
´
š
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.
2‰3 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]:
‡‰ˆŠ ‡˜›
˜
Z
J
—
:
™
š
‹"Œgv U?‘ 2‰
Ž3u< x ’b3…“n”–•
†
†
(5)
K
2‰3 ‡
‡
œ‹$Œgv ?‘ 24
Ž3uw x’.3…“g”Xž
ƒŸ ™ Ÿ
cŸ ™ ¡ ˜
where ™ refers to the optimization variable and Ÿ¢
¤£ ¥U ~ƒ¦ §
g¨g©¨
fgǻ . 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
2‰other
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
Ã
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
dÁƒ . 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 šÙ š Ù µ š # µ óôõ
µ š
# ö
µ (8)
ٚ µ
where
and š # µ , respectively, are the wanted state and an auxiliary
š µ 2 š µ
÷ G
state. It is easy to prove that applying š # µ Ï
Ñ the output of
the oracle will be
š µ 2 š µ
Ù
šÙ µ
÷ G
µ
if is not in the database
û
š
š
µ
Ï
Ñ
ٚ µ
O
óôùøúú
Ï
š µ 2 š µ
Ñ
Ù
Ù
(
÷ G
š µ Ï
if is in the database Ñ
úúü
or in more common compact mathematical form
š Ù µ ô O $ý Ôwþ Ö š Ù µ (9)
š
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
2‰3
5
š ÿ1µ Í š µ šÙ µ Ä
(10)
B
7
8
( d þ
g
where š µ has a dimension of _ , š µ š z {-| } µ . The expression š ÿ1µ in
Ã
(10) has uniform probability amplitudes, then the output can be computed as follows. The Hadamard and the controlled phase shifter inverts the state š ÿ1µ about its mean:
ñ
ͨΠÍ
Í š µ šÍ
Í Í
(11)
òœ
¹
ò
where is the _ dimensional identity operation and
and so utilizing (10), (11) becomes [6]
ñ
š ÿ1µ ÿ'š ¹ Uò
Í
Í
¸ (12)
B. Required Number of Evaluations in Grover’s Search Algorithm
š¾ µ ( d
šÀ µ 5
þ
(
5
þ
Ù
) states š 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 š 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
šÙ µ (13)
šÙ µ (14)
5
š ÿ1µ šÙ µ ( d
d
d
5
( d
š¾ µ šÙ µ d
šÀ µ (15)
The projection of š ÿ1µ on the axes š ¾ µ and š 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 š ÿ1µ has unity length.
The optimal number of iterations depends on the initial angle
between š ÿ1µ and š ¾ µ , as well as the number of the identical entries in the database. In [13] it was showed that the rotation of state š ÿ1µ to
the desired state š À µ after © evaluations of quantum search is
ñ š ÿBµ
s Ï
X
š¾ µ Ñ
suwx Ï
Ñ
šÀ 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
!"
‹$Œ
$%'&&
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 š µ without š ´.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.
šÙ µ
Furthermore,
,=4 , which turns out as
Ù µ we define a function =
š
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 š ´.µ : š ´ 3 µ and
š ´ 2‰3 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 š ´.µ either? In this case it would be enough
to use only one
qregister and the length of š ´.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 š ´ 3 µ or š ´ 2‰3 µ in a proper situation,
which indicates that either the symbol or was transmitted by
user .
Ù
Unfortunately, the function = š 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 š ´ 6 µ , £ ª with attention, the operations on both
š ´ 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 š Ù µ Ù
where š 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 š ´.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 š ´.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 œ£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 ; ž EGF . With
straitforward manner, W–
the above mentioned description the size of qregister š ´.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
ƒ
€ ‚
„
€ ‚
„
€ ‚
0
ƒ
€ ‚
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 = š 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
š ´ 6 µ ,
O
d
š µ
O šÀ µ š ´ 6 µ O
(22)
¾ h
d
d
Ù
where š ¾ µ consists of such configurations of š µ , which does not results
i
Z , while š 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 š ´ 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˜
Ž
r
Š‹
…
r
Œ
†
õ
‡ˆ
š ›Zœž¤ ¥ o¦„
Ÿ ¡ ¢ £ §ž« ¬ ¨o­ © ª
™
®C¯´ µ °Q¶0± ² ³ ·ž» 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 ô•õ ö ÷ p(2 j) øAú ù
û üÉý|þòÿ p(2 j)
V. C ONCLUSIONS
0
ÀÁ ¿
¾
p(2 j)
Â
ê ëQì
! " #
ÛÝ Ü
g
ÃDÄÆÇÉÈ Å
Ï\ÐCÒ•Ó 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 š ÿ1µ
0/
š 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 š 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 š µ after the controlled Grover
iterations is
G6587:9
G
Ò 6 . š µš 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 š ´ 6 µ , if it is equal to
zero, it is not included in.
š % 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 š 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 š 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.