Priminive logical elements synthesis by QCE emulator
Мета роботи - синтез і дослідження в середовищі емулятора QCE ряду найпростіших традиційних і нових квантових логічних елементів та алгоритмів.
Gespeichert in:
| Datum: | 2009 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2009
|
| Schriftenreihe: | Оптико-електронні інформаційно-енергетичні технології |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/32229 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Priminive logical elements synthesis by QCE emulator / V.G. Deibuk, G.P. Gorskyi // Оптико-електронні інформаційно-енергетичні технології. — 2009. — № 1 (17). — С. 132-136. — Бібліогр.: 4 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-32229 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-322292012-04-15T12:27:49Z Priminive logical elements synthesis by QCE emulator Deibuk, V.G. Gorskyi, G.P. Оптична і квантова електроніка в комп’ютерних та інтелектуальних технологіях Мета роботи - синтез і дослідження в середовищі емулятора QCE ряду найпростіших традиційних і нових квантових логічних елементів та алгоритмів. The aim of present paper is emulation and exploration of primitive quantum logical elements by QCE emulator tools. This emulator is based on Izing quantum computer hardware model. 2009 Article Priminive logical elements synthesis by QCE emulator / V.G. Deibuk, G.P. Gorskyi // Оптико-електронні інформаційно-енергетичні технології. — 2009. — № 1 (17). — С. 132-136. — Бібліогр.: 4 назв. — англ. 1681-7893 http://dspace.nbuv.gov.ua/handle/123456789/32229 537.611 en Оптико-електронні інформаційно-енергетичні технології Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Оптична і квантова електроніка в комп’ютерних та інтелектуальних технологіях Оптична і квантова електроніка в комп’ютерних та інтелектуальних технологіях |
| spellingShingle |
Оптична і квантова електроніка в комп’ютерних та інтелектуальних технологіях Оптична і квантова електроніка в комп’ютерних та інтелектуальних технологіях Deibuk, V.G. Gorskyi, G.P. Priminive logical elements synthesis by QCE emulator Оптико-електронні інформаційно-енергетичні технології |
| description |
Мета роботи - синтез і дослідження в середовищі емулятора QCE ряду найпростіших традиційних і нових квантових логічних елементів та алгоритмів. |
| format |
Article |
| author |
Deibuk, V.G. Gorskyi, G.P. |
| author_facet |
Deibuk, V.G. Gorskyi, G.P. |
| author_sort |
Deibuk, V.G. |
| title |
Priminive logical elements synthesis by QCE emulator |
| title_short |
Priminive logical elements synthesis by QCE emulator |
| title_full |
Priminive logical elements synthesis by QCE emulator |
| title_fullStr |
Priminive logical elements synthesis by QCE emulator |
| title_full_unstemmed |
Priminive logical elements synthesis by QCE emulator |
| title_sort |
priminive logical elements synthesis by qce emulator |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2009 |
| topic_facet |
Оптична і квантова електроніка в комп’ютерних та інтелектуальних технологіях |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/32229 |
| citation_txt |
Priminive logical elements synthesis by QCE emulator / V.G. Deibuk, G.P. Gorskyi // Оптико-електронні інформаційно-енергетичні технології. — 2009. — № 1 (17). — С. 132-136. — Бібліогр.: 4 назв. — англ. |
| series |
Оптико-електронні інформаційно-енергетичні технології |
| work_keys_str_mv |
AT deibukvg priminivelogicalelementssynthesisbyqceemulator AT gorskyigp priminivelogicalelementssynthesisbyqceemulator |
| first_indexed |
2025-07-03T12:45:29Z |
| last_indexed |
2025-07-03T12:45:29Z |
| _version_ |
1836629863663927296 |
| fulltext |
5
УДК 537.611
V.G. DEIBUK, G.P. GORSKYI
PRIMINIVE LOGICAL ELEMENTS SYNTHESIS BY QCE EMULATOR
Yu.Fedkovych Chernivtsi National University,
2, Kotsiubynsky str., 58012, Chernivtsi, Ukraine,
E-mail: gena_grim@mail.ru
Анотація. Метою статті є синтез і дослідження в середовищі емулятора QCE ряду найпростіших
традиційних і нових квантових логічних елементів та алгоритмів.
Аннотация. Целью статьи является синтез и исследование в среде эмулятора QCE ряда простейших
традиционных и новых квантовых логических элементов и алгоритмов.
Abstract. The aim of present paper is emulation and exploration of primitive quantum logical elements by
QCE emulator tools. This emulator is based on Izing quantum computer hardware model.
Keywords: quantum computer, emulator, logical elements.
PROBLEM ACTUALITY AND INVESTIGATION AIM
Today quantum computation problem is one of key problems of computer sciences [1]. It became actual
problem after proving of n -tuple quantum register parallelism. Due to this phenomenon n -tuple quantum
register during single machine instruction may contain and process not one but
n2 n-digit numbers
simultaneously. Sense of parallelism is in fact, that quantum bit (QB) is not in fixed state logical 0 or 1 (if its
state is not fixed especially), but uniformly in two states, thus average bit value is 0.5. Such quantum parallelism
may be used for quick solving of many problems, such as search in large disordered databases (Grover’s
algorithm), or big integers factorization (Shore’s algorithm). Actuality of last problem lies in sense of high-
protected computer cryptosystems design. Quantum bits, which role may play different objects, such as electron
or nuclear spin aggregates, quantum dots massifs, Josephson junctions in superconductors and other, may be
guided by electric or magnetic fields, optical radiation (e.g. laser beam), etc. But now direct experimental
researches with such objects are expensive and require special equipment. In other hand theoretical computation
and physical modeling are essential of every new microprocessor generation design. The aim of present article is
synthesis and approbation of some quantum logical elements and quantum algorithms.
QCE PRINCIPLES AND BASIC OPERATIONS
Especially for quantum computers (QC) and quantum algorithms (QA) designing, study and
investigation purposes some variants of quantum computer emulators (QCE) are developed. One of it is emulator
QC on nuclear magnetic resonance (NMR) [1]. It is realized by C++ programming language. Essence of
emulation process is solving of time-dependent Schrödinger equation (TDSE) by Suzuki product method. This
method is approximate, but its results are enough exact for most of practical purposes. TDSE is solving for
aggregate of spins, which are in strong static and radio-frequency (RF) weak magnetic fields and interact one
another by exchange field, which causes phenomenon of ferromagnetism.
As model Hamiltonian we use Izing Hamiltonian, which may be represented as:
( ) ( ) ( )∑ ∑ ∑ ∑
≠ = =
−−=
ji zyx i zyx
jjjiij StHSStJtH
,, ,,α α
ααααα , (1)
where
α
ijJ – exchange coupling integral between spins with numbers i and j , ( ) −tH
j
α
total magnetic
field acting on spin number j along −α axis direction. ( )tH j
α dependence is:
( ) ( )ααααα ϕπ jjjjj tfHHtH ++= 2cos10
, (2)
where α
jH 0
– static part of magnetic field, −ααα ϕ jjj fH ,,1
respectively amplitude, frequency and initial
phase of RF magnetic field, which acts on spin number j along −α axis direction.
All QA, which may be designed and checked by this QCE, are sets of microinstructions (MI) in which
framework the external static and RF and internal exchange magnetic fields acting on spins (which represent
V.G. DEIBUK, G.P. GORSKYI, 2009
ПРИНЦИПОВІ КОНЦЕПЦІЇ ТА СТРУКТУРУВАННЯ РІЗНИХ РІВНІВ ОСВІТИ З ОПТИКО-ЕЛЕКТРОННИХ ІНФОРМАЦІЙНО-
ЕНЕРГЕТИЧНИХ ТЕХНОЛОГІЙ
6
QB) and duration of their acting are determined.
MIs are the “electronic blanks”, in which corresponding cells we type values of Hamiltonian parameters
according to desired action on spins. Action of external and internal magnetic fields on spins may be represented
in terms of their rotations by user defined angles around fixed axes. Model Hamiltonian parameters are not
enough for rotation angle definition. We must define the MI action time. For this purpose in “electronic blank”
are the corresponding check boxes named “Main step”, “Intermediate steps” and “Time step”. Usually we take
Main step=Intermediate steps=1. Thus MI action time is determined by Time step value.
Basic QC operation is spin (QB) number j rotation by arbitrary angle around fixed axisα . Note, that
this rotation may be realized if magnetic field is directed along rotation axis. If time is measured in full phase
units, i.e. π2 , we may take corresponding static magnetic field 10 =α
jH ( zyx ,,=α ). Thus executing time of
MI πϕτ 20= , where
0ϕ is rotation angle in radians.
In order to illustrate interaction between QBs we describe operation ( )012 ϕR , which realizes controlled
spin rotation by predetermined angle around z -axis. It acts in such manner. Guided spin (QB) 1 rotates around
z-axis by
0
ϕ angle if guiding spin (QB) 2 is in logical 1 state and remains in initial state in opposite case. In
order to include interaction between QBs at first we turn guiding QB by 2π angle around y-axis clock wise.
This is first МІ 2Y . Second MI ( )012 ϕA realizes controlled rotation of QB 1.Its parameters
are: 112 −=
x
J , 5.001 =x
H , and action time is πϕτ 20= . After that guiding QB must be returned to initial state.
This may be realized by inverse transformation 2Y . Thus we have ( ) ( ) 20122012 YAYR ϕϕ = .
Another important QC operation is controlled phase shift. Its parameters
are JJHHJJ
z
j
z
i
z
ij 000 ,2, ϕτ −=−=== . This operation doesn’t influence on logical 0 or 1 probabilities for
acting QB’s. This property is important for quantum interference using in QA based on discrete Fourier
transformation (DFT). Quantum interference has full analogy with interference of waves in optics.
Attractive peculiarities of this emulator are its physical clearness and free of charge circulation in
Internet for education purposes.
Emulator has demonstration set of algorithms, but user may design and approbate different QA by using
graphic user interface and option of generation approbation results as numerical data file for further mathematic
processing.
MAIN RESULTS AND ITS DISCUSSION
Now we describe some synthesized traditional and new primitive elements. They are: CNOT gate,
swapping gate, 3-QB Toffoli gate, which may be used as AND gate, coincidence circuit and XOR gate on 3-QB
rotation operation basic, OR gate, trigger and “smart” swapping gate. Yet we synthesized and approbated
Grover’s database search algorithm for 8 records database and Na
x mod period “indicator”.
Consider, e.g. CNOT gate. It’s one of basic quantum gates. If QB 1 is controlling and QB 2 is target,
that CNOT operation may be represented as 212212 YIYCNOT = , where 22 ,YY are QB rotations by angles
2π± around y axis, 12I - operation with parameters .5.0,5.0,1 0112 ==−= τzz
HJ Its physical sense is QB 2
by angle π− rotation around z-axis. Magnetic field compensates exchange field if controlling QB is in logical 0
state and amplifies it if controlling QB is in logical 1 state. Thus QB 2 inverts if QB 1 is in logical 1 state and
doesn’t invert in opposite case.
By CNOT gates combination we can design ordinary SWAP gate as
12211212 CNOTCNOTCNOTSWAP = .
We may design the inverse CNOT gate. It may be represented as
2122
)(
12 YIYCNOT
inv
= , where
12I is
operation with parameters .5.0,5.0,1 0112 =−== τzz
HJ This inverse gate inverts target QB if control QB is in
logical 0 state and not inverts target QB in opposite case.
Toffoli gate repeats first and second input QB and inverts third input QB if first and second inputs QB
both are in logical 1 state. In other cases Toffoli gate repeats third input QB too. If input QBs are 1,2,3 and
output ones are 4,5,6 in reduced form Toffoli gate may be represented as
( )( ) ( )( ) ( )( ) ( )( ) 414452556366266166266456123 4444 YIYYIYYIRIRIRIRLOGTOF
yyyy
−=−
, (3)
where ( )46
y
R and ( )46 −
y
R are rotations of QB 6 by angles 4π or 4π− respectively.
ПРИНЦИПОВІ КОНЦЕПЦІЇ ТА СТРУКТУРУВАННЯ РІЗНИХ РІВНІВ ОСВІТИ З ОПТИКО-ЕЛЕКТРОННИХ ІНФОРМАЦІЙНО-
ЕНЕРГЕТИЧНИХ ТЕХНОЛОГІЙ
7
Fig. 1. CNOT-gate Fig.2. Ordinary swapping gate Fig.3. Toffoli gate for 6 QBs
If initial state of output QB 3 is logical 0 then XOR-NOT gate (coincidence circuit) may be represented as:
31233123)( YIYNOTXOR =− , (4)
where operation 123I has parameters .5.0,12313 === τzz
JJ It means, that ordinary XOR gate may
be represented as:
3123331233123 YIYYIYXOR == . (5)
Fig.4. Coincidence circuit Fig. 5. XOR gate
By CNOT and Toffoli gate using we may synthesize elements OR, AND as well as trigger. Let QBs 1,
2 are input and QB 3 is output. Install QB 3 into logical 0.In this case AND gate may be represented as:
32,1 −= TOFAND . (6)
Now install QB 3 into logical 1.In this case OR gate may be represented as:
2
2
2
13)2,1(
2
2
2
1 YYTOFYYOR −= . (7)
Fig. 6. AND or simple Toffoli gate Fig. 7. OR gate
Now we describe the trigger. Let QB 1 is information input (D), QB 2 is enable input (E), QB 3 is direct
output (Q) and QB 4 is inverse output ( Q ). If enable input is in logical 0 state trigger repeats input information
(QB1) on direct output and inverts it on inverse output. In opposite case outputs 3 and 4 are swapping. This may
be represented as:
3133414432334244)4,3()2,1( YIYYIYYIYYIYTRG =−
. (8)
Corresponding between input and output states of trigger may be represented by table 1.
Let us consider yet “smart swapping gate”, which changes values of two QBs, if they are different and
not changes it in opposite case. It is really XOR-circuit with feedback. QB 1 and 2 are input.
If they are different, that XOR(1,2)=1 and XOR(1,2)=0 in opposite case. If QB3 (auxiliary), which is
output for XOR we use as controlling QB for QBs 1 and 2, we obtain such resulting representation for “smart
swapping gate”:
3123322321131123231 YIYYIYYIYXORCNOTCNOTSmSWAP == . (9)
Table 1.
Corresponding between input and output states of trigger
QB1 (D) QB2 (E) QB3 (Q) QB4 ( Q )
0 0 0 1
1 0 1 0
ПРИНЦИПОВІ КОНЦЕПЦІЇ ТА СТРУКТУРУВАННЯ РІЗНИХ РІВНІВ ОСВІТИ З ОПТИКО-ЕЛЕКТРОННИХ ІНФОРМАЦІЙНО-
ЕНЕРГЕТИЧНИХ ТЕХНОЛОГІЙ
8
0 1 1 0
1 1 0 1
Fig. 8. Trigger Fig. 9. “Smart” swapping gate
Now we consider Grover’s algorithm [2] for 3-QB database, which contains 8 3-QB words (records). Its
basic operation is controlled phase shift by 8π angle. Parameters of corresponding MI
12g are 112 =
z
J ,
.25.0=τ QA consist of 3 stages: preparing 3-tuple register superposition state, labeling of records and search of
records. This may be represented as:
2
33
2
22
2
11
3
12
3
12112233 XYXYXYgfgXYXYXYGROV jj = . (10)
Here
jf are the labeling functions for records with numbers 0, 1…7. They may be represented as it’s shown in table 2
Table 2.
Labeling functions for different records
Number of record, j Labeling function jf representation
0
112233 XYXYXY
1
112233 XYXYXY
2
112233 XYXYXY
3
112233 XYXYXY
4
112233 XYXYXY
5
112233 XYXYXY
6
112233
XYXYXY
7
112233 XYXYXY
Now consider alternative version of Grover’s algorithm for 3 QBs. It has the form:
2
33
2
22
2
11231312231312112233
~
XYXYXYgggfgggXYXYXYGROV jj = . (11)
Labeling functions for it are represented in table 3.
Table 3.
Labeling functions for records in alternative algorithm
Number of record, j Labeling function representation
jf
~
0
112233 XYXYXY
1
112233 XYXYXY
2
112233 XYXYXY
3
112233 XYXYXY
4
112233 XYXYXY
5
112233 XYXYXY
6
112233 XYXYXY
7
112233 XYXYXY
But last algorithm is not exact. Simulation shows, that algorithm represented by (10) generates numbers
of records { }7,6,5,4,3,2,1,0 in binary notation. Algorithm (11) generates “average” numbers {0.875, 1.625,
2.375, 3.125, 3.875, 4.625, 5.375, 6.125}, or numbers { }7,6,5,4,3,2,1,0 with probability 0.67. It’s quantum
effect, which hasn’t classical analog.
Consider yet “quantum emulator” of classical Shore’s algorithm. It’s not quantum Shore’s algorithm in
the true sense. But it’s “indicator” of period Na
x mod function. Since 1mod =Na
x , if 0=x , then period of
ПРИНЦИПОВІ КОНЦЕПЦІЇ ТА СТРУКТУРУВАННЯ РІЗНИХ РІВНІВ ОСВІТИ З ОПТИКО-ЕЛЕКТРОННИХ ІНФОРМАЦІЙНО-
ЕНЕРГЕТИЧНИХ ТЕХНОЛОГІЙ
9
this function is first positive integer, for which 1mod =Na
x . Let ,15=N 2=a . Hence we have .4=x Now
we obtain 35)12)(12(15 22 ⋅=−+= . In binary notation 1004 = , 0011 = . Let use QBs 4, 5, 6 for indication of
Na
x mod period and QBs 1,2,3 for indication Na
x mod value if 1mod =Na
x . Thus for the case ,15=N
2=a indicator performance algorithm may be represented as:
2
2
3
363325221411 YYIYYIYYIYf = . (12)
Values of cubits for this case are represented in table 4.
Table 4.
Values of cubits for different x if ( ) 15mod2 xxf =
x QB1 QB2 QB3 QB4 QB5 QB6
1 0 0 1 1 0 0
2 1 1 1 0 1 0
3 0 1 1 1 1 0
4 1 0 0 0 0 1
From this table we see, that period of ( ) 15mod2 xxf = is equal to 4 because only for this case we
obtain 1 (in binary notation) in “indicating” QBs 1, 2, 3.
CONCLUSIONS
1. Іn Izing quantum computer hardware model framework we emulated and investigated some primitive
quantum logical elements. Standard ones are CNOT and TOFFOLI GATE. Last one may be considered as
universal logical element. New ones are “traditional” classical computer logical elements, as coincidence
circuit XOR-NOT, XOR, AND, OR, trigger and introduced by authors “smart” swapping gate.
2. For XOR-NOT, XOR and “smart” swapping gate synthesis we introduced the new 3-QB rotation
operation 123I .
3. Yet we emulated and explored some algorithms, as two Grover’s algorithm versions for 8 records
database and period indication algorithm for Na
x mod function in case 15,2 == Na .
REFERENCES
4. Валиев К.А. Квантовые компьютеры и квантовые вычисления / К.А.Валиев // Успехи физических
наук. – 2005. – Т.175, №1. – С.5 – 39. Библиогр.: с.38-39.
5. Mishielsen K., De Raedt H. QCE: A Simulator For Quantum Computer Hardware / K. Mishielsen, H. De
Raedt // Turk. J. Phys. – 2003. – V.27. – P.1 – 29. Ref.:p.28-29.
6. Крупичка С. Физика ферритов и родственных им магнитных окислов: в 2 т. / С. Крупичка; [пер. с
нем. под ред. А.С. Пахомова]. – М: Мир, 1976. – Перевод по изд. Svatopluk Krupička. Physic der
ferrite und verwandten magnetischen oxide (Prag: Academia, 1973),Т.1. – 1976. – 354 с. Библиогр. в
конце глав.
7. Бауместер К. Физика квантовой информации. – М: Постмаркет, 2002.
Надійшла до редакції 23.11.2008р.
DEIBUK V. G. — Sс.D., Professor of Computer Systems and Networks Department, Chernivtsi
National University, Chernivtsi , Ukraine, vdei@chnu.edu.ua.
GORSKYI G. P. — master’s degree holder, Chernivtsi National University, Chernivtsi National
University, Chernivtsi , Ukraine, gena_grim@mail.ru.
|