CC BY
30
Step 2. To construct an initial mesh, we pick out points from the set {b 3n3k }■ We want the initial mesh to be close to the aspect ratio
demand. The subroutine, say, “Initial”, starts from the rectangle A 00 A 01A 10 A n , consider six pairs of 3-dimensional triangles
corresponding to 8 points:
AAAAAAAA л 00 01,yi 0 2 10' л w'1 12’ л 2021 ’
and choose one pair which is more close to the aspect ratio condition. After that the subroutine goes to either a rectangle A oiA 02 A n A 12 or a rectangle A 10 A n A 20 A 21 ■ Thus, we obtain a net
B 00 , B 01, B 10- - - B lm, • where B ij e {A nk }, L1 ^L M 1 ^M
So, we have constructed a triangular mesh g(B ..) that satisfies conditions 1), 2), and conv (B ..) = R.
iJ iJ
Step 3. For a subdivision process, we suggest an interpolating Modified Butterfly scheme ([5], Ch.4). We can suppose here that, loosely speaking, the subdivision surfaces f k (R) approach to a given surface S. Here f k —— f , and f (R) is the subdivision surface. After k-th step
k k
of subdivision, we obtain a triangular net {B .. } CR, and a corresponding mesh {b .. }, where
4 4
f (B kij) = b kj- g(B j )e {b ij } k=0,1, ... .
A user can interactively choose a level of subdivision in different domains. Let the subroutine be called “Subdivision”.
k k k
Step 4. After k-th level of subdivision we project (by a subroutine, say, “Projection”) a mesh {b .. } onto the surface S. Let a .. = P(b ,,
4 ч 4
k
) , where P is a projection, a j £ S.
Step 5. Now one should verify a condition 3. We suggest here to use a distance d j between a barycenter of corresponding triangle and
S (instead of d), and verify a condition d j < £ /2. Let the subroutine be called “Distance”.
4. Conclusions and future work
In this paper we suggested a method for finite element mesh generation by a program consisting of 5 subroutines. However, there are some unsolved problems in this project. For example, what is the best algorithm in the step 2? How to show mathematically, that a difference between f and g is sufficient small in the step 3? How to connect nodes in the final mesh in the case of adaptive subdivision? These problems and others, together with computer implementation, are subjects of future research.
References
1. “Fuji technical research” company. Private communications, Tokyo, 2000.
2. Gerald Farin “Curves and surfaces for CAGD”. Academic press, 1993.
3. Ichiro Hagiwara. Private communications, Tokyo Institute of Technology, 2000.
4. K.-J. Bathe “Finite Element Procedures”. Prentice-Hall, 1996
5. “Subdivision for Modeling and Animation”. SIGGRAPH 99 Course Notes.
Берзин Д.В.
Кандидат физико-математических наук, доцент, Финансовый университет при Правительстве Российской Федерации,
Москва
КВАНТОВЫЕ ВЫЧИСЛЕНИЯ И АВТОМАТИЗИРОВАННЫЕ СИСТЕМЫ ПРОЕКТИРОВАНИЯ
Аннотация
В данной статье мы показываем, что применение квантовых вычислений в автоматизированных системах проектирования может существенно повысить их эффективность.
Ключевые слова: квантовые вычисления, системы автоматизированного проектирования.
Berzin D.V.
PhD, Associate Professor,
Financial University under the Government of the Russian Federation, Moscow QUANTUM COMPUTATIONS AND CAD
Abstract
In this paper we show how quantum computations can enhance CAD systems performance.
Keywords: quantum computations, CAD.
1. Motivation
Information processing (computing) is the dynamical evolution of a highly organized physical system produced by technology (computer) or nature (brain). The initial state of this system is (determined by) its input; its final state is the output. Physics describes nature in two complementary modes: classical and quantum. Up to the nineties, the basic mathematical models of computing, Turing machines, were classical objects, although the first suggestions for studying quantum models date back at least to 1980 ([Ma]). Roughly speaking, the motivation to study quantum computing comes from several sources: physics and technology, cognitive science, and mathematics. We will briefly discuss them in turn.
1) Physically, the quantum mode of description is more fundamental than the classical one. In the seventies and eighties it was remarked that, because of the superposition principle, it is computationally unfeasible to simulate quantum processes on classical computers ([Po], [Fe1]). Roughly speaking, quantizing a classical system with N states we obtain a quantum system whose state space is an (N-1)-dimensional complex projective space whose volume grows exponentially with N. One can argue that the main preoccupation of quantum chemistry is the struggle with resulting difficulties. Reversing this argument, one might expect that quantum computers, if they can be built at all, will be considerably more powerful than classical ones ([Fe1], [Ma]). Progress in the microfabrication techniques of modern computers has already led us to the level where quantum noise becomes an essential hindrance to the error-free functioning of microchips. It is only logical to start exploiting the essential quantum mechanical behavior of small objects in devising computers, instead of neutralizing it.
2) As another motivation, one can invoke highly speculative, but intriguing, conjectures that our brain is in fact a quantum computer (see a recent paper [Ha]). For example, the progress in writing efficient chess playing software (“Deep Blue”) shows that to simulate the
world championship level using only classical algorithms, one has to be able to analyze about 10 6 positions/sec and use about 1010 memory
-3
bytes. Since the characteristic time of neuronal processing is about 10 sec, it is very difficult to explain how the classical brain could possibly do the job and play chess as successfully as Karpov does. A less spectacular, but not less resource consuming task, is speech generation and perception, which is routinely done by billions of human brains, but still presents a challenge for modern computers using
9
classical algorithms. The implementation of efficient quantum algorithms which have been studied so far can be provided by one, or several, quantum chips (registers) controlled by a classical computer. A very considerable part of the overall computing job, besides controlling quantum chips, is also assigned to the classical computer. Analyzing a physical device of such architecture, we would have direct access to its classical component (electrical or neuronal network), whereas locating its quantum components might constitute a considerable challenge. For example, quantum chips in the brain might be represented by macromolecules of the type that were considered in some theoretical models for high temperature superconductivity. It would be extremely interesting to devise an experimental setting purporting to show that some fragments of the central nervous system relevant for information processing can in fact be in a quantum superposition of classical states.
3) Finally, we turn to mathematics. One can argue that nowadays one does not even need additional motivation, given the predominant mood prescribing the quantization of “everything that moves''. Quantum groups, quantum cohomology, quantum invariants of knots etc come to mind.
2. Brief review of classic works
Problems of energy dissipation during computations were investigated in Landauer’s paper in 1961 ([La]). It was shown that von Neumann’s evaluation is valid for logically irreversible operations realization, but, if a computer deals with logically reversible operations, it is difficult to say about real energy dissipation. Thus, it is necessary to teach computers to compute reversibly.
C.Bennet constructed a logically reversible computation scheme ([Bnt]). Soon, P.Benniof represented a procedure of reversible computations in Hamiltonian form and described a quantum Turing machine ([Beni], [Ben2]).
New epoch in the history of quantum computers was started with R.Feinmann papers ([Fei], [Fe2]), in which a brilliant introduction to the problem of quantum computers was given. He noted that the number of possible conditions in quantum systems is exponentially great in comparison with a classical system.
It should be noted that the necessity of quantum computations development was proclaimed in Yu.Manin’s work [Ma] before R.Feynman. Yu.Manin wrote that the main problem in the theory of quantum machines is an abstract formulation of their work, using general principles of quantum theory and description of systems evolution by means of unitary operators in Hilbert space. This problem was solved by D.Deutsch ([Deu]). Manin’s principles led him to a natural, but a somewhat unexpected discovery of rapid computing methods, impossible to classical schemes.
Theoretical quantum computing entered its modern stage in 1994, when P. Shor ([Sh]) devised the first quantum algorithm showing that prime factorization can be done on quantum computers in polynomial time, that is, considerably faster than by any known classical algorithm. (P.Shor's work was inspired by the earlier work [Si] of D.Simon). Shor's paper gave a new boost to the subject. Another beautiful
result due to L.Grover ([Gro]) is that a quantum search among N objects can be done in c VN steps. A.Kitaev [Kii] devised new quantum algorithms for computing stabilizers of abelian group actions; his work was preceded by that of D.Boneh and R.Lipton [BoL], who treated the more general problem by a modification of Shor's method (cf. also [Gri]). At least as important as the results themselves are the tools invented by Shor, Grover, and Kitaev.
Currently, nobody knows how to build a quantum computer, although it seems as though it might be possible within the laws of quantum mechanics. Some suggestions have been made as to possible designs for such computers [Te], [Ldi], [Ld2], [CZ], [Vi], [SW], [Yam]. In addition, we want to mention most recent papers [Oh], [Ko], [Ld3]. However, there will be substantial difficulty in building any of these. The most difficult obstacles appear to involve the decoherence of quantum superpositions through the interaction of the computer with the environment, and the implementation of quantum state transformations with enough precision to give accurate results after many computation steps. Both of these obstacles become more difficult as the size of the computer grows, so it turns out to be possible to build small quantum computers, while scaling up to machines large enough to do interesting computations may present fundamental difficulties.
Nevertheless, physical laws do not prohibit diminishing the size of computers until bits size approaches to the atoms one, and the quantum behavior become to be dominant.
3. Superposition principle and quibits
Quantum mechanical phenomena are difficult to understand since most of our everyday experiences are not applicable. Quantum mechanics is a theory in the mathematical sense: it is governed by a set of axioms. The consequences of the axioms describe the behavior of quantum systems. In this short paper we cannot describe even a basic ideas of quantum mechanics. For more detailed explanation see, for example, [RP], [Ki2].
From the physical point of view the basic unit of a computer is a physical system that provides storage for data of computation, that is intermediate values of variables necessary for the calculation. Since it is essentially a physical, material thing, it must behave in accord with the laws of physical theory that describes its functioning. Suppose it be the quantum theory. Then the conditions are to be fulfilled:
Q1 The logical state of calculation must correspond to the quantum state of a register;
Q2 Transitions between states of the register must be the quantum ones.
At this point we have to recall a few basic principles of quantum mechanics:
51 The state of a system Q is represented by a vector > of a complex linear space H, which has the structure of Hilbert space. Vectors a | ф > , a being a complex number, a Ф 0, correspond to the same state of the system. The scalar product of two vectors | у > and | ф > is given by <Щ \ ф> .
52 Dynamic variables of the system Q are hermitian operators with respect to the scalar product < | >, given above, acting in H.
53 Dynamic transformations of the system Q are unitary operators with respect to the scalar product < | > acting on vectors of H.
54 Measurement is a physical process which effect on state vectors is given by the action of a projection operator, P; the value P
| у > is the result of mesuaring process applied to the state | у > with probability P \ Щ > .
The most important corollary of points Si- S3 is
Superposition principle. If the system may exist in states | у > and | ф >, it may also exist in states that correspond to the linear combinations a | у > + b \ ф > , in which a,b are complex numbers.
From S3 we infer that dynamical transformations in quantum theory are reversible operations. Hence, we shall have important constraints imposed on logical operations, if we suppose that the action of an algorithm relies on quantum physical properties. Indeed, they must be reversible as well. Thus the quantum logic is reversible. To see the point let us consider the main difference between the classical and the quantum registers. Suppose that the classical register comprises only one memory cell, or it has only one bit of memory. The contents of the cell may be visualized as a set of two elements 0 and 1; it is the smallest memory storage possible in the classical world. What is its counterpart in the quantum world? It should contain at least two elements so as to accommodate a meaningful storage of states. Thus we may suggest that there are two states | у > and | ф >, but according to the superposition principle given above there are also states
corresponding to a \ у > + b | ф >. Consequently, the smallest storage of quantum states is a two-dimensional complex linear space, or qubit.
10
We see that the abstract requirement of the superposition principle turns out to be extremely powerful. It will require novel instruments to deal with the quantum storage. The superposition principle is the main law of microscopical world of quantum mechanics; it declares that if a system may occupy two different states, it may also occupy a continuum of other states in some sense generated by them. Unlike classical bits, a qubit can be put in a superposition state that encodes both 0 and 1. There is no good classical explanation of superpositions: a quantum bit representing 0 and 1 can neither be viewed as “between” 0 and 1 nor it can be viewed as a hidden unknown state that represents either 0 or 1 with a certain probability. Even single quantum bits enable interesting applications, for example, to secure key distribution.
4. Advantages of quantum computations
In classical physics, the possible states of a system of n particles, whose individual states can be described by a vector in a twodimensional vector space, form a vector space of 2n dimensions. However, in a quantum system the resulting state space is much larger: a
system of n qubits has a state space of 2 n dimensions. It is this exponential growth of the state space with the number of particles that suggests a possible exponential speed-up of computation on quantum computers over classical computers.
Individual state spaces of n particles combine classically through the Cartesian product. Quantum states, however, combine through the tensor product. Let us look briefly at distinctions between the Cartesian product and the tensor product that will be crucial to understanding quantum computations.
Let V and W be two 2-dimensional complex vector spaces with bases {v ^ ,v 2 } and {w ^ ,w 2 } respectively. The Cartesian product of
these two spaces can take as its basis the union of the bases of its component spaces {v ^ ,v 2 , w ^ ,w 2 }. The dimension of the state space of multiple classical particles grows linearly with the number of particles, since dim(X*Y)=dim(X)+dim(Y). The tensor product V 0 W of V and W has basis { v ^ 0 w^, v^ 0 w 2 , v 2 0 w^, v 2 0 w 2 }. So the state space for two qubits, each with basis {|0>,|1>}, has basis {|0> 0 |0>, |0> 0 |1>, |1> 0 |0>, |1> 0 |1>} which can be written more compactly as {|00>,|01>,|10>,|11>}. More generally, we write |a>
to mean |a0a^...a >, where aare the binary digits of the number a. A basis for a three qubit system is
{|000>,|001>, |010>, |011>,|100>,|101>,|110>,|111>}
and in general an n qubit system has 2 basis vectors. We now can see the exponential growth of the state space with the number of quantum particles. The tensor product X 0 Y has dimension dim(X)*dim(Y).
Measurement of a single qubit projects the quantum state on to one of the basis states associated with the measuring device. The result of a measurement is probabilistic and the process of measurement changes the state to that measured. Through measurements we intervene into a system. According to the principles of quantum mechanics, the action is accompanied by a projection of the state vector of the system, or its collapse, which is essentially irreversible process, so that the measurement is the final step after which the system is to all practical purposes destroyed. In this sense the action of a measurement can provide only partial information.
Now we are going to see that the superposition principle gives us an opportunity for quantum parallelism. Firstly, we can make a simple observation that the vector
|a> =|aо>0 | a j>0 ... 0 | a >
in which a(. = 0,1 ; i=0,1, ..., n-1, provides a means for the storage of integer numbers written in 2-adic notation:
_ _ n-1
a = a о + ax2 + ...+ a 2
In fact, there is a vector that contains all integers up to 2
I 2" -1
!*> = ~j= 21 a >
V2" a=0
The vector can be obtained from the standard vector
it reads
|0> = |0>0|0>0... 0|0> with the help of the transformation
H| 0> = H|0> 0 H|0> 0 H|0> = |t> in which the matrix H reads
1 1 1
H= -j=( )
V2 1 -1
(1)
Equation (1) is an example of the power of quantum computations. Indeed, we have generated all the integers up to 2 n in one step, and stored them in a state of the register. They suggest that the quantum parallelism is at the crux of the matter for quantum computations. It has become possible, because the quantum mechanics enables us to store information by means of vectors in linear space.
Let us consider a specific example, the calculation of a function. Suppose f(a) is a function of integer argument, taking values 0,1. We
may construct a unitary transformation U f , or in physical terms a quantum process affecting the register, that will effect its calculation for
all a = 0,1, ... , 2n -1, in one step. For |t> defined above we write down
1 2n -l
u f (|t> 0|0>) = —= ^ | a >01/(a) > (2)
V2" a=0
To find the value of f(a) we have to perform a measurement that amounts, in mathematical terms, to employing a projection operator to the right hand side of equation (2) so as to obtain the summand
|a> 0 |f(a)>
The examples and arguments given above make clear the two most important advantages of quantum computations, that is the exponential growth of storage space as the number of qubits increases and the quantum parallelism. Both are the corollaries of the quantum nature of computational device, or quantum computer. We see that in contrast to the use of classical computers when we have to tackle bits of information one at a time, the quantum computations based on the superposition principle enable us to deal with ensembles, state vectors, and result in quantum parallelism.
5. Applications to CAD and future research
11
There are at least two set of applied problems for which the use of quantum computers will bring certain advantages.
1. Search algorithms.
The following problem is alleged to be known: given an unstructured list of items a 0, a ^,..., a N find a particular item a. The
common example is looking for a particular telephone number in the telephone directory, for someone whose name you do not know. The classical algorithm requires N /2 steps for a list of N items. The quantum algorithm found by Grover ([Gro]) requires only the number of
states of order VN.
2. Factorization of integers.
The classical problem of factorizing a given integer N into the product of its prime factors. By now the best algorithm requires a number of computational steps of order given by the formula
r_T 1/3 , 2/3
E = exp[2L (log L) ]
in which L= log N. From the estimate one may infer that factoring a number of 130 digits, that is L :
3,
; 300, amounts to E ~ 1018 . The
quantum algorithm found by Shor requires only O(L ~ ) steps.
Thus the Q-approach promises to solve problems difficult for the classical one. It is only natural to expect that quantum algorithms may turn out to be useful for problems outside search and arithmetics. As far as we can see, Grover’s algorithm can be very useful for CAD computations. For example, consider a well-known CAGD procedure called “Z-buffer”. There is an unsorted set consisting of N points in R
3
. A problem is to sort them along z-axis. Suppose, for example, that N=1000. Using quantum computation, we will fulfil the procedure approximately 16 times faster then using ordinary computer. Undoubtedly, we can solve CAD problems much more efficiently by means of quantum computations instead of classical algorithms.
Of course, quantum computations need specific devices, or quantum computers, to become a working tool for science and business. The research in this direction is gathering momentum, and one may expect that the Q-computer will be a feature of foreseeable future.
References
1. [Ben1] P.Benioff. “The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines.” // J. Stat.Phys., 22 (1980), 563-591.
2. [Ben2] P.Benioff. “Quantum mechanical Hamiltonian models of Turing machines that dissipate no energy.” // Phys.Rev.Lett., 48 (1980), 1581-1585.
3. [BoL] D.Boneh, R.Lipton. “Quantum cryptoanalysis of hidden linear functions.” // Proc. of Advances in Cryptology - CRYPTO '95, Springer LN in Computer Science, vol. 963 (1995), 424-437.
4. [Bnt] Charles H. Bennett, IBM J.Res.Develop., 17, 525 (1973)
5. [CZ] J.Cirac, P.Zoller. “Quantum computation with cold trapped ions.” // Phys. Rev. Lett., 74:20 (1995), 4091--4094.
6. [Deu] D.Deutsch. “Quantum theory, the Church--Turing principle and the universal quantum computer”. // Proc. R. Soc. Lond. A 400 (1985), 97--117.
7. [Fe1] R.Feynman. “Simulating physics with computers.” // Int. J. of Theor. Phys., 21 (1982), 467-488.
8. [Fe2] R.Feynman. “Quantum mechanical computers.” // Found. Phys., 16 (1986), 507-531.
9. [Gri] D.Grigoriev. “Testing the shift-equivalence of polynomials using quantum mechanics.” // In: Manin's Festschrift, Journ. of Math. Sci., 82:1 (1996), 3184-3193.
10. [Gro] L.K.Grover. “Quantum mechanics helps in searching for a needle in a haystack.” // Phys.Rev.Lett. 79 (1997), 325-328.
11. [Ha] S.Hagan, S.R.Hameroff, Tuszhynski J.A. “Quantum computations in brain microtubules? Decoherence and biological feasibility.” // Quant-ph/0005025, May 2000
12. [Ki1] A.Kitaev. “Quantum computations: algorithms and error correction.” // Russian Math. Surveys, 52:6 (1997), 53--112.
13. [Ki2] A.Kitaev. “Classical and quantum computations.” // Lecture notes, Independent University, Moscow, 1998.
14. [Ko] A.Kokin (Institute of Phisics and Technology, Russian Academy of Science) “A model for ensemble NMR quantum computer using antiferromagnetic structure.” // quant-ph/0002034, February 2000.
15. [La] R.Landauer, IBM J.Res.Develop., 3, 183 (1961)
16. [Ld1] S. Lloyd (MIT Mechanical Engineering) “A potentially realizable quantum computer.” // Science 261, pp.1569-1571, 1993
17. [Ld2] S. Lloyd (MIT Mechanical Engineering) “Envisioning a quantum supercomputer.” // Science 263, p. 695, 1994
18. [Ld3] S. Lloyd (MIT Mechanical Engineering) “Unconventional Quantum Computing Devices.” // Quant-ph/0003151, March 2000.
19. [Ma] Yu.Manin. “Computable and uncomputable” (in Russian). // Moscow, Sovetskoye Radio, 1980.
20. [Oh] Toshio Ohshima (Fujitsu Laboratories Ltd.) “All optical cellular quantum computer having ancilla bits for operations in each cell.” // Quant-ph/0002004, February 2000.
21. [Po] R.P.Poplavskii. “Thermodynamical models of information processing (in Russian).” // Uspekhi Fizicheskikh Nauk, 115:3 (1975), 465-501.
22. [RP] Eleanor G. Rieffel, Wolfgang Polak “An Introduction to Quantum Computing for Non-Physicists” // quant-ph/9809016, 1998
23. [Sh] P.W.Shor. “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer.” // SIAM J. Comput., 26:5 (1997), 1484-1509.
24. [SW] T.Sleator , H.Weinfurther “Realizable universal quantum logic gates.” // Phys. Rev. Lett., 74, pp.4087-4090, 1995
25. [Te] W.G. Teigh, K.Obermayer, G.Mahler “Structural basis of multistationary quantum systems.” // Phys. Rev. B., 37, pp.81118121,1988
26. [Vi] DiVincenzo “Two-bit gates are universal for quantum computation.” // Phys.Rev. A, 51, pp. 1015-1022, 1996
27. [Yam] Y.Yamamoto, I.L. Chuang “A simple quantum computer” // Quant-ph/9505011, May 1995
Битюцкая Н.И.1, Руденко В.Г.2
1 Доцент, кандидат физико-математических наук, 2 доцент, кандидат физико-математических наук, Северо-Кавказский
федеральный университет, филиал в г. Пятигорске
РЕШЕНИЕ СИСТЕМ РЕКУРРЕНТНЫХ МУЛЬТИПЛИКАТИВНЫХ УРАВНЕНИЙ ПЕРВОГО ПОРЯДКА
Аннотация
Предложены два метода решения систем рекуррентных мультипликативных уравнений первого порядка. Получено в явном виде решение для системы двух уравнений.
Ключевые слова: числовые последовательности, рекуррентные мультипликативные соотношения, системы рекуррентных уравнений.
Bityutskaya N.I.1, Rudenko V.G.2
1Candidate of physics and mathematics, associate professor, 2 candidate of physics and mathematics, associate professor, North-
Caucasian Federal University, Pyatigorsk branch
SOLUTION SYSTEMS OF RECURRENT MULTIPLICATIVE FIRST-ORDER EQUATION
12
