Von Neumann Quantum Logic vs. Classical von Neumann Architecture?

4 Pages·0.11 MB·

by  Alexander Yu. Vlasov

vs.
Von Neumann Quantum Logic Classical von Neumann
Architecture?
A. Yu. Vlasov
Conference
0 Federal Radiological Center, IRH
’2000
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Mira Street 8, St.-Petersburg, Russia
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n
a
J Abstract. The name of John von Neumann is common both in quantum mechanics and

3 computer science. Are they really two absolutely unconnected areas? Many works devoted to
quantum computations and communications are serious argument to suggest about existence
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H of such a relation, but it is impossible to touch the new and active theme in a short review.
O In the paper are described the structures and models of linear algebra and just due to their
. generality itispossibletouseuniversaldescriptionofverydifferentareasasquantummechanics
s
c and theory of Bayesian image analysis, associative memory, neural networks, fuzzy logic.
[

1
J. von Neumann is considered as one of “fa- integer numbers.
v
1 thers” of modern computers due to his theoreti- 1) x : dot = [1 .. N, 1 .. N];
0 calworks andelaborationof computer EDVAC at Here x is represented as point on 2D space
0
1 1945–50,but already20year beforeit, at 1926–31 for simplicity, because of digitization we can con-
0 he participated in drawing yet another kind of sider space of any dimension asinterval of natural
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logic—logicofquantum mechanics andonly now numbers x : 1 .. N2. In this example x is visual
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/ thedifferentareasofresearchofonescientist seem model for n+n = 2n-bits register.
s
c going to meet and born new family of ultrascale 2) I : set 2D = set of dot;
:
v cybernetic devices — quantum computers. or
i
X Currently the quantum information science I : set 2D = array [dot] of Boolean;
r exists mainly as theoretical area of research and The set, “image” can be considered as black
a
size of quantum registers does not exceed of 3–5 and white picture — black set on white sheet. It
quantum bits (qubits), but in the paper is con- is necessary 2N·N i.e. 2 (2 (2n)) bits for the set.
↑ ↑
sidered question: does the quantum logic has 3) F : fuzzyset = array [dot] of real;
someusefulapplicationasan abstract mathemati- The real is interval [0.0 .. 1.0] of real num-
cal model in computer science? Such applications bers represented with some finite precision d. It
of physical models nowadays are not unusual, for can be considered as an analogue of gray-scaled
example Boltzmann machines and other methods picture. The fuzzy set also can be standardized
came to area of artificial neural networks from by condition PxF[x] = 1 where Px in example
statistical physics [1, 2]. under consideration is double sum:
It is also useful to draw some analogy with
N
fuzzy sets and logic [3]. Here is used discrete rep-
X Fij = 1 (1)
resentation of 2D sets on some lattice due to un-
i,j=1
derstanding analogy with bit-map pictures.
(in continuous case an integration used instead of
Let us introduce few data types in some
the summation and distributions F(x) are used
“Pascal-like” notation to come from fuzzy sets
to quantum registers. Here n and N = 2n are instead of arrays).
The condition (1) is used in statistical inter- lengths (due to normalization condition) and so
pretation of fuzzy set, when it is suggested to the function really can be unit only for equivalent
choose some point of set with probability propor- vectors.
tional of adequacy function F[x], and sum of all It should be mentioned also, that the formula
the probabilities is unit. Let us use notation p[x] (3) is given for real qK for simplicity and in com-
for the probabilities. plex case it contains terms with complex conjuga-
tion: q q¯′ . It is Hermitian norm [7]. Such case
Now let us introduce notion of quantum set K K
also has applications, for example then we work
(it is not standard term, mathematical object dis-
with complex Fourier transform of some image.
cussed further corresponds to 2n-qubit register in
The discussed property of quantum set as
quantum information science [6] or quantum me-
“square root of fuzzy set” makes clear, why it can
chanical system with N2 states [7]):
be useful in such abstract area, as image recog-
4) Q : qu set = array [dot] of complex;
nition [4, 5], quite far from initial appearance in
Where complex is q = u + iw, q 1. In-
| | ≤ quantum mechanics.
stead of standardization condition here is used
Let us discuss now some question related with
normalization: Px|Q[x]|2 = 1 i.e.: “hardware”. We had few data structures:
N 1) x : dot, 2) I[x] : set 2D, 3) p[x] : fuzzyset
X Qij 2 = 1 (2) It is sequence with more and more complicated
| |
i,j=1 structure with occupation of more and more com-
puter memory. But let us suggest, that register
for example under consideration.
x is permanently changing its value by such a
In quantum mechanics the complex numbers
law, that after enough period of time T it can
q[x] is called amplitudes, related with classical
probabilities as p = q 2 = u2 + w2, i.e. there be found, the x had value v during time t[v] and
is standardized fuzzy |se|t related with given nor- lim t[v]/T = p[v].
T→∞
malized quantum set via formula: p[x] = q[x] 2. The algorithm can be implemented by soft-
| |
It ispossibleforsimplification toconsider case ware, but it also can be considered as some
with real amplitudes (w = 0), and let us explain hardware register (like implementation of random
why an “auxiliary” fuzzy set with q[x] = √p[x] number generator in some computers with main
has some independent useful application. difference, that p[v] depends on v; or input port of
Let us consider two sets p1[x], p2[x] and look some analog-to-digitalconverter is scanning some
for some likelihood function H(p ,p ) with prop- “physical model” of fuzzy set p).
1 2
erties: H(p ,p ) < 1 for p = p , H(p,p) = 1 Then each access to the register produces
1 2 1 2
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[1, 3]. For standardized sets such function can some value of x with probability p[x], so one 2n-
be chosen as H(p1,p2) = Px(p1[x]p2[x])1/2. If we bits stochastic register is enough to implement
use quantum sets q , q (p = q2, p = q2), the statistical model of standardized fuzzy set dis-
1 2 1 1 2 2
cussed earlier.
formula is H(q1,q2) = Pxq1[x]q2[x].
In our example x was chosen as multi-index of Now let us come to qu set data type. Why
2D array mostly for simple visualization and any
it can be modeled by one quantum register? The
array can be described as one-dimensional. Here example withstochasticregister asmodeloffuzzy
x = (i,j) can be substituted by index of the 1D
set is some analogy (see also [8]). But procedure
array like K = (i 1)N +j, K [1 .. N2], then of access to such register due to laws of quantum
− ∈
formula can be written as: mechanics has some differences with classical sta-
tistical register discussed below.
N N2
H(q,q′) = X qijqi′j ≡ X qKqK′ (3) Let us consider some value q ∈ qu set of
i,j=1 K=1 the register, it is array of numbers q[x], we may
not read all the numbers, but if we access to
and it shows, that H(q,q′) is simply scalar prod-
the register, we read number x with probability
uct of two vectors with N2 elements and unit
p[x] = q[x] 2 and it coincides with functionality of Similarly it is possible to define operations
| |
stochastic register described above. with fuzzy sets by definition of real analogs of
It should be mentioned only, that any access Boolean operations not ( ), and ( ), or ( ).
¬ ∧ ∨
to q destroys the quantum register by substitu- For example a 1 a, a b min(a,b),
¬ 7→ − ∧ 7→
tion instead of q new array with 1 in element with a b max(a,b) is a good choice, but here is
∨ 7→
index x and with all other is 0, and so the regis- used a second one, more algebraic a b a b,
∧ 7→ ·
ter should be reset (preparation in terminology of a b a+b ab = (( a) ( b)).
∨ 7→ − ¬ ¬ ∧ ¬
quantum mechanics) in q after each access (quan- But qu set introduced above is not directly
tum measurement). used in quantum logic — the linear operators are
But the quantum register has other useful used here instead of vectors:
property, it is possible instead of simple access L : qu map = array [dot,dot] of complex;
described above toperform another operation, we It is matrix for linear map: qu set qu set:
→
prepare some given q′ qu set and read the reg-
ister q withusing q′ as∈some“quantum bit-mask”, N N2
′ ′ ′ ′
then with probability H(q,q′) 2 (see Eq. 3) op- qij = X Lij,klqkl or qI = X LIKqK (4)
| | k,l=1 K=1
eration is successful and so by repeating it more
times we may found H with more precision. where indexes I, K are used instead of multi-
| |
Because the H has useful application as like- indexes [i,j] and [k,l] (let us for simplicity use
| |
lihoodfunction, the quantum register canbeused further the indexes like I,K : [1 .. M], M = N2).
as some hardware accelerator for image analysis. The operators A,B,... qu map form an
∈
Currently such hardware is not accessible and so algebra with usual matrix multiplication C=AB:
it was interesting to research advantages and dis-
M
advantages of the particular function H in usual
| | CKJ = XAKIBIJ (5)
software applications.
I=1
It is promising not only because such kind of
A special kind of operators, projectors, make
software would suffer giant speed-up after cre-
possible comparison of the algebra with logic, i.e.
ation specific quantum hardware, but also be-
Boolean algebra. The projector is operator with
cause the used mathematical constructions and
property P2 = P. Let us consider set of orthog-
methods of linear algebra are quite convenient
onal projectors, i.e. P P = 0, i = j, then the
i j
and powerful. 6
operators produce Boolean algebra in respect of
It should be mentioned, that similar mathe-
operations:
matical methods already was used in models of
associative memory [9]andformalneural network P 1 P, P R PR, P R P+R PR (6)
¬ ≡ − ∧ ≡ ∨ ≡ −
[10] without any relation with quantum mechan-
ical models. Only noticeable difference was us- Elements of the algebra have form P there
(S)
ing real linear spaces instead of complex and Eu- S : set of 1 .. M:
clidean norm (Eq. 3) instead of Hermitian (with
complex conjugation). P(S) = X PI (7)
I∈(S)
Let us now consider some operations with
fuzzy and quantum sets, discuss fuzzy and quan- There is relation between q qu set and
∈
tum logic. some projector Pq qu map: Pq[I,J] = q[I]q[J].
∈
For usual sets we have basic operations for To describe properties of the projector it is con-
A, B: set 2D, — intersection: A B, union: venient together with q considered as row with M
∩
A B, complementation: Ac. With using presen- elements to consider transposed column q+ (con-
∪
tation of set as Boolean array the operations in jugated for complex case).
components can be written: (ac)[x] = not a[x], Then Pq = q+ q and q q+ = H(q,q) 2 = 1
· · | |
(a b)[x] = a[x] b[x], (a b)[x] = a[x] b[x]. (the row q can be considered as 1 M ma-
∩ ∧ ∪ ∨ ×
trix, column q+ as M 1 matrix and due to
×
law of multiplication the q q+ is 1 1 matrix [4] A. Yu. Vlasov, “Quantum Computations and Im-
· ×
i.e. number and q+ q is M M matrix) and ages Recognition,” Conference QCM’96, full paper:
P P = q+ q q+ q =·q+ 1 q×= q+ q = P . quant-ph/9703010
q q q
· · · · · ·
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1
q ,...,q then P = q+ q are orthogonal projec- Data Analysis,” quant-ph/9802028
2 k i i · i
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tation,” Physics Today 48 (1995) 24
have rather relations with usual logic than with
fuzzy one. [7] A.I.Kostrikin,Yu.I.Manin,Linear Algebra and Ge-
ometry, Nauka, M. 1986 [Russ.]
For more clear explanation of properties of P
i
it is possible to use existence of some orthogo- [8] R. R. Zapatrin, “Logic Programming as Quantum
Measurement”, Int. J. Theor. Phys. 34 (1995) 1813
nal (unitary for complex case) matrix U same for
all P such, that all P′ = UP U−1 are diagonal [9] T. Kohonen, Associative Memory: A System- Theo-
i i i
and have very simple form: P′ = diag(1,0,...,0), retical Approach, Springer 1978 [Mir, M. 1980]
1
P′ = diag(0,1,...,0), etc.. [10] E. N. Sokolov, G. G. Vatkyavichus, The neuro-
2
intelligence: from neuron – towards neurocomputer,
The classical Boolean structure of the opera-
Nauka, M. 1988 [Russ.]
tors P and their sums P (see Eq. 7) is because
i (S)
[11] A.A.Grib,Violation of Bell’s Inequalities and Prob-
of all the operators commute [11]. If to choose
lem of Measurements in QuantumTheory, JINRP2–
projectors P, R: PR = RP the Eq. 6 do not
6 92–211,Dubna 1992 [Russ.]
produce Boolean algebra, but it is other kind of
[12] A.Yu.Vlasov,“RepresentationandProcessingofIn-
non-Boolean logic, than fuzzy one.
formationinQuantumComputers,”Tech.Phys.Lett.
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tion with fuzzy set have so-called mixed quantum
[13] Landau and Lifshitz, Course of Theoretical Physics,
states R = w P , w = 1 where w have
Pi i i Pi i i III (Quantum Mechanics) Nauka, M. 1989 [Russ.]
statistical nature and so here is written analog of
[14] R. P. Feynman, “Quantum-Mechanical Computers,”
standardized fuzzy set. Found. Phys. 16 (1986) 507
A representation of some kind of fuzzy opera-
[15] D. Deutsch, “Quantum Theory, the Church-Turing
tions is example with family of commuting op-
Principle and the Universal Quantum Computer,”
erators [12], but not projectors. They are de- Proc. R. Soc. London A 400 (1985), 97
scribed by diagonal matrices: Dq[I,I] = q[I], [16] D. Deutsch, “Quantum Computational Networks,”
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q
6
Bibliographicalnotes: Inadditiontoreferences [17] D. Deutsch, A. Ekert, R. Lupacchini, “Machines,
[7, 11] some general handbook on quantum me- Logic and Quantum Physics,” math/9911150
chanics like [13] is appropriate for most text of [18] G. Birkhoff, J. v. Neumann, “The logic of quantum
the paper. New area of quantum computation mechanics,” Ann. Math. 37 (1936) 823
is presented in [6, 14–17]. Two works of J. von [19] A. Burks, H. Goldstine, J. v. Neumann, Preliminary
Neumann devoted to quantum logic [18] and elec- discussion of the logical design of an electronic com-
puting instrument, Princeton 1946
tronic computers [19] are included for complete-
ness.
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