arXiv:cs/0001001v1 [cs.OH] 3 Jan 2000 Conference ieo unu eitr osntece f3–5 of exceed ( not bits does and quantum registers research quantum of of area size theoretical as mainly exists ultrascale of family — new devices born cybernetic and meet seem to scientist one going of of research now of only areas kind and different mechanics the another quantum of yet logic — drawing logic in 1926–31 at participated it, before he year 20 already at EDVAC but theoreti- computer 1945–50, his of elaboration to and due works cal computers modern of thers” Psa-ie oaint oefo uz sets Here fuzzy registers. from quantum come to to notation “Pascal-like” un- to pictures. due bit-map lattice with some rep- analogy on discrete derstanding used sets is 2D Here of resentation [3]. logic and sets fuzzy from networks neural 2]. artificial [1, physics of statistical area to methods came other for and unusual, machines not Boltzmann example are nowadays models physical has of logic model quantum cal the as application does useful some question: sidered o emn unu Logic Quantum Neumann Von urnl the Currently “fa- of one as considered is Neumann von J. e sitouefwdt ye nsome in types data few introduce us Let with analogy some draw to useful also is It ’2000 n hoyo aeiniaeaayi,ascaiememory, associative linea analysis, ac of image and ve Bayesian models of new of description and universal theory the use structures and touch to the to possible is described impossible it are generality is paper it the but In relation, argum a serious such are unconnect of communications absolutely and two really computations they quantum Are science. computer Abstract. ncmue cec?Sc applications Such science? computer in qubits h aeo onvnNuani omnbt nqatmmechani quantum in both common is Neumann von John of name The unu nomto science information quantum ,bti h ae scon- is paper the in but ), unu computers quantum nasrc mathemati- abstract an iaSre ,S.Ptrbr,Russia St.-Petersburg, 8, Street Mira n eea ailgclCne,IRH Center, Radiological Federal and N Architecture? .Y.Vlasov Yu. 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[0.0 interval is P ler n utdet their to due just and algebra r 2D 2D n osgetaotexistence about suggest to ent n 1. ,1. N]; .. 1 N, .. [1 = · N ieteei hr review. short a in theme tive x 2 = set N F i.e. = = [ i,j 2 x X n nti example this In . N ] array dot of set =1 = bt register. -bits 2 where 1 = ↑ F array (2 ij (1) 1 = ↑ [ dot (2 n [ dot ; )bt o h set. the for bits )) ] fBoolean of P F ] x freal of ( standardized sand cs x nexample in r used are ) x svisual is ; d It . ; 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- ′ Hermitian Now let us introduce notion of quantum set tion: qK q¯K . It is norm [7]. Such case 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 quantum information science [6] or quantum me- The discussed property of quantum set as “square root of fuzzy set” makes clear, why it can chanical system with N 2 states [7]): 4) Q : qu set = array [dot] of complex; be useful in such abstract area, as image recog- 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 2 Let us discuss now some question related with normalization: Q[x] =1 i.e.: Px | | “hardware”. We had few data structures: N 1) x : dot, 2) I[x] : set 2D, 3) p[x] : fuzzy set 2 X Qij =1 (2) It is sequence with more and more complicated | | i,j=1 structure with occupation of more and more com- for example under consideration. puter memory. But let us suggest, that register In quantum mechanics the complex numbers x is permanently changing its value by such a 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 | | lim t[v]/T = p[v]. is standardized fuzzy set related with given nor- T →∞ 2 malized quantum set via formula: p[x] = q[x] . The algorithm can be implemented by soft- | | It is possible for simplification to consider 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-digital converter is scanning some for some likelihood function H(p1,p2) with prop- “physical model” of fuzzy set p). erties: H(p ,p ) < 1 for p = p , H(p,p)=1 Then each access to the register produces 1 2 1 6 2 [1, 3]. For standardized sets such function can some value of x with probability p[x], so one 2n- 1/2 bits stochastic register is enough to implement be chosen as H(p1,p2) = Px (p1[x]p2[x]) . If we 2 2 statistical model of standardized fuzzy set dis- use quantum sets q1, q2 (p1 = q1 , p2 = q2), the cussed earlier. formula is H(q1, q2)= Px q1[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 with stochastic register as model of fuzzy 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 .. N 2], then − ∈ of access to such register due to laws of quantum formula can be written as: mechanics has some differences with classical sta-

N N 2 tistical register discussed below. H(q, q′)= q q′ q q′ (3) Let us consider some value q qu set of X ij ij X K K ∈ 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 N 2 elements and unit 2 p[x] = q[x] 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 index x and with all other is 0, and so the regis- used∨ a7→ second one, more algebraic a b a b, ∧ 7→ · ter should be reset (preparation in terminology of a b a + b ab = (( a) ( b)). quantum mechanics) in q after each access (quan- ∨But7→ 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 to perform another operation, we It is matrix for linear map: qu set qu set: prepare some given q′ qu set and read the reg- → ′ ∈ N N 2 ister q with using q as some “quantum bit-mask”, ′ ′ ′ ′ ′ 2 qij = X Lij,klqkl or qI = X LIKqK (4) then with probability H(q, q ) (see Eq. 3) op- 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 lihood function,| the| quantum register can be used further the indexes like I,K : [1 .. M], M = N 2). 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 = X AKIBIJ (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 P 2 = 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 6 and powerful. 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] and formal neural network P 1 P, P R PR, P R P +R P R (6) [10] without any relation with quantum mechan- ¬ ≡ − ∧ ≡ ∨ ≡ − ical models. Only noticeable difference was us- Elements of the algebra have form P(S) there 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). c + + 2 components can be written: (a )[x] = not a[x], Then Pq = q q and q q = H(q, q) = 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 If we consider family of nonintersected sets q1, [5] A. Yu. Vlasov, “Analogue Quantum Computers for + Data Analysis,” quant-ph/9802028 q2,...,qk then Pi = qi qi are orthogonal projec- tors and so quantum sets· in such representations [6] C. H. Bennett, “Quantum Information and Compu- 48 have rather relations with usual logic than with tation,” Physics Today (1995) 24 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 Pi it is possible to use existence of some orthogo- [8] R. R. Zapatrin, “Logic Programming as Quantum 34 nal (unitary for complex case) matrix U same for Measurement”, Int. J. Theor. Phys. (1995) 1813 ′ −1 [9] T. Kohonen, Associative Memory: A System- Theo- all Pi such, that all Pi = UPiU are diagonal ′ retical Approach, Springer 1978 [Mir, M. 1980] and have very simple form: P1 = diag(1, 0,... , 0), ′ [10] E. N. Sokolov, G. G. Vatkyavichus, The neuro- P2 = diag(0, 1,..., 0), etc.. The classical Boolean structure of the opera- intelligence: from neuron – towards neurocomputer, Nauka, M. 1988 [Russ.] tors Pi and their sums P(S) (see Eq. 7) is because of all the operators commute [11]. If to choose [11] A. A. Grib, Violation of Bell’s Inequalities and Prob- lem of Measurements in Quantum Theory, JINR P2– projectors P , R: P R = RP the Eq. 6 do not 92–211, Dubna 1992 [Russ.] produce Boolean algebra,6 but it is other kind of [12] A. Yu. Vlasov, “Representation and Processing of In- non-Boolean logic, than fuzzy one. formation in Quantum Computers,” Tech. Phys. Lett. It should be mentioned, that more direct rela- 20 (1994) 992 [Pis’ma Zh. Tekh. Fiz. 20 (1994) 45] tion with fuzzy set have so-called mixed quantum [13] Landau and Lifshitz, Course of Theoretical Physics, states R = Pi wiPi, Pi wi = 1 where wi have 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,” D [I,J] = 0, I = J and already shown Eq. 6. Proc. R. Soc. London A 425 (1989), 73 q 6 Bibliographical notes: In addition to references [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- tronic computers [19] are included for complete- puting instrument, Princeton 1946 ness.

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