Representing fuzzy structures in quantum computation with mixed states

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1 Representing fuzzy structures in quantum computation with mixed states Hector Freytes Fellow of CONICET-Argentina Antonio Arico Department of Mathematics Viale Merello 9, 913 Caglari Roberto Giuntini Giuseppe Sergioli Abstract In this work we introduce a particular kind of quantum operations called polynomial quantum operations that allow us to represent the basic operations of the standard Product MV -algebra Consequently, these operations can be treated as quantum computational gates in the powerful model of quantum computation given by quantum operations - density operators Keywords: Quantum computation, quantum operations, fuzzy logic I INTRODUCTION In the usual representation of quantum computational processes, a quantum circuit is identified with an appropriate composition of quantum gates, ie unitary operators acting on pure states of a convenient (n-fold tensor product Hilbert space Consequently, quantum gates represent time reversible evolutions of pure states of the system Nonetheless, this constrain is unduly restrictive Apparently, it does not encompass states whose information is non-maximal, such as states whose preparation is unknown Moreover, there are relevant physical processes that cannot be represented by unitary evolutions, such as measurements in the middle of a process, decoherence and so on Several authors [1], [3], [5], [8], [1] considered a more general model of quantum computational processes called quantum computation with mixed states, where pure states and unitary operators are replaced by density operators and quantum operations, respectively In this case, time evolution is, in general, no longer reversible Further, such a model will allow us to represent, in a probabilistic way, the operations of the standard P MV -algebra [6], [1] as quantum operations This paper is organized as follows: in Section II we provide some basics; in Section III we introduce the polynomial quantum operations and we prove a representation theorem which allows us associate a polynomial quantum operation to every element of a particular class of polynomials; finally, in Section IV, we show how to represent the Product t-norm, the Łukasiewicz negation and the Łukasiewicz sum as quantum operations April 16, 1 II BASIC NOTIONS Let H be an arbitrary complex Hilbert space We denote by L(H the vector space of all linear operators on H and by D(H the set of all density operators Standard quantum computing is based on quantum systems with finite dimensional Hilbert spaces, specially C, the two-dimensional state space of a quantum bit A quantum bit or qbit, the fundamental concept of quantum computation, is a pure state in the Hilbert space C The standard orthonormal basis {, 1 } of C where (1, and 1 (, 1 is called the logical basis Thus, pure states ψ in C are coherent superpositions of the the basis vectors: ψ c + c 1 1, with complex coefficients such that c + c 1 1 Generalizing for a positive integer n, n-qbits are pure states represented by unit vectors in n C A special basis, called the n -computational basis, is chosen for C n More precisely, it consists of the n orthogonal states ι, ι n where ι is in binary representation and ι can be seen as the tensor product of states ι ι 1 ι ι n where ι j {, 1} A pure state ψ n C is a superposition of the basis vectors ψ n ι1 c ι ι with n ι1 c ι 1 In general, a quantum system is not in a pure state For, a physical system is always interacting with the environment, and therefore phenomena such as decoherence and noise come into play These physical systems whose information is nonmaximal are named mixed states, and they are described by density operators A density operator is represented on the n -dimensional complex Hilbert space by an Hermitian (ie ρ ρ positive operator with unit trace, tr(ρ 1 Due to the fact that Pauli matrices:

2 ( 1 σ I; σ x 1 σ z ( i ; σ y i ( 1 1 ; ; where I I ( is the identity matrix, form a basis for the set of operators on C, an arbitrary density operator ρ for n-qbits may be represented in terms of tensor products of them in the following way: ρ 1 n P µ1µ n (σ µ1 σ µn µ 1µ n where µ i {, x, y, z} for each i 1 n As usual, we have chosen units such that 1 The real expansion coefficients P µ1µ n are given by P µ1µ n tr(σ µ1 σ µn ρ Since the eigenvalues of the Pauli matrices are ±1, the expansion coefficients satisfy P µ1µ n 1 Taking into account the Born roule, for ρ D( n C we define the probabilty value of ρ as p(ρ tr(p (n 1 ρ, ie the expectation value of ρ in the state P (n 1, where P (n 1 ( n 1 I 1 1 Let ρ D(C such that ρ 1 (I +r xσ x + r y σ y + r z σ z, ie ρ 1 ( 1 + rz r x ir y r x + ir y 1 r z ( 1 α β β α Interestingly enough, any real number λ ( λ 1 uniquely determines a density operator ρ λ of the following form: ρ λ (1 λp +λp 1 1 (I+(1 λσ z ( 1 λ λ It is easy to see that, if ρ D(C, then p(ρ 1 rz and p(ρ λ λ Thus each density operator ρ in D(C can be written as ( 1 p(ρ a ρ a p(ρ In the usual representation of quantum computational processes, a quantum circuit is identified with an appropriate composition of quantum gates, mathematically represented by unitary operators acting on pure states of a convenient (n-fold tensor product Hilbert space n C [11] In other words, standard quantum computation is mathematically founded on qbits-unitary operators A quantum operation [9] is a linear operator E : L(H 1 L(H representable as E(ρ i A iρa i where A i are operators satisfying i A i A i I (Kraus representation Theorem [9] If A i are unitary operators, the corespondent quatum operation is named unitary quantum operation It can be seen that a quantum operation maps density operators into density operators The new model density operatorsquantum operations also called quantum computation with mixed states ([1], [1] is equivalent in computational power to the standard one but gives a place to irreversible processes as measurements in the middle of the computation III POLYNOMIAL QUANTUM OPERATIONS In this Section, we represent in a probabilistic way some classes of polynomials as quantum operations First of all, we introduce some notations and preliminary definitions The term multi-index denotes an ordered n-tuple α (α 1, α n of non negative integers α i The order of α is given by α α α n If x (x 1,, x n is an n-tuple of variables and α (α 1, α n a multi-index, the monomial x α is defined by x α x α1 1 xα xαn n In this language a real polynomial of order k is a function p(x α k a αx α such that a α R Let x (x 1,, x n and k be a natural number Then, we consider the set D k (x defined as D k (x {(1 x 1 α1 x β1 1 (1 x n αn x βn n : α i +β i k, 1 i n} Lemma 31: Let X 1,, X n be a family of matrices such that ( 1 xi b X i i b i and let us consider a tensor product X ( k X 1 ( k X ( k X n Then we have that Diag(X D k (x 1,, x n Proof: By induction on k we can prove that Diag( k X i {h 1 h h k : h j {(1 x i, x i }, 1 j k} {(1 x 1 α x β 1 : α + β k} Thus, Diag(( k X 1 ( k X ( k X n {(1 x 1 α1 x β1 1 (1 x n αn x βn n : α i + β i k, i {1,, n}} Whence our claim follows Lemma 3: Let x (x 1,, x n and k be a natural number Given any monomial x α such that α k, we have that: 1 x α y D k (x δ yy; 1 x α y D k (x γ yy; where δ y and γ y lies in {, 1} Proof: For each i {1,, n}, consider the matrix X i given by ( 1 xi X i x i Let us prove 1 Let x α x α1 1 xα xαn n such that α k Thus, there exist s 1,, s n such that α i + s i k Let W ( s1 X 1 ( s X ( sn X n and let consider the matrix Wx α In view of Lemma 31, Diag(Wx α D k (x 1,, x n since every element in x i

3 Diag(Wx α is a monomial of order nk Further, since tr(wx α (trwx α 1x α x α, we have that x α tr(wx α is the required polynomial expansion Now we prove Let X ( k X 1 ( k X ( k X n By Lemma 31 Diag(X D k (x 1,, x n and tr(x 1 Taking into account that x α y D k (x δ yy, we define γ y 1 if δ y and γ y if δ y 1 Therefore, 1 tr(x y D k (x δ yy + y D k (x γ yy x α + y D k (x γ yy and 1 x α y D k (x γ yy Definition 31: A quantum operation P : L( nk C L( nk C is called polynomial quantum operation iff there exists a polynomial P (x 1,, x n such that for each n-tuple (σ 1,, σ n in D(C we have that: p(p(( k σ 1 ( k σ n P (p(σ 1,, p(σ n Theorem 31: Let x (x 1,, x n be an n-tuple of variables and consider the set D k (x Let P (x y D k (x a yy be a polynomial such that y D k (x, a y 1 and the restriction P (x [,1] n satisfies that P (x [,1] n 1 Then there exists a polynomial quantum operation P : L( nk C L( nk C such that for each n-tuple σ (σ 1,, σ n in D(C p(p(( k σ 1 ( k σ n P (p(σ 1,, p(σ n Moreover, P(( k σ 1 ( k σ n ( 1 nk 1 nk 1 I ρ P (p(σ1,,p(σ n Proof: Let σ 1,, σ n density operators of C Assume that for any σ i ( 1 xi b σ i i b i x i Hence, p(σ i x i It is clear that σ ( k σ 1 ( k σ n is a matrix of order nk nk and, by Lemma 31, Diag(σ D k (x 1,, x n Thus, each y D k (x can be seen as the (i, i-th entry of Diag(σ Further, the polynomial P (x y D k (x a yy nk j1 a jy j is such that every y j is the (j, j-th entry of Diag(σ Let, now, y j Diag(σ a We want to place a j y j in the (s, s-th entries of a nk nk matrix Let us consider the nk nk matrix A s such that D s aj D s nk 1 has 1 just in the (s, -th entry and in any other entry It is not difficult to check that A s σ(a s required matrix Moreover, one can verify that: s A s σ(a s 1 nk 1 is the a j y j a j y j b Taking into account that 1 y D k (x y nk j1 y j, we have that: 1 a j y j y j a j y j (1 a j y j nk j1 nk j1 nk j1 nk j1 We want to place (1 a j y j in the (s 1, s 1-th entries of a nk nk matrix Let us consider the nk nk matrix A s 1 1 aj D s 1 nk 1 such that D s 1 have 1 just in the (s 1, -th entry and in any other entry It is not difficult to check that A s 1 σ(a s 1 is the required matrix Moreover, one can verify that: A s 1 σ(a s 1 1 nk 1 s 1 (1 a j y j (1 a j y j Thus, we have that P s As σ(a s + σ(a s 1 s 1 As 1 ( 1 1 nk ( kn 1 nk 1 j1 I a jy nk j1 a jy j Let us consider A s (As A s + s+1 (As+1 A s+1 Our task is now to verify that A I c First of all, notice that the matrix (A s A s has the value a j just in the (j nk 1, -th entry and in any other entry Therefore, the matrix s (As A s has the value nk 1 a j nk 1 a j in the (, -th entry and all the other entries are equal to Hence: a 1 a (A s A s a 3 s a 4 d On the other hand the matrix (A s 1 A s 1 has the value 1 a just in the (j nk 1, -th entry and in any other entry Therefore, the matrix s 1 (As 1 A s 1 has the value nk 1 (1 a j 1 a nk 1 j in the (, -th entry and all the other entries are equal to Hence: (A s 1 s 1 A s 1

4 1 a 1 1 a 1 a 3 1 a 4 Thus s 1 (As 1 operation s (As A s 1 + A s 1 I and P is a quantum IV REPRESENTING THE STANDARD P M V -OPERATIONS The standard P M V -algebra [6], [1] is the algebra [, 1] P MV [, 1],,,,, 1, where [, 1] is the real unit segment, x y min(1, x + y, is the usual product, and x 1 x This structure plays a notable role in quantum computing, in that it decribes, in a probabilistic way, a relevant system of quantum gates named Poincarè irreversible quantum computational algebra [], [4] Of course, can be given as a polynomial in the generator system D 1 (x, whence by Theorem 31, it is representable as a polynomial quantum operation A possible representation can be the following: NOT (ρ σ x ρσ x It is worth noting that p(not (ρ 1 p(ρ Furthermore, can be represented by a polynomial in the generator system D (x, y According with the construction presented in Theorem 31, the following representation obtains Let us consider the following matrices: Fig 1 The Łukasiewicz conorm G 1 1 G 1 Fig P (x, y G 3 G 5 G G 4 G 6 G It is straightforward to check that 8 i1 G i(τ σg i 1 I ρ p(τp(σ where σ, τ D(C Thus, by Kraus representation Theorem [9], 8 i1 G i(τ σg i is a quantum operation, and p( 8 i1 G i(τ σg i p(τ p(σ This quantum operation represents the well known quantum gate IAND modulo a tensor power [], [11] Actually, the Łukasiewicz conorm is not a polynomial, Figure 1 Therefore, our idea is to give a polynomial P (x, y in some generator system D k (x, y, such that P (x, y can approximate the Łukasiewicz sum By using numerical methods, we obtain the following polynomial approximant of in [, 1]: P (x, y 5 1 x(1 x+ 5 1 y(1 x+ 5 1 x(1 y+ 5 1 y(1 y+1 x+1 y, whose graph is depicted in Figure

5 [9] Kraus K, States, effects and operations, Springer-Verlag, Berlin, 1983 [1] D Mundici and B Riecǎn, Probability on MV-algebras In: Handbook of Measure Theory (E Pap Ed, , North Holland, Amsterdam, ( [11] Nielsen MA, Chuang IL: Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, ( [1] Tarasov V Quantum computer with Mixed States and Four-Valued Logic, Journal of Physics A, 35, pp Fig 3 (x y P (x, y Let us remark that P (x, y x y Therefore, e max [,1] {(x y P (x, y} 8 (see also Figure 3 It can be seen that P (x, y is a polynomial given by the generator system D (x, y, and it satisfies also the hypothesis of Theorem 31 Thus, P (x, y is representable as a polynomial quantum operation P, where p(p (τ σ (p(τ p(σ± 8 V CONCLUSION We can conclude that the accuracy of the obtained approximation is extremely high This result provides a strong quantum computational motivation for the investigation of algebraic structures equipped with the Łukasiewicz sum and build a bridge between classical fuzzy logic and quantum computational logics ACKNOWLEDGEMENTS We warmly thank Antonio Ledda for his precious suggestions during the preparation of this paper REFERENCES [1] Aharanov D, Kitaev A, Nisan N Quantum circuits with mixed states Proc 13th Annual ACM Symp on Theory of Computation, STOC, 1997, pp -3 [] Cattaneo G, Dalla Chiara M L, Giuntini R, Leporini R, Quantum computational structures, Mathematica Slovaca 54, (4, pp [3] Dalla Chiara M L, Giuntini R, Greechie R, Reasoning in Quantum Theory, Sharp and Unsharp Quantum Logics, Kluwer, Dordrecht- Boston-London, 4 [4] G Domenech and H Freytes, Fuzzy propositional logic associated with quantum computational gates Int J Theor Phys 34, 8-61 (6 [5] Duan R, Ji Z, Feng Y, Ying M: Quantum operation, quantum Fourier transform and semi-definite programming, Phys Lett A 33 (4, [6] F Esteva, L Godo and F Montagna, The LΠ and LΠ 1 : two complete fuzzy systems joining Łukasiewicz and Product Logic Arch Math Logic 4, (1 [7] Giuntini R, Ledda A, Paoli F, Expanding quasi-mv algebras by a quantum operator, Studia Logica, 87, 1, 7, pp [8] Gudder S, Quantum computational logic, International Journal of Theoretical Physics, 4, 3, pp 39-47