New Quantum Algorithm Solving the NP Complete Problem

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1 ISSN , p-adic Numbers, Ultrametric Analysis and Applications, 01, Vol. 4, No., pp c Pleiades Publishing, Ltd., 01. SHORT COMMUNICATIONS New Quantum Algorithm Solving the NP Complete Problem M. Ohya ** Department of Information Sciences Tokyo University of Science, Japan Received February 1, 01 Abstract We review the quantum chaos algorithm solving the NP-complete problems in polynomial time. This work has been done in the series of papers with Professor Igor Volovich for nearly ten years. DOI: /S Key words: quantum algorithm, NP-complete problem. Dedicated to 65th birthday of Professor Igor Volovich 1. INTRODUCTION I met Professor Igor Volovich nearly 0 years ago in Roma, Italy. Since then we discussed and worked together on quantum information and mathematical physics. He has open eye and flexible mind to catch the essence of existence, so he is one of the most important mathematical physicists in our time. I could always enjoy working together with him. Our most important joint work is to find the algorithm solving the NP-complete (NPC) problem, as reviewed in this paper. Any problem that can be solved in polynomial time by a nondeterministic Turing machine is polynomially transformed to an NPC problem [1]. These NPC problems are not known whether there exists an algorithm to solve this problem in a polynomial time for more than thirty years. Among the NP-complete problems, there are famous problems such as the knapsack problem, the travelling salesman problem, the integer programming problem, the subgraph isomorphism problem, the satisfiability (SAT) problem that have been studied for several decades, for which all known algorithms do not have polynomial running time in the length of the input. These NP-complete problems are known to be equivalent, and it has been considered that such problems are very difficult and probably need exponential time to solve them. In the series of papers [6 9, 11, 13, 16] we found the algorithms to solve the NPC problems in polynomial time. In this paper we review the essence of these works done with I. Volovich. It is widely believed that quantum computers are more efficient than classical computers. In particular, Shor [] gave a quantum algorithm to solve the factoring problem in polynomial time which is not NP-complete but NP-intermediate. Ohya and Volovich [6, 7] proposed a new method of quantum computation including a chaotic dynamics (amplifier). Our quantum algorithm succeeded to solve the SAT problem in a polynomial time. It was first shown in [8] that the SAT problem can be solved in polynomial time by using a quantum computing under the assumption that a special superposition of two orthogonal vectors can be physically detected. However, the problem one has to overcome there is that the desired output of computations could be a very small and one needs to amplify it to a reasonable large quantity so as to detect it. For this purpose, we proposed that the amplification in chaotic dynamics plays a constructive role in computations [6, 7]. Chaos and quantum decoherence are considered normally as the degrading effects which lead to an unwelcome increase of the error rate with the input size. However, in our case, it is not so. Our algorithm is constructive [10, 13] and uses the chaotic dynamics only as an amplifier of a certain amplitude. In Section, the SAT problem is explained. The quntum part of our algorithm is briefly reviewedin Section3,andthepartofamplification process by means of a chaos dynamics is discussed in Section 4. In the last section, we discuss further results on our algorithm [15, 16] and some application of ours to solve other problems such as factoring and searching problems [17 0]. The text was submitted by the author in English. ** mohy3331@yahoo.co.jp 161

2 16 OHYA. SAT PROBLEM For a given set X {x 1,,x n },anelementx k and its negation x k (k =1,,,n) are called literals and the set of all such literals is denoted by X = {x 1, x 1,,x n, x n }.AnelementC F(X ), the set of all subsets of X, is called a clause. We take a truth assignment to all variables x k. If one can assign the truth value to at least one element of C, then C is called satisfiable. When C is satisfiable, the truth value t(c) of C is regarded as true, otherwise, that of C is false. Take the truth values as true 1, false 0. Then C is satisfiable iff t(c) =1. Let L = {0, 1} be a Boolean lattice with usual join and meet ; ε ε := max{ε, ε },ε ε := min{ε, ε }, and t(x) be the truth value of a literal x in X. Namely, t is a Boolean homomorphism from X to {0, 1} with the property: Then the truth value of a clause C is written as t(x) t(x) =1, x X. t(c) x C t(x). Further the set C of all clauses C j (j =1,, m) is called satisfiable iff the meet of all truth values of C j is 1; t(c) m j=1t(c j )=1. Thus the SAT problem is written as follows: Definition 6. SAT Problem: Given a set X {x 1,,x n } and a set C = {C 1,C,,C m } of clauses, determine whether C is satisfiable or not. That is, this problem is to ask whether there exists a truth assignment t to make C satisfiable. 3. QUANTUM ALGORITHM Let 0 and 1 of the Boolean lattice L be denoted by the vectors 0 1 and 1 0 in the 0 1 Hilbert space C, respectively. That is, the vector 0 corresponds to falseness and 1 does to truth. An element x X can be denoted by 0 or 1, so by 0 or 1. In order to describe a clause C with at most n length by a quantum state, we need the n-tuple tensor product Hilbert space H n 1 C. For instance, in the case of n =, given C = {x 1,x } with an assignment x 1 =0and x =1, then the corresponding quantum state vector is 0 1, so that the quantum state vector describing C is generally written by C = x 1 x Hwith x k =0or 1 (k=1,). The quantum computationis performed by a unitary gate constructed from several fundamental gates such as Not gate, Controlled-Not gate, Controlled-Controlled Not gate [11]. Once X {x 1,,x n } and C = {C 1,C,,C m } are given, the SAT is to find the vector t (C) m j=1 x Cj t(x), where t(x) is 0 or 1 when x =0or 1, respectively, and t(x) t(y) t(x y), t(x) t(y) t(x y). We consider the quantum algorithm for the SAT problem. Since we have n variables x k (k =1,,n) and a quantum computation produces some dust bits, the assignments of the n variables and the l dusts are represented by n qubits and l qubits, respectively, in the Hilbert space n 1 C l 1 C. Moreover the resulting state vector t (C) should be added in the last qubit, so that the total Hilbert space is H n 1 C l 1 C C.

3 NEW QUANTUM ALGORITHM 163 Let us start the quantum computation of SAT problem from an initial vector v 0 n 1 0 l when C is composed of n Boolean variables x 1,,x n. We apply the discrete Fourier transformation denoted by U t n to the part of the Boolean variables of the vector v 0, then the 1 1 resulting state vector becomes v ( ) U t l 1 I v 0 = 1 n 1 ( ) l 1 0 0, where I is the identity matrix in C. This vector can be written as v = 1 n j=1 x j l Now, we perform the quantum computation to check the satisfiability,which willbe done by a unitary operator U t properly constructed by unitary gates. Then after the computation by U t,thevector v goes to v t U t v = 1 = 1 U t n j=1 x j l n j=1 x j l i=1 y i t (x 1,,x n ), where t (x 1,,x n ) t (C) because C contains x 1, x n, and y i are the dust bits produced by the computation. Note that the unitary operator U t is concretely constructed [10, 11, 13]. The final step to check the satisfiability of C is to apply the projection E n+l 1 I 1 1 to the state v t, mathematically equivalent, to compute the value v t E v t. If the vector E v t exists or the value v t E v t is not 0, then we conclude that C is satisfiable. The value of v t E v t is r/ with an integer r less than, which is the probability to find the result t(x) =1, and it is often very small and is difficult to be detected even when it is not zero, so that we desire to have some way to amplify this value, if it is not zero. Let us simplify our notations. After the step (ii) the quantum computer will be in the state v f = 1 q ϕ q ϕ 1 1, where ϕ 1 and ϕ 0 are normalized n + l qubit states and q = r/. Effectively our problem is reduced to the following 1 qubit problem. We have the state ψ = 1 q 0 + q 1 and we want to distinguish between the cases q =0and q>0 (small positive number). In order to amplify the no-zero q, we proposed to combine quantum computer with a chaotic dynamics amplifier. Such a quantum chaos computer is a new model of computations and we demonstrated that the amplification is possible in the polynomial time [6, 7, 11]. 4. AMPLIFICATION BY CHAOTIC DYNAMICS Various aspects of classical and quantum chaos have been the subject of numerous studies, see [11] and references therein. Chaotic behavior in a classical system usually is considered as an exponential sensitivity to initial conditions. It is this sensitivity that we would like to use to distinguish between the cases q =0and q>0 from the previous section.

4 164 OHYA Consider the so called logistic map which is given by the equation x n+1 = ax n (1 x n ) f(x), x n [0, 1]. The properties of the map depend on the parameter a. If we take, for example, a =3.71, then the Lyapunov exponent is positive, the trajectory is very sensitive to the initial value and one has the chaotic behavior [7]. It is important to notice that if the initial value x 0 =0, then x n =0for all n in the logistic map. It is known [3] that any classical algorithm can be implemented on quantum computer. Our quantum chaos computer will be consisting from two blocks. One block is the ordinary quantum computer performing computations with the output ψ = 1 q 0 + q 1. The second block is a computer performing computations of the classical logistic map. This two blocks should be connected in such a way that the state ψ first be transformed into the density matrix of the form ρ = q P 1 + ( 1 q ) P 0, where P 1 and P 0 are projectors to the state vectors 1 and 0. This connection is in fact nontrivial and actually it should be considered as the third block. One has to notice that P 1 and P 0 generate an Abelian algebra which can be considered as a classical system. In the second block the density matrix ρ above is interpreted as the initial data, and we apply the logistic map as ρ m = (I + f m (ρ)σ 3 ), where I is the identity matrix and σ 3 is the z-component of the Pauli matrix on C. To find a proper value m we finally measure the value of σ 3 in the state ρ m such that M m trρ m σ 3. After simple computation we obtain ρ m = (I + f m (q )σ 3 ), and M m = f m (q ). Thus the question is whether we can find such a m in polynomial steps of n satisfying the inequality M m 1 for very small but non-zero q. Here we have to remark that if one has q =0then ρ 0 = P 0 and we obtain M m =0for all m. If q 0, the stochastic dynamics leads to the amplification of the small magnitude q in such a way that it can be detected as is explained below. The transition from ρ 0 to ρ m is nonlinear and can be considered as a classical evolution because our algebra generated by P 0 and P 1 is abelian. The amplification can be done within at most n steps due to the following theorems. Since f m (q ) is x m of the logistic map x m+1 = f(x m ) with x 0 = q, we use the notation x m in the logistic map for simplicity. Theorem 1. For the logistic map x n+1 = ax n (1 x n ) with a [0, 4] and x 0 [0, 1], let x 0 be 1 and a set J be {0, 1,,,n, n}. If a is 3.71, then there exists an integer m in J satisfying x m > 1. Theorem. Let a and n be the same as in the above theorem. If there exists m 0 in J such that x m0 > 1, then m 0 > n 1 log According to these theorems, it isenoughtocheckthevaluex m (i.e., M m ) around the above m 0 when q is 1 for a large n. More generally, when q= k n with some integer k, it is similarly checked that n the value x m (M m ) becomes over 1 within at most n steps. So we can amplify the small q in polynominal steps. The proofs and other details are given in the book [11]. 5. CONCLUSION The complexity of the quantum algorithm for the SAT problem was considered by constructing unitary gates to compute the SAT problem. We also needed to consider the number of steps in the classical amplification process by using the logistic map performed on the resulting state obtained by quantum computation, which is the probabilistic part of the algorithm to distingish the cases q =0and q>0. Thus we found that all processes, i.e., quantum computation and amplification, has been done within polynomial steps to solve the SAT problem.

5 NEW QUANTUM ALGORITHM FURTHER STUDIES The above discussion could be descrbed by Language of the generalized qantum Turing machine, in which non-unitary dynamics can be involoved [15,16]. Recently, we found some incomplete points in the proof of Shor s factoring problem [18]. In that paper, we discussed the computational complexity of Shor s quantum algorithm, and we pointed out that there is a problem with the unitary implementation of the modular exponentiation and show that the complexity of the original Shor s algorithm is in fact not polynomial but exponential. To estimate the complexity of the modular exponentiation Shor counts only the number of squarings. However each squaring operation is rather complicated, and we have to compute the number of quantum gates to perform each squaring. Then we get the exponential complexity for the modular exponentiation. Then the complexity becomes exponential in the worst case. Moreover we found similar incompleteness even for the discussions using quantum circuits done by several authors, e.g., [4], after Shor. We are now working to prove the Shor s factoring problem by means of our quantum chaos algorithm [0]. REFERENCES 1. M. Garey and D. Johnson, Computers and Intractability a guide to the theory of NP-completeness (Freeman, 1979).. P. W. Shor, Algorithm for quantum computation : Discrete logarithm and factoring algorithm, in Proc. 35th Annual IEEE Symposium on Foundation of Computer Science, (1994). 3. D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. Royal Soc. London A 400, (1985). 4. A. Ekert and R. Jozsa, Quantum computation and Shor s factoring algorithm, Rev. Mod. Phys. 68 (3), (1996). 5. E. Bernstein and U. Vazirani, Quantum complexity theory, Proc. 5th ACM Symp. Theory Comput., 11 0 (1993). 6. M.OhyaandI.V.Volovich, Quantum computing and chaotic amplification, J. Opt. B 5 (6), (003). 7. M.Ohya and I.V.Volovich, New quantum algorithm for studying NP-complete problems, Rep. Math. Phys. 5 (1), 5 33 (003). 8. M. Ohya and N. Masuda, NP problem in quantum algorithm, Open Syst. Inform. Dyn. 7 (1), (000). 9. L. Accardi and M. Ohya, A stochastic limit approach to the SAT problem, Open Syst. Inform. Dyn. 11, 1 16 (004). 10. L. Accardi and R. Sabbadini, On the Ohya-Masuda quantum SAT algorithm, Volterra Prepr. 43 (000). 11. M.OhyaandI.V.Volovich,Mathematical Foundation of Quantum Information and Computation and its Applications to Nano- and Bio-Systems (Springer-Verlag, TMP Series, 011). 1. C. H. Bennett, E. Bernstein, G. Brassard and U. Vazirani, Strengths and weaknesses of quantum computing, SICOMP 6 (5), (1997). 13. S. Iriyama and M. Ohya, Rigorous estimation of the computational complexity for OV SAT algorithm, Open Syst. Inform. Dyn. 15 (), (008). 14. S. Iriyama, T. Miyadera and M. Ohya, Note on a universal quantum Turing machine, Phys. Lett. A 37, (008). 15. S. Iriyama and M. Ohya, Language classes defined by generalized quantum Turing machine, Open Syst. Inform. Dyn. 15 (4), (008). 16. S. Iriyama, M. Ohya and I. V. Volovich, Generalized quantum Turing machine and its application to the SAT chaos algorithm, QP-PQ: Quantum Prob. White Noise Anal., Quantum Inform. Comput. 19, 04 5 (World Sci. Publ., 006). 17. S. Iriyama, M. Ohya and I. V. Volovich, On computational complexity of Shor s quantum factoring algorithm, TUS Preprint (01). 18. L. Accardi, S. Iriyama, M. Ohya and I. V. Volovich, Note on Shor s quantum factoring algorithm, TUS Preprint (01). 19. S. Iriyama, M. Ohya and I. V. Volovich, On quantum algorithm for binary search and its computational complexity, TUS Preprint (01). 0. Goto, S. Iriyama, M. Ohya and I. V. Volovich, New proof of the factoring algorithm, in preparation. 1. S. Iriyama and M. Ohya, The problem to construct unitary quantum Turing machine computing partial recursive functions, TUS Preprint (009).