Handouts P=NP

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P=NPSolving SAT with a Chaotic Quantum Computer

Wilson Rosa de Oliveira Jr.(Based on Ohya&Volovichs Work)

DEInfo-UFRPE

November 15, 2012

Wilson Rosa (DEInfo-UFRPE) P=NP November 15, 2012 1 / 33


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Outline:

1 Introduction

2 SAT Problem

3 Chaotic Dynamics

4 Main Results

5 ProofsProof Theorem 1Proof Theorem 2Comments

6 Conclusion

7 Perspective for further research



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Introduction

NP-complete problems: the napsack problem, the traveling salesmanproblem, the integer programming problem, the subgraph

isomorphism problem, the satisfiability problem;have been studied for decades and for which all known algorithmshave a running time that is exponential in the length of the input.

Any problem that can be solved in polynomial time on a

nondeterministic Turing machine is polynomially transformed to anNP-complete problem [1].

it seems that such problems are very difficult and probablyexponential.

If so, solutions are still needed, and in this talk we consider anapproach to these problems based on quantum computing andchaotic dynamics.



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Introduction (cont.)

It is widely believed that quantum computers are more efficient thanclassical computers.

In particular Shor [2] gave a remarkable quantum polynomial-timealgorithm for the factoring problem.

However, it is unknown whether this problem is NP-complete.

The computational power of quantum computers has been explored in

a number of papers. Bernstein and Vasirani [3] proved thatBPPBQP PSPACE.The question whether NPBQP, is open.It was proved in [4] that relative to an oracle chosen uniformly at

random, with probability 1, the class NP can not be solved on aquantum Turing machine in time o

2n/2

.

these results are not immediately applicable to the chaotic quantumcomputer.



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Introduction (cont.)

For a recent discussion of computational complexity in quantumcomputing see [5, 6, 7].

Mathematical features of quantum computing and quantuminformation theory are summarized in [8].

A possibility to exploit nonlinear quantum mechanics so that the classof problems NP may be solved in polynomial time has been

considered by Abrams and Lloyd in [9].It is mentioned in [9] that such nonlinearity is purely hypotetical; allknown experiments confirm the linearity of quantum mechanics.



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Introduction

In this talk we argue that chaotic dynamics plays a constructive rolein computations.

Chaos and quantum decoherence are considered normally as thedegrading effects which lead to an unwelcome increase of the errorrate with the input size.

However, in this talk we argue that under some circumstances chaoscan play a constructive role in computer science.

In particular we show how to combine quantum computer with thechaotic dynamics amplifier.

We will argue that such a chaotic quantum computer can solve theSAT problem in polynomial time.

As a possible specific implementation of chaotic quantumcomputations it has been recently proposed atomic quantumcomputer [11].

It has been argued [11] that in the atomic quantum computer one can

build also nonlinear quantum gatesWilson Rosa (DEInfo-UFRPE) P=NP November 15, 2012 7 / 33


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SAT Problem

Let X = {x1, , xn} be a set.Then xk and its negation xk (k = 1, 2,

, n) are called literals

the set of all such literals is denoted by X

= {x1, x1, , xn, xn}.The set of all subsets of X

is denoted by F(X) and an elementC F(X) is called a clause.We take a truth assignment to all variables xk.

If at least one element of C is asssigned true, then C is calledsatisfiable.

When C is satisfiable, the truth value t(C) of C is regarded as true,otherwise, that of C is false.

ThenC is satisfiable iff t(C) = 1.



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SAT Problem (cont.)

Let L = {0, 1} be a Boolean lattice with usual join and meet ,and t(x) be the truth value of a literal x in X.

Then the truth value of a clause C can be written as

t(C) xCt(x).

A set C of all clauses Cj (j = 1, 2, m) is called satisfiable iff themeet of all truth values of Cj is 1;

t(C) mj=1t(Cj) = 1.



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SAT Problem (cont.)

Definition

SAT Problem: Given a set X {x1, , xn} and a setC = {C1, C2, , Cm} of clauses, determine whether C is satisfiable or not.

That is, this problem is to ask whether there exsits a truth assignment

to make C satisfiable.it is polynomial time to check the satisfiability when a specific truthassignment is given, but it is not known how to determine thesatisfiability in polynomial time when an assignment is not specified.

Note that a formula made by the product (AND

) of the disjunction

(OR ) of literals is said to be in the product of sums (POS) form.For example:

(x1 x2) (x1) (x2 x3)



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SAT Problem (cont.)

SAT problem can be regarded as determining whether or not aformula in POS form is satisfiable.

The following analytical formulation of SAT problem is useful: definea family of Boolean polynomials fA, indexed by A

A = {S1,...,SN, T1,...,TN} ,

where Si, Ti {1,...,n} , and fA is defined as

fA(x1,

, xn) =

N

i=1

1 +

aSi

(1

xa)

bTi

xb

.

We assume here the addition modulo 2.

The SAT problem now is to determine whether or not there exists avalue ofx = (x

1,

, xn

) such that fA(x) = 1.



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Quantum Algorithm

we will work in the (n + 1)-tuple tensor product Hilbert space H

n+11 C

2 with the computational basis

|x1,...,xn, y = ni=1 |xi |y

where x1,...,xn, y = 0 or 1.

We denote

|x1,...,xn, y

=

|x, y

.

The quantum version of the function f(x) := fA(x) is given by theunitary operator Uf |x, y = |x, y + f(x) .We assume that the unitary matrix Uf can be build in the polynomialtime, see [10].



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Quantum Algorithm (cont.)

If r is suitably large to detect it, then the SAT problem is solved inpolynominal time.

However, for small r, the probability is very small and this means wein fact dont get information about the existence of the solution off(x) = 1, so we need further work.

Let us simplify our notations. After the step (ii) the quantum

computer will be in the state

|vf =

1 q2 |0 |0 + q|1 |1

where |1 and |0 are normalized n qubit states and q =

r/2n.

So, our problem is reduced to the following 1 qubit problem. We havethe state

| =

1 q2 |0 + q|1and we want to distinguish between the cases q = 0 and q > 0(smallpositive number).



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Quantum Algorithm (cont.)

The amplification could not be possible in the standard model ofquantum computations with a unitary evolution.

The idea here is to combine quantum computer with a chaoticdynamics amplifier.

Such a quantum chaos computer is a new model of computationsgoing beyond usual scheme of quantum computation and we

demonstrate that the amplification is possible in the polynomial time.One could object that we dont suggest a practical realization of the

new model of computations. But at the moment nobody knows of

how to make a practically useful implementation of the standard

model of quantum computing ever.

Quantum circuit or quantum Turing machine is a mathematical modelthough convincing one. It seems to us that the quantum chaoscomputer deserves an investigation and has a potential to berealizable.



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Chaotic Dynamics

Chaotic behaviour in a classical system usually is considered as anexponential sensitivity to initial conditions.

It is this sensitivity we would like to use to distinquish between thecases q = 0 and q > 0.

Consider the logistic map:

xn+1 = axn(1

xn)

g(x), xn

[0, 1] .

The properties of the map depend on the parameter a.

If we take, for example, a = 3.71, then the Lyapunov exponent ispositive, the trajectory is very sensitive to the initial value and one

has the chaotic behaviour.It is important to notice that if the initial value x0 = 0, then xn = 0for all n.



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Chaotic Dynamics (cont.)

Any classical algorithm can be implemented on quantum computer.

Our quantum chaos computer will be consisting of two blocks. One

block is the ordinary quantum computer performing computationswith the output

| =

1 q2 |0 + q|1.

The second block is a computer performing computations of theclassical logistic map.

This two blocks should be connected in such a way that the state |is transformed into the density matrix of the form

= q2P1 +

1 q2P0where P1 and P0 are projectors to the state vectors |1 and |0 .



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This connection is in fact nontrivial and actually it should beconsidered as the third block.

Notice that P1 and P0 generate an Abelian algebra which can beconsidered as a classical system.

In the second block the density matrix above is interpreted as theinitial data 0, and we apply the logistic map as

m =(I + gm(0)3)

2

where I is the identity matrix and 3 is the z-component of Paulimatrix on C2.

To find a proper value m we finally measure the value of 3 in thestate m such that

Mm trm3.



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The question is whether we can find such a m in polynomial steps ofn satisfying the inequality Mm

12

for very small but non-zero q2.

Here we have to remark that if one has q = 0 then 0 = P0 and weobtain Mm = 0 for all m. If q= 0, the stochastic dynamics leads tothe amplification of the small magnitude q in such a way that it canbe detected as is explained below.

The transition from 0 to m is nonlinear and can be considered as aclassical evolution because our algebra generated by P0 and P1 isabelian.

Since gm(q2) is xm of the logistic map xm+1 = g(xm) with x0 = q2,

we use the notation xm in the logistic map for simplicity.



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Main Results

The amplification can be done within atmost 2n steps due to thefollowing theorems.

Theorem (1)

For the logistic map xn+1 = axn (1 xn) with a [0, 4] and x0 [0, 1] ,let x0 be

12n

and a set J be {0, 1, 2, , n, 2n} . Ifa is 3.71, then thereexists an integer m in J satisfying xm >

1

2

.

Theorem (2)

Let a and n be the same in the above proposition. If there exists m0 in Jsuch that xm0 >

12

, then m0 >n1

log2 3.71.


P f Th 1


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Proof Theorem 1

Proof.

Suppose that there does not exist such m in J. Then xm 1

2 for anym J. The inequality xm 12 implies

xm = 3.71(1 xm1)xm1 3.712

xm1.

Thus we have

1

2 xm 3.71

2xm1

3.71

2

mx0 =

3.71

2

m 12n

,

from which we get

2n+m1 (3.71)m .


P f Th 1


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Proof Theorem 1

Proof. (cont)

According to the above inequality, we obtain

m n 1log2 3.71 1

.

Since log2 3.71 1.8912, we have

m n 1log2 3.71 1

12

.


P f Th 2


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Proof Theorem 2

Proof.

Since 0

xm

1, we have

xm = 3.71(1 xm1)xm1 3.71xm1,which reduces to

xm

(3.71)m x0.

For m0 in J satisfying xm0 >12

, it holds

x0 1(3.71)m0

xm0 >1

2 (3.71)m0 .


Proof Theorem 2


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Proof Theorem 2

Proof. (cont)

It follows that from x0 =1

2n

log2 2 (3.71)m0 > n,which implies

m0 >n 1

log2 3.71.


C


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Comments

According to these propositions, it is enough to check the value xm(Mm) around the above m0 when q is

12n

for a large n.

More generally, when q=k2n with some integer k, it is easily checked

that the above two propositions are held and the value xm (Mm)becomes over 1

2around the m0 above.

One can think about various possible implementations of the idea ofusing chaotic dynamics for computations and/or realization ofnonlinear quantum gates, which is open problem

In Fig.1 we can see how to amplify the small q in several steps.

n

xn

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 40 45 50

Fig.1. Change of xn w.r.t. time n


C


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Conclusion

The complexity of the quantum algorithm for SAT was shown to bepolynomial time.

It is the probabilistic part of the construction that requires one tocompute several times to be able to distingish the cases q = 0 andq > 0.

the quantum chaos algorithm is proposed combines the ordinaryquantum algorithm with quantum chaotic dynamics amplifier.

The physical implementation of this algorithm is another question ...


P i f f h h


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Perspective for further research

What is so special about the logistic map? Vanishes at 0 and amplifies 0 < q


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References I

M. Garey and D. Johnson, Computers and Intractability - a guide tothe theory of NP-completeness, Freeman, 1979.

P.W. Shor, Algorithm for quantum computation : Discrete logarithmand factoring algorithm, Proceedings of the 35th Annual IEEESymposium on Foundation of Computer Science, pp.124-134, 1994.

E. Bernstein and U. Vazirani, Quantum Complexity Theory, in: Proc.

of the 25th Annual ACM Symposium on Theory of Comuting, (ACMPress, New York,1993), pp.11-20.

C. H. Bennett, E. Bernstein, G. Brassard, U. Vazirani,Strengths and Weaknesses of Quantum Computing, quant-ph/9701001

L. Fortnow and J. Rogers, Complexity Limitations on QuantumComputation, cs.CC/9811023.


References II


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References II

R. Cleve, An Introduction to Quantum Complexity Theory,quant-ph/9906111.

E. Hemaspaandra, L.A. Hemaspaandra and M. Zimand,Almost-Everywhere Superiority for Quantum Polynomial Time,quant-ph/9910033.

M. Ohya, Mathematical Foundation of Quantum Computer, Maruzen

Publ. Company, 1998

D. S. Abrams and S. Lloyd, Nonlinear quantum mechanics impliespolynomial-time solution for NP-complete and #P problems,

quant-ph/9801041.

M. Ohya and N. Masuda, NP problem in Quantum Algorithm,quant-ph/9809075.

I.V. Volovich, Atomic Quantum Computer, quant-ph/9911062.


References III


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References III

M. Ohya, Complexities and Their Applications to Characterization ofChaos, Int. Journ. of Theoret. Physics, 37 (1998) 495.

S.A. Gardiner, J.I. Cirac and P. Zoller, Phys. Rev. Lett. 79(1997) 4790.

R. Schack, Phys. Rev. A57 (1998) 1634; T. Brun and R. Schack,quant-ph/9807050.

I. Kim and G. Mahler, Quantum Chaos in Quantum Turing Machine,quant-ph/9910068.

D. Deutsch, Quantum theory, the Church-Turing principle and theuniversal quantum computer, Proc. of Royal Society of London series

A, 400, pp.97-117, 1985.J.I. Cirac and P. Zoller, Phys. Rev. Lett., 74 (1995) 74.

N.A. Gershenfeld and I.L. Chuang, Science, 275 (1997) 350.


References IV


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References IV

G. Burkard, D. Loss and D.P. DiVincenzo, cond-mat/9808026.

A. Ekert and R. Jozsa, Quantum computation and Shors factoringalgorithm, Reviews of Modern Physics, 68 No.3,pp.733-753, 1996.

I.I. Sobelman, Atomic Spectra and Radiative Transitions,Springer-Verlag, 1991.

Accardi, L. and Ohya, M.: Teleportation of general quantum states,quant-ph/9912087.

Fichtner, K.-H. and Ohya, M.:Quantum Teleportation with EntangledStates given by Beam Splittings, quant-ph/9912083.



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Thanks for your attention!


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