Thesis Kreso Bil An

7/31/2019 Thesis Kreso Bil An

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Contents

1 Introduction 21.1 Public key cryptography standard . . . . . . . . . . . . . . . . . 21.2 Breaking RSA encryption with quantum computer . . . . . . . . 31.3 Classes of problems . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 SQUID and quantum neural networks . . . . . . . . . . . . . . . 121.5 Some basic notions of graph theory . . . . . . . . . . . . . . . . . 16

2 Methods 202.1 Maximum independent set portfolio . . . . . . . . . . . . . . . . 20

3 Results 343.1 Orion returned results . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Further pruning of diversied portfolio - graph theory . . . . . . 363.3 Is learning rate in classical neural network related to Adiabatic

quantum computing? . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Discussion and conclusion 394.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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Chapter 1

Introduction

1.1 Public key cryptography standard

Lets illustrate the importance of Quantum Information Theory on the simpleexample of breaking cryptographic protection. Most of the secure type internetconnections from 2000s in the commercial and private use are using Public KeyCryptography Standard, which was initially started by the company RSA DataSecurity Inc. Now, the work is being transferred to the IETF PKIX group.Thereare two different branches of algorithms that are used today for establishing andusing secure connections:

symmetric (private) key algorithms

asymmetric public key algorithmsSymmetric (private) key algorithms are faster, but they require pre-sharing of the private key between parties. This is not convenient as a single internet servercan have secure connections with millions or more of its client. Asymmetricpublic key algorithms enable secure connections between parties previously notknown to each other. The drawback in the latter algorithms is that they aremuch more computationally intensive.

Lets assume Alice has fallen in love with celebrity Bob, who doesnt knowAlice. Alice has a key of an apartment which she wants to share with andonly Bob. Alice can simply send the key to Bob, if only Bob will obtain theletter. One of the problems is that Eve could eavesdrop on the Alices letterwith key, could take the key of the apartment and send it to Bob herself. So,Alice proceeds in the following way:

Bob takes two big prime numbers, multiplies them, and asks authorities toissue him the passport with exactly the same number he also picks anotherrelative prime number and writes it down to the passport (which will serveas a public encoding asymmetric key). Bob also keeps for himself a privateasymmetric decoding key.

Alice can check validity of the passport with the authorities (which ex-tends to the Home Office and Government) which are organised in a Chainof Trust. Alice then makes a photo of the apartment key (this is a private

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symmetric key). She then encrypts the photo with Bobs public asymmet-

ric key. Alice sends to Bob encrypted photo of her apartment key.

Upon receiving message from Alice, Bob uses his private decoding asym-metric key to obtain the photo of the apartment key, so he can gut it (thisis private symmetric key)

Eve can eavesdrop only if she is able to guess the private decoding asym-metric key based on the passport number. But if the passport number isbig, that is difficult. (E.g.

143 = 11 13, which is easy to check but more difficult to guess.) Eve cannot forge theBobs passport to deceive Alice because of the Chain of Trust.

From the above, it is visible that the crucial is the existence of the problemfor which once a given solution, it is easy to check, but difficult to nd thesolution. More in the section 3. Classes of problems. All widespread secureconnections, like VPN (virtual private networks) or HTTPS (secure http) usethe same principle. VPNs though perform mutual party authentication, whileour above example usually perform a single party authentication. HTTPS canoptionally perform mutual authentication, and not only a single party authenti-cation. In short, to perform authentication internet server and client would usea predened cryptographic suite with (a) integrity algorithm (or authenti-cation) algorithm (e.g. RSA - Ronald Rivest, Adi Shamir, Leonard Adleman,1977, or AES or 3DES), (b) pseudo-random (or hash ) algorithm (in theexample above it was digital photo of the actual key; usually hash is used to

produce a unique number - handle for some content) and with (c) condential-ity or ciphering algorithm (which uses exchanged symmetric key). The publicdeciphering asymmetric key is contained in the X.509 certicate veriablethrough a Chain of trust .

1.2 Breaking RSA encryption with quantum com-puter

It will be shown how to efficiently nd an algorithm that determines period r of the function f(x). This is crucial for factoring large numbers as a product of prime numbers. Finding of the period r is not easy, because not all periodicfunctions have a simple shape, as e.g. sinusoid. The shape within the periodcan uctuate widely, making it difficult to discover the period r . The bestknown classical algorithms (as of now) scale exponentially with n1/ 3 . In 1994Peter Schor discovered that quantum computer can do it faster than n3 . Schorsalgorithm can be applied to successfully break an RSA algorithm.

A set of positive integers less than N constitutes under a multiplication mod-ulo N if the set satises three conditions any group has to satisfy: (a) it possessesan identity element, (b) each element has an inverse and (c) multiplication op-eration is closed within the group. The order of the group is the number of theelements of the group. The order of the subgroup (any group contained withinthe original group) is a divisor of the order of the original group. That can beestablished as follows.

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Figure 1.1: A typical ow when establishing secure connections

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For any element a of group G and subgroup S we dene a coset space aS

which contains all elements of the form g = as , where s is any member in S .Any coset aS has the same number of members as the subgroup S.If two cosets aS and bS have a common member then they are identical,

which is seen from as = bs then ( as )S = ( bs )S and a(sS ) = b(s S ) so thataS = bS .

Since 1 is in S, then any member of G is in some coset S. Then since thecosets have the same number of elements as the subgroup S, since they aredisjoint, then the order of the subgroup S divides the order of the group G.

In difference to the order of the group, the order of an element a is given asthe smallest non-zero k for which ak = 1. The sequence a, a 2 , a 3 , ...a k1 formsa subgroup of k elements, therefore k divides the order of the group G.

Lets pick up two prime numbers p and q and pq = N . One can pick allnumbers relatively prime to N to form the group modulo N GN . It is easy toshow Little Fermats theorem starting from:

k

i =1a i

p

= a p1 + a p2 + a

pk +

p!n1!n2! . . . n k !

an 11 an 22 . . . a

n kk

and noting that the sum on the right hand side is divisible by p to

(a1 + a2 + ak ) p

a p1 + a

p2 + a

pk (mod p)

from which (assuming a i = 1

a p a(mod p)Then for the same reason:

aq1 p1 1(mod p)and

a p1 q1 1(mod p)Since a p1q1 1 divides both p and q, then

a ( p1)( q1) 1(mod pq)It is easy to see that also it must hold:

a1+ s ( p1)( q1) a(mod pq)If we now take c being a relative prime to (p-1)(q-1) (the probability that twolarge random number are relatively prime numbers is greater then 1/2), then cis an element G( p1)( q1) , such that it has an inverse d and

cd 1 (mod ( p1) (q 1))The following three equations (which follow from the above) are then the basisof the RSA encryption:

acd a mod ( pq),

b ac (mod pq) bd a (mod pq)

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If one would now return briey to the section 1, Bob would pick up two big

(2048 bits, en example) prime numbers, p and q, and would communicate toAlice N = pq. Bob would also pick up a public encoding key c = (p-1)(q-1).He keeps d strictly to himself. When Alice sends the message a she encrypts itand sends b ac (mod pq). Only Bob can decoded by performing bd mod ( pq).Were Eve to eavesdrop, she would have to guess d, in other words she wouldhave to nd p and q, and then nd d as an inverse of c ( p 1)(q 1).If Eve would have an efficient period nding function, here is how she wouldproceed. Eve nds the period of the intercepted message b. I.e. Eve nds r, sothat br b. This the order of b in the G pq . But this the order of a as well sinceb ac (mod pq). Since r is the order of G pq , it must divide (p-1)(q-1) which isthe number of the elements in G pq . Therefore, r and c do not have commondivisor. Then, there must exist d so that

cd = mod rEve can then calculate (because a r 1)

bd acd = a1+ mr = a (a r )m

a (mod pq)One wants to nd the period of the function

f (x) = bx mod (N )f (x + r ) = f (x)

which is not easy classically because the values of f uctuate (visually almostrandomly). For start, one can use super fast Quantum Fourier transform denedon the computational basis (see the simple explanation for this and notation

used below the equation)

U F T |x n =1

2n/ 2

2n 1

y=0e2ixy/ 2

n

|y nHere, the notation means the following:

| nn is the number of the Qbits used. Each Qbit represent a quantum subsystemthat can be in any state a0 0 + a1 1 where a0 and a1 are complex numbers whichsquared modules |a0|

2 and |a1|2 give the probability that the Qbit will be in

states 0 or 1, respectively. Dirac introduced instead of an arrow for the vector,the

|. So, e.g.

|5 could be written in base-2 as

|5 =

|1

|0

|1 =

|101 . Which

means that we would have 3 Qbits (3 quantum subsystems, each having twopossible states) and in this particular example 1st Qbit is in the state |1 , thesecond Qbit is in the state |0 and the third Qbit is in the state |1 . Note thatin this example one doesnt mention probabilities as we denitely know we arein state |5 .

e2ixy/ 2n

2n/ 2is the numerical factor - amplitude, whose modules squared will give the prob-ability of nding the quantum (system) computer in the state |y

U F T

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is a complex linear operator that one wants to physically realise, so that the

transformation from the state |x n 12n/ 22n

1

y=0e2ixy/ 2n |y n is done by quan-

tum computer. (What one wants is similar trying to design an analog computerto solve a given problem).

Lets introduce some helping operators Z and H

Z |y n = e2iy/ 2n

|y nand

H =12

1 11 1

where we assume 0 = |0 =10 and

1 = |1 =01 , so that e.g. H |1 =

1 2 (|1 |0 ) One denotes the operation of H on each and all Qbits by H nand then one hasH n |0 n =

12n/ 2

2n 1

y=0|y n

And nally putting both helping operators together

U F T |x n = Z x H n |0 nWe proceed with our attempt to construct U F T and restrict ourselves to thecase of n = 4. Now with subscripts next to H denoting on which Qbit doesit act. Subscripts next to x identify Qbit (note that these subscripts wouldcorrespond to the powers of 2, if we would be writing binary digits in the usual

computer science/ telecommunication literature).U F T |x3 |x2 |x1 |x0 = Z x H 3H 2H 1H 0 |0 n

With the introduction of the third helping operator, number operator n denedby n |1 = 1 |1 = |1 and n |0 = 0 |0 = 0, Z is then given by

Z = ei

8 (8 n 3 +4 n 2 +2 n 1 + n 0 )

andZ x = e

i8 (8 x 3 +4 x 2 +2 x 1 + x 0 )(8 n 3 +4 n 2 +2 n 1 + n 0 )

Using the obvious e(2 in ) = 1 one obtains

Z x = ei [n 3 x 0 + (x 1 +12 x 0 )n 1 + (x 2 + 12 x 1 + 14 x 0 )n 2 + (x 3 + 12 x 2 + 14 x 1 + 18 x 0 )n 3 ]

Since it is also obvious e( ix n ) H |0 = H |x we can writeei (x 0 n 3 + x 1 n 2 + x 2 n 1 + x 3 n 0 ) H 3H 2H 1H 0 |0 |0 |0 |0 =ei (x 0 n 3 ) H 3ei (x 1 n 2 ) H 2ei (x 2 n 1 ) H 1ei (x 3 n 0 ) H 0 |0 |0 |0 |0 =H 3H 2H 1H 0 |x0 |x1 |x2 |x3

Not to forget, we are trying to nd out out how to construct quantum Fourier transform and we have practically nished. We shall re-arrange the terms andintroduce two more simple helping operators.

U F T |x3 |x2 |x1 |x0 = ei [12 x 0 n 2 + ( 12 x 1 + 14 x 0 )n 1 + ( 12 x 2 + 14 x 1 + 18 x 0 )n 0 ]H 3H 2H 1H 0 |x0 |x1 |x2 |x3

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or

U F T |x3 |x2 |x1 |x0 = H 3ei n 212 x 0 H 2ei n 1 (

12 x 1 + 14 x 0 )H 1ei n 0 (

12 x 2 + 14 x 1 + 18 x 0 )H 0 |x0 |x1 |x2 |x3

or using again e( ix n ) H |0 = H |x one obtains (note that the order of x sub-scripts is reversed)U F T |x3 |x2 |x1 |x0 = H 3ei

12 n 2 n 3 H 2ei n 1 (

12 n 2 + 14 n 3 )H 1ei n 0 (

12 n 1 + 14 x 2 + 18 n 3 )H 0 |x0 |x1 |x2 |x3

With the introduction of the helping operators V and permutation P we arenally arriving at the compact expression for U F T which can be easily gener-alised to the cases other then n = 4.

V ij = ei n i n j / 2| i j |

P |x3 |x2 |x1 |x0 = |x0 |x1 |x2 |x3U F T |x3 |x2 |x1 |x0 = H 3 (V 32 H 2) (V 32 V 31 H 1) (V 32 V 31 V 30 H 0) P |x3 |x2 |x1 |x0

Coming back to the question of nding the period r . We are preparingan initial state of the quantum computer as (as mentioned previously withf (x) = bx mod (N ) ) which is not difficult. Such an initial state preparation ismore powerful with quantum computer then with classical computer as functionf can be made on all computational basis vectors in one go (this is the rstcrucial fact in this algorithm). We just mention here that the rst n Qbits inthe equation below are input Qbits, and the n0 are the output Qbits.

12n/ 2

2n 1

x =0 |x n |f (x) n0

When we perform measurement on the output Qbits, these Qbits will collapsein a single state with the value f (x0) = f 0 . Generalised Born rule for thecollapse of the quantum states are then saying that the input Qbits will be asuperposition

| n =1

mm

k=0|x0 + kr n

where m depends on the size of the input register. To retrieve the value of theperiod r, we shall apply quantum Fourier transform to this input Qbits (butafter we let the operator of the function f to work upon both input and output

Qbits and after we measured output Qbits)

U F T 1 mm 1

k =0 |x0 + kr n =1

2n/ 22n 1

y=0

1 mm 1

k=0e2i (x 0 + kr )y/ 2

n

|y=

2n 1

y=0e2ix 0 y/ 2

n 1 2n mm 1

k =0e2ikr/ 2

n

|y

We now see that the probability of nding the input Qbits in the state |y is

p (y) =1

2n m

m 1

k =0

e2ikr/ 2n

2

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Figure 1.2: Quantum gate model for UFT operator - consisting of only 1-gateand 2-gates

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Figure 1.3: Initial state and Uf operator

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Note that we dont have any longer dependency on x0 (this is the second crucial

step in the algorithm. It corresponds to the Heisenberg uncertainty principle,since we dont know for which x we are investigating period r. I.e. we have lostthe absolute information, but we are getting the relative information of r forsome - unknown x0) Lets assume a special value for y ( with absolute j lessthan 1/2 )

y = yj = j2n

r+ j

we get

p (yj ) =1

2n msin2 ( j mr/ 2n )sin2 ( j r/ 2n )

Since m is implicitly chosen according to the size of the register and period rwith mr/ 2n close as possible to 1, one gets

p (yj ) =1

2n msin2 ( j )

( j r/ 2n )2=

1r

sin2 ( j )( j )2

From where for absolute j < 1/ 2 follows

p(yj ) 4

21r

Since there are at least r-1 different yj values, and r a big number, we see thatwe shall end up with one of the special values yj very often ( 4 2 approx. 0.4).This analysis can be rened further, so to obtain almost certain value of theperiod r in the rst go of the quantum computer.

We shall stop here. Many aspects of the quantum computing are not men-tioned in this short exposition. Nevertheless, the RSA breaking mechanism hasbeen shown step-by-step. There are many questions arises: what are the ap-propriate problems for quantum computers? and although theoretically sound,how feasible quantum computers are? The rst question cannot be answeredtoday, but more information in the next section. The second question is thenaddressed after the next section.

1.3 Classes of problems

The issue is complexity of the problem (time and space resources required tond the solution of the problem). How efficiently one can solve the problem?Whether classical computer is worse/ equal/ or even better than quantum com-puter. Are there problems for which classical computer is better, and someproblems for which quantum computer is better?

The number of complexity classes in which one can classify problems (andalgorithms) is too big (and growing every day) to be described here, in anydetail. So, we shall mention P and NP problems, as we shall be dealing withNP problems later.

Regarding the Schors period nding algorithm from the previous section,obviously it is very powerful. And, at the moment, it seems regarding periodnding, that the quantum computer is much more powerful then a classicalcomputer, we dont yet have the proof of that. It is entirely possible, thatsomeone nds an equally efficient classical algorithm.

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Roughly speaking P class of problems have algorithms that can nd solution

quickly. NP class has algorithms where the solution can only be checked quickly,but not solved quickly.Obviously P is a subset of NP. What is not clear is whether NP is a proper

superset, i.e. if there are problems that cannot be solved quickly. E.g. in fac-torisation, it is clear that if we have two big prime factors, it is easy to checkthey are prime factors of the given number. What is not clear, is whether ourproblem when factorising big numbers is due to our inability to nd quick (clas-sical) algorithm to solve the factorisation problem, or such algorithm doesntexist. If the efficient factorisation algorithm wouldnt exist, we would be havingP NP . But, at the moment:

P ?NP

There is an interesting subclass of NP problems, called NP-complete. NP-

complete are in a sense at least as hard as any other NP problem (NP prob-lem including NP-complete problems). It is an interesting fact that knowingan efficient NP-complete problem solving algorithm, could solve (with a smalloverhead) any other NP problem. Funny, Sudoku games are NP-complete, andnding an efficient Sudoku solving algorithm would conrm P NP To include quantum computers in this discussion, we dene a new classBQP - Bounded quantum P class. We included the word bounded, since manyquantum algorithms will give a result with some probability of an error (due toquantum probability considerations). We can dene and demand this error tobe as small as we wish (that might change a space or time requirement on theBQP algorithm, but is not crucial).

We shall now try to tentatively relate BAP to P and NP, but we need anotherclass of the problems. This is a PSPACE which represents algorithms that dontrequire a huge space, but might require any amount of time to solve the problem(in NP problems we usually have very bad time resource required, i.e. a lot of time required, but we know the upper limit of this resource. In PSPACE wedont care about this time resource limit at all - we have all the time of theUniverse.) What is known is that BQP is obviously bigger then P. What isknown is that BQP is less than PSPACE. But the relation between BQP andNP is completely unknown.

Lets just mention that the nding of Hamiltons cycle or path is NP-complete and the nding of Maximum independent set is also NP-completeproblem. We shall encounter these problems in later sections.

1.4 SQUID and quantum neural networksWhen trying to physically realise the quantum computer, one of the naturalchoices is to use superconductors. In superconductors electron wave functionsare highly coherent, i.e. the phase between different superconducting wavevectors is very well dened. Another important fact for us is that above somecritical current I c superconductivity seizes.

In 1962 Josephson noticed that when letting current through a constrictionin the circuit, the supercurrent I s through the constriction is given by:

I s = I c sin

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Figure 1.4: Where does BQP ts?

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way: one would x initially some J ij and/ or i . This choice of xing would

depend on the problem to be solved. Then one would slowly let some or allof this xed values lose, and let the system relax to a new energy level, butmaking sure that the system remains in the ground state. I.e. ground statechanges the energy but the superconducting processor remains in this groundstate. One can also turn on slowly some other J ij and/ or i towards the endof the calculation. At the end of the calculation the read-outs of J ij and i givethe solution to the problem.

Regarding Ising Hamiltonian, it is also connected with Neural network the-ory. E.g. the function that describes stability of neural networks, Lyapunovfunction (Lyapunov function generally describes stability of the systems) is givenfor some neural networks as:

L = H In such neural networks it holds:

i (t + 1) = sgnj

J ij j (t) + i

where if i 1 neutron is ring. J ij is synaptic weight. And i is renamed h iand represents external input = negative threshold.

1.5 Some basic notions of graph theory

Graph G consists of vertices V i and edges connecting these vertices E ij . Verticesconnected by an edge are co-incident .

Path connects two vertices through other vertices and edges, not goingacross the same edge or vertex.

Cycle is the path where initial and nal vertices are the same.Graph is connected if any two vertices are connected by some path.Connected acyclic graph is tree .Spanning tree (or spine ) of graph G is a spanning sub-graph of G, i.e. it

contains all the vertices.Hamiltons path of graph G is a spanning path, i.e. path that contains all

vertices of the graph. Hamiltons cycle is a spanning cycle of the graph.Graph is called Hamiltons graph if it has Hamiltons cycle.The graph is -connected if there different paths between any two vertices

in the graph, such that that none of the paths shares any vertices (except of

course, the initial and nal vertex).The set of vertices S of graph G is stable or independent if none of two ver-tices of S are co-incident. The maximum cardinal number of some independentset in G is called stability or independance of the graph G and is denotedby (G). Determination of (G) is NP-complete.

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Figure 1.6: Some basic notions of the graph theory - part 1

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Chapter 2

Methods

2.1 Maximum independent set portfolio

We closely follow D-wave systems Inc. ideas on how to use Maximum indepen-dent set approach to create portfolio. (This approach has been known to theauthor half year before published by D-wave systems).

We start with representing our stocks (or other nancial instruments by thegraph G. Each nancial instrument represented by the vertex V j in the graph,and the relationship between different nancial instrument i and j representedby the edge E ij .

If we restrict the nancial instruments to be the tickers included in the DowJones Wilshire 5000 Composite index, we can associate each ticker symbol with

a graph vertex. If the value of the ticker is P i , then we can approximate thereturn on stock (similar but not the same as dividend) by:

R i (t) = ln [ P i (t) /P i (t 1)]Now we can identify edges in the graph by measuring correlation between dif-ferent returns on the stock. I.e. E ij is identied with

Qij =R i R j R i R j

R2i R i 2 R2j R j 2But, lets further assume that below certain threshold Q ij we assume the cor-relation to be purely noise (we do allow negative correlation above the noise,nevertheless). Since we are not interested in the correlations of the small abso-lute value, we shall connect vertices V j and V j with edges E ij only if the absolutevalue Q ij is above some threshold K.

Since we can have a huge number of vertices, and the number of all possible

edges n2 can easily be in the range of the millions. To make the prob-

lem tractable, the threshold K on Q ij to reduce the number of edges seemsreasonable.

What we are achieving with having a threshold and constructing the graph,is that highly correlated ( Q ij 1 ) and anti-correlated ( Qij 1 ) nan-cial instruments are always connected while weekly correlated (anti-correlate)

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nancial instrument are not connected. By nding the maximum number of

the vertices - nancial instruments, where none of the vertices are directly con-nected we have identied the most diversied portfolio. This portfolio is themaximum independent set of the graph G, and nancial instruments in it havethe smallest correlations between themselves.

The example below shall be given for some stocks and other types of thenancial instruments.

Vertex Ticker Name Comment1 GOOG Google2 CRA Celera Genomics company. First to se-

quence the human genome3 GE General Electric Similar to Siemens in Europe,

but more diversied4 GLD SPDR gold trust Exchange traded fund of gold5 HSBA HSBC bank holding company The biggest UK bank6 MON Monsanto Agriculture giant7 PFE Pzer Pharmacology8 RTP Rio Tinto Gold miner9 XOM Exxon Oil producer10 BA BAE Systems Defence contractor

Using Google nance e.g. we obtained the following data:

Date closeP 1(10) 8 Sep 2009 458.62

P 1(9) 4 Sep 2009 461.30P 1(8) 3 Sep 2009 457.52P 1(7) 1 Sep 2009 455.76P 1(6) 28 Aug 2009 464.75P 1(5) 27 Aug 2009 466.06P 1(4) 26 Aug 2009 468.00P 1(3) 25 Aug 2009 471.37P 1(2) 24 Aug 2009 468.73P 1(1) 21 Aug 2009 465.24P 1(0) 20 Aug 2009 460.41

Date close

P 2(10) 8 Sep 2009 6.50P 2(9) 4 Sep 2009 6.50P 2(8) 3 Sep 2009 6.35P 2(7) 1 Sep 2009 6.34P 2(6) 28 Aug 2009 6.55P 2(5) 27 Aug 2009 6.61P 2(4) 26 Aug 2009 6.68P 2(3) 25 Aug 2009 6.74P 2(2) 24 Aug 2009 6.58P 2(1) 21 Aug 2009 6.70P 2(0) 20 Aug 2009 6.51

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Date closeP 3(10) 8 Sep 2009 14.50P 3(9) 4 Sep 2009 13.87P 3(8) 3 Sep 2009 13.45P 3(7) 1 Sep 2009 13.34P 3(6) 28 Aug 2009 14.08P 3(5) 27 Aug 2009 14.19P 3(4) 26 Aug 2009 14.11P 3(3) 25 Aug 2009 14.30P 3(2) 24 Aug 2009 14.20P 3(1) 21 Aug 2009 14.21P 3(0) 20 Aug 2009 13.81


Date closeP 5(10) 8 Sep 2009 654.30P 5(9) 4 Sep 2009 658.30P 5(8) 3 Sep 2009 644.00P 5(7) 1 Sep 2009 643.90P 5(6) 28 Aug 2009 672.10P 5(5) 27 Aug 2009 663.60P 5(4) 26 Aug 2009 668.60P 5(3) 25 Aug 2009 671.00P 5(2) 24 Aug 2009 664.10P 5(1) 21 Aug 2009 659.10

P 5(0) 20 Aug 2009 645.80

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Date close




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Date close



Now it is easy to calculate stock returns R i

Date closeR1(10) 8 Sep 2009 -0.0058R1(9) 4 Sep 2009 0.0082R1(8) 3 Sep 2009 0.0039R1(7) 1 Sep 2009 -0.0195R1(6) 28 Aug 2009 -0.0028R1(5) 27 Aug 2009 -0.0042R1(4) 26 Aug 2009 -0.0072R1(3) 25 Aug 2009 0.0056

R1(2) 24 Aug 2009 0.0075R1(1) 21 Aug 2009 0.0104

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Date close

R2(10) 8 Sep 2009 0.0000R2(9) 4 Sep 2009 0.0233R2(8) 3 Sep 2009 0.0016R2(7) 1 Sep 2009 -0.0326R2(6) 28 Aug 2009 -0.0091R2(5) 27 Aug 2009 -0.0105R2(4) 26 Aug 2009 -0.0089R2(3) 25 Aug 2009 0.0240R2(2) 24 Aug 2009 -0.0181R2(1) 21 Aug 2009 0.0288

Date closeR3(10) 8 Sep 2009 0.0444R3(9) 4 Sep 2009 0.0307R3(8) 3 Sep 2009 0.0082R3(7) 1 Sep 2009 -0.0540R3(6) 28 Aug 2009 -0.0078R3(5) 27 Aug 2009 0.0057R3(4) 26 Aug 2009 -0.0134R3(3) 25 Aug 2009 0.0070R3(2) 24 Aug 2009 -0.0007R3(1) 21 Aug 2009 0.0286

Date closeR4(10) 8 Sep 2009 -0.0010R4(9) 4 Sep 2009 0.0007R4(8) 3 Sep 2009 0.0372R4(7) 1 Sep 2009 0.0053R4(6) 28 Aug 2009 0.0023R4(5) 27 Aug 2009 0.0043R4(4) 26 Aug 2009 0.0003R4(3) 25 Aug 2009 0.0045R4(2) 24 Aug 2009 -0.0141R4(1) 21 Aug 2009 0.0148

Date closeR5(10) 8 Sep 2009 -0.0061R5(9) 4 Sep 2009 0.0220R5(8) 3 Sep 2009 0.0002R5(7) 1 Sep 2009 -0.0429R5(6) 28 Aug 2009 0.0127R5(5) 27 Aug 2009 -0.0075R5(4) 26 Aug 2009 -0.0036R5(3) 25 Aug 2009 0.0103R5(2) 24 Aug 2009 0.0076R5(1) 21 Aug 2009 0.0204

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Date close

R6(10) 8 Sep 2009 0.0140R6(9) 4 Sep 2009 0.0082R6(8) 3 Sep 2009 -0.0045R6(7) 1 Sep 2009 -0.0129R6(6) 28 Aug 2009 -0.0193R6(5) 27 Aug 2009 0.0076R6(4) 26 Aug 2009 0.0042R6(3) 25 Aug 2009 -0.0010R6(2) 24 Aug 2009 -0.0008R6(1) 21 Aug 2009 0.0156

Date closeR7(10) 8 Sep 2009 -0.0110R7(9) 4 Sep 2009 0.0191R7(8) 3 Sep 2009 -0.0185R7(7) 1 Sep 2009 -0.0259R7(6) 28 Aug 2009 -0.0030R7(5) 27 Aug 2009 0.0054R7(4) 26 Aug 2009 -0.0018R7(3) 25 Aug 2009 0.0042R7(2) 24 Aug 2009 0.0054R7(1) 21 Aug 2009 0.0249

Date closeR8(10) 8 Sep 2009 0.0470R8(9) 4 Sep 2009 0.0223R8(8) 3 Sep 2009 0.0257R8(7) 1 Sep 2009 -0.0373R8(6) 28 Aug 2009 0.0051R8(5) 27 Aug 2009 0.0020R8(4) 26 Aug 2009 -0.0284R8(3) 25 Aug 2009 -0.0059R8(2) 24 Aug 2009 0.0203R8(1) 21 Aug 2009 0.0344

Date closeR9(10) 8 Sep 2009 0.0210R9(9) 4 Sep 2009 0.0134R9(8) 3 Sep 2009 -0.0022R9(7) 1 Sep 2009 -0.0247R9(6) 28 Aug 2009 -0.0105R9(5) 27 Aug 2009 -0.0072R9(4) 26 Aug 2009 0.0097R9(3) 25 Aug 2009 -0.0087R9(2) 24 Aug 2009 0.0195R9(1) 21 Aug 2009 0.0192

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Date close

R10(10) 8 Sep 2009 0.0059R10(9) 4 Sep 2009 0.0094R10(8) 3 Sep 2009 0.0159R10(7) 1 Sep 2009 0.0016R10(6) 28 Aug 2009 0.0201R10(5) 27 Aug 2009 -0.0572R10(4) 26 Aug 2009 -0.0247R10(3) 25 Aug 2009 0.0003R10(2) 24 Aug 2009 -0.0051R10(1) 21 Aug 2009 0.0145

Now it is easy to calculate Q ij

Q ij 1 2 3 4 5 6 7 8 912 0.763 0.67 0.734 0.11 0.26 0.065 0.9 0.77 0.69 -0.026 0.39 0.55 0.74 -0.06 0.337 0.75 0.7 0.56 -0.25 0.8 0.568 0.64 0.51 0.88 0.15 0.58 0.53 0.379 0.58 0.42 0.74 -0.25 0.57 0.74 0.55 0.710 0.28 0.32 0.17 0.28 0.3 -0.26 -0.06 0.37 0.11

Lets set an arbitrary threshold K. The value that is reasonable in some senseshall be given in the later section. If we set the absolute value of threshold equal0.5, then we obtain the existence of the edges E ij for the following edges:

E ij 1 2 3 4 5 6 7 8 91 X X X X X X2 X X X X X X3 X X X X X X X45 X X X X X X X

6 X X X X X7 X X X X X X X8 X X X X X9 X X X X X X10

Lets proceed according to the D-wave systems API development guide. D-wave systems offers a service of quantum processor solvers through an Orionweb services. So the Orion web service is a front end towards the customerthat can use quantum computing by using appropriate client software which

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Figure 2.1: Graph for which we are looking Maximum independent set - maxi-mally diversied portfolio

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Figure 2.2: Orion authentication C program

implements client side of Orion web services.Here we present only the C client for interfacing towards Orion web services.Three libraries are necessary (the last one only for testing) libcurl, libxml2

and libcheck. From here it is visible that client is running on Unix like operatingsystem (possibly Linux).

After downloading the installation follows the usual procedure for congur-ing, making and installing the client.

The code below describes how to submit the problem as a job. (Note: this

listing contains as the problem SAT, while we are looking for MIS - maximumindependent set problem)

The SAT Problem Data describes a SAT problem with 2 variables and 2clauses: (x1 or x2) and !x2 The Problem Data is a String, so it has a mimetype of text/plain. In our case of MIS problem we would have a string sayingsomething like: p edge 10 24 e 1 2 e 1 3 ... starting with the number of vertices, then edges, and then identifying edges themselves... We continue asfor the SAT problem.

When we use Orion web services and through these web services accessquantum processor, we need to take care of four things: Problem data, problemspecication, job and answer.

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Figure 2.3: Orion problem submission C program - part 1

Problem data for Maximum independent set contains string with the numberof vertices, edges, and denition of edges themselves.

Problem specication details which algorithm are we running. In our case itis Maximum independent set.

Job considerations are mainly with the known issues that the jobs can bein queues with different priorities. Also, job might detail some of the operatingparameters for the quantum processor, or in the case of algorithm testing, mightcontain some data for classical simulator of the quantum processor (as it isexpected that the processing time of quantum processor will be expensive). Eachwill have state COMPLETED, FAILED, CANCELLED or IN-PROGRESS.

The answer depends on the problem specication. In the case of the Max-imum independent set, the answer will be the binary string with 0s indicatingvertices that are not part of the Maximum independent set.

Lets just mention without further explanation the possible problem types

that can be solved: Binary Quadratic Program (BQP) problem/bqp Maximum Satisability (MAX-SAT) problem/maxsat Cardinality SAT problem/cardsat Weighted Maximum Satisability (Weighted MAX-SAT) problem/maxweightedsat Model Expansion (MX) problem/mx Maximum Independent Set (MIS) problem/mis

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Figure 2.5: Orion problem submission C program - part 3

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Clique (CLQ)

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Chapter 3

Results

3.1 Orion returned results

Although C program works, it was easier (and might be easier for any otherinterested person, as a rst try) to manually submit the problem to Orion webservices. That has been done, as shown in the gure below.

The submitted problem data was:p edge 10 24e 1 2e 1 3e 1 5e 1 7

e 1 8e 1 9e 2 3e 2 5e 2 6e 2 7e 2 8e 3 5e 3 6e 3 7e 3 8e 3 9e 5 7e 5 8e 5 9e 6 7e 6 8e 6 9e 7 9e 8 9

for which the answer is: 0001001101 4, indicating 4 vertices in the Maximum in-dependent set, and vertices in the Maximum independent set are: V 4 , V 7 , V 8andV 10.

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Figure 3.1: Web services for Orion returned the Maximum independent set

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Which are GLD, PFE, RTP and BA ( gold ETF fund, pharma giant, gold miner

and defence contractor ) and together they represent the most diversied port-folio.

3.2 Further pruning of diversied portfolio - graphtheory

Lets give without the proof the followingTHEOREM (Chvatal-Erdos, 1972). If G is a graph with at least 3 vertices, sothat its stability (independence) is not bigger then its connectedness , i.e. . Then G is a Hamiltons graph.So in the case if is very big, which would indicate very strongly couplednancial instruments, if we would get the result of less than , then we wouldbe able to connected all the nancial instrument in one path. I.e. we would beable to conclude what is the logical connection between different nancial in-struments and which one inuences the other one. In other words, the seemingcomplexity of correlations - or someone might say volatility - would disappearand each nancial instrument would be connected with only two other nancialinstruments.

It doesnt seem strange that theorem in that case implies that such connec-tion between instruments must be cyclic (since the graph is Hamiltons graph).So the causation in this Hamiltons cycle runs in both directions, and also -nancial instrument through the cycle impacts itself.

3.3 Is learning rate in classical neural networkrelated to Adiabatic quantum computing?

Here, one likes to suggest the way forward when analysing performance of Adi-abatic quantum computer built on the bases of SQUIDs. As the system of SQUIDs corresponds to the neural networks, one would expect, since SQUIDsshould interact in a quantum coherent manner that the corresponding theoryshould be of quantum neural networks. Because no consistent quantum theoryof neural networks exists, one could try to use Adiabatic quantum computingas an inspiration. In the other direction, since there are lot of unknown perfor-mance issues related to the Adiabatic quantum computing, one could use - asthe worst case - the conclusions of classical neural networks performance.

E.g. to determine the time required for Adiabatic computing algorithm tonish computing, one could start by using the rate of learning algorithm forclassical neural network.

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Figure 3.2: Maximum independent set

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Chapter 4

Discussion and conclusion

4.1 Discussion

In the section Public key cryptography standards, we started with the motiva-tion of a considerable military and economic interest. The possibility of breakingone of the most secure encryption algorithms today. The description of the ba-sic security mechanisms is widely available. For those wanting to know more,author would suggest some introductory in IPsec protocol.

In the section Breaking RSA with quantum computer we constructed theo-retically quantum computer that can break RSA ciphering. We used so calledquantum gate approach. Although theoretically very elegant approach, it doesntseem feasible technically. The exposition in this section closely follows [ND].

In the section Classes of problems we introduced some notions from com-plexity theory. The exposition closely follows [NC].In the section SQUIDs and quantum neural networks we shortly introduced

SQUIDs as the basis of the quantum processors, and also shown a tentativeconnection to the neural networks. One can read more in e.g. [MT] for SQUIDsand [CKS] for neural networks. The idea of quantum neural networks is stillnot very developed, lacking the aim, consistency in the theoretical approach andunderstanding how to include quantum effects.

In the section Some basic notions of graph theory we only introduced somebasic graph theory notions. The reader can check any book on graph theory forfurther reading.

In the section Maximum independent set portfolio we used the ideas from[GR] to create a very well diversied portfolio. The interface to the D-wavesystems Orion web services for quantum processing is explained. Maximumindependent set and portfolio are determined.

In the section Further pruning of diversied portfolio - graph theory, weare giving a somewhat unexpected result hinting that in moderately volatileconditions one can always nd an order, and that each nancial instrumentdepends exactly on two other nancial instruments.

In the section Is learning rate in classical neural network related to Adiabaticquantum computing?, we are trying to argue that it makes sense for researchersin both elds to be acquainted with the both theories as they should inter-change many results. Again for Adiabatic quantum computing there is not a

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lot of information so the reader is instructed to check the currently submitted

papers. Again for neural networks, for start, [CKS] is good.

4.2 Conclusion

The main attractiveness of the existing superconducting quantum processors istheoretical. After Sir Roger Penroses claim that the consciousness is an quan-tum effect originating in the cell organelle microtubule, Max Tegmark refutedit, showing that the coherence time for the quantum effects is too short for anyquantum effect to arise. Then it is interesting that we know that the efficiencyof the photosynthesis is due to quantum effects, and we also know that the Adi-abatic quantum computing doesnt need long coherence time to be successful.In that respect, it is interesting to see further development of superconducting

quantum processors.From the practical point of view, while it is possible that some importantalgorithms can be found, which would be very efficiently solved by Adiabaticquantum processors, that doesnt seem to be of great interest, at the moment.

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Bibliography

[DM] N. David Mermin, Quantum computer science: An introduc-tion, Cambridge University Press 2007

[NC] Michael A. Nielsen, Isaac L. Chuang, Quantum computationand quantum information, Cambridge University Press 2007

[MT] Michael Tinkham, Introduction to superconductivity 2nd ed.,Dover publications 2006

[CKS] A. C. C. Coolen, R. Kuhn, P. Sollich, Theory of neural infor-mation processing systems, Oxford University Press 2005

[DV] Darko Veljan, Kombinatorna i diskretna matematika (Combi-natorial and discrete mathematics), Algoritam 2001

[GR] Geordie Rose, dwave.wordpress.com, September 2008

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