1D Ising model simulation of the complexity growth bound in the `complexity equals action'-conjecture

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UNIVERSITEIT VAN AMSTERDAM

1D Ising model simulation of the

complexity growth bound in the

‘complexity equals action’-conjecture

by

Boudewijn Poot

10686355

Supervisor: Examiner:

Dr. B. W. Freivogel Prof. Dr. C.J.M. Schoutens

Report Bachelor Project Physics and Astronomy, size 15 EC, conducted between 01 – 03 – 2016 and 01 – 07 – 2016

A thesis submitted in partial fulfillment for the degree of Bachelor of Science, Beta-Gamma

in the Faculty of Science

Institute for Theoretical Physics Amsterdam

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UNIVERSITEIT VAN AMSTERDAM

Abstract

Faculty of Science

Institute for Theoretical Physics Amsterdam

Bachelor of Science, Beta-Gamma by Boudewijn Poot

10686355

Ever since the proposal of Hawking radiation, there has been a huge discussion among physicists how to resolve the ‘black hole information paradox’. It has been proposed that analysing black holes using computational complexity could help shed light on the paradox. Recently, a complexity growth bound for black holes has been conjectured. This thesis analyses this theory by simulating systems other than black holes using a 1D Ising model and time evolving it using quantum gates in an attempt to find such a complexity growth bound. It was concluded that the simulation found a complexity growth bound dependent on the total number of qubits in the system. Also, signs for the presence of tighter complexity growth bounds were found. However, the simulation showed little resemblance with physical energetic behavior making its findings questionable. The simulation did show potential in making theories about complex objects tractable and further research might prove useful in solving the ‘black hole information paradox’.

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Popular Dutch Summary

Figuur 1: Kunstenaars weergave van een zwart gat; Bron: NASA

Voor 1975 dacht de wereld dat de gi-gantische zwaartekracht van een zwart gat het onmogelijk maakte voor deeltjes om uitgezonden te worden vanuit het zwarte gat. In 1975 publiceerde Stephen Hawking een paper waaruit bleek dat een zwart gat toch straling leek uit te zenden. Dit wordt ook wel Hawking radiatie genoemd. Deze straling leidt echter tot een paradox. In de quantummechanica is er namelijk een

aantal belangrijke principes. Enerzijds kan informatie niet gekopieerd worden, anderzi-jds kan er met de zogenaamde toestand van een quantum-systeem het complete verleden en de complete toekomst van dat systeem berekend worden.

Als we dit toepassen op het zwarte gat, kan de straling dus geen kopie van de informatie bevatten -informatie kan immers niet gekopieerd worden- en moet er in het zwarte gat steeds minder en minder informatie zijn over het systeem. Dit staat echter weer in contrast met het tweede principe: als er minder informatie in het systeem is, is het onmogelijk dat alle informatie over het verleden en de toekomst bewaard is. Dit levert dus een probleem op.

Tot de dag van vandaag zijn natuurkundigen bezig om dit probleem op te lossen. Ze hebben zelfs de hulp van informatici erbij geroepen. Zo kijken ze hoe moeilijk het precies zou zijn om de informatie in een zwart gat, en in de straling van het zwarte gat uit te lezen. Wellicht helpt dit met het oplossen van het probleem. Aan het einde van vorig jaar is er bedacht dat deze moeilijkheidsgraad van het uitlezen van de informatie in een zwart gat slechts met een beperkte snelheid kan stijgen. Deze hangt af van de gemiddelde energie van het zwarte gat.

Dit stuk probeert een zwart gat te simuleren door een makkelijker systeem vergeli-jkbaar met een zwart gat te laten groeien en gedragen. Als blijkt dat ook dit simpelere systeem zich aan de beperkte snelheid houdt, dan is dit alleen maar meer bewijs dat de opgelegde snelheidslimiet van natuurkundigen lijkt te kloppen. Jammer genoeg is gebleken dat de simulatie simpelweg te slecht een zwart gat simuleert en dat de con-clusie ervan dus weinig bewijs levert voor de echte wereld. Wel werd er een vergelijkbare snelheidslimiet gevonden zoals deze in de theorie beschreven werd.

Ook bleek het simuleren van de theorie een makkelijke en toegankelijke manier om de theorie te testen en kunnen toekomstige onderzoekers, door het model te verbeteren, wellicht meer informatie over zwarte gaten vinden.

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Acknowledgements

I want to take this opportunity to express my gratitude to Dr. B.W. Freivogel for his valuable weekly guidance and sharing of expertise extended to me in order to fullfill this project.

Next, I would like to give a sincere thank you to R.A. Jefferson MSc for sitting in on every meeting and helping with the entire thought process.

Last, I wish to express my sincere thanks to Prof. Dr. C.J.M. Schoutens for being the second reader for this project.

I also mention my sense of gratitude to all, who directly or indirectly, have lent their hand in this thesis.

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2.2.1 Time evolution . . . 9 2.2.2 Time steps . . . 9 2.2.3 Simulation . . . 11 3 Results 12 3.1 Random . . . 12 3.2 1 bit . . . 14 3.3 Fully aligned . . . 15 3.4 Half aligned . . . 16 3.5 Alternating . . . 16 3.6 Dependency on size. . . 17 3.7 Simulation superpositions . . . 17 4 Conclusion 19 5 Discussion 21 iv

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Contents v

A Quantum gates 22

A.1 Hadamard gate . . . 22

A.2 Pauli-X gate. . . 23

A.3 Phase shift gate. . . 23

A.4 Controlled Not gate . . . 23

B Simulations code 24 B.1 Random distribution . . . 24

B.2 1 bit in the middle . . . 27

B.3 Fully aligned . . . 28

B.4 Half aligned . . . 28

B.5 Alternating . . . 28

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List of Figures

1 Representation of artist of black hole . . . ii

3.1 Complexity of 10 random bits . . . 12

3.2 Rate of complexity of 10 random bits. . . 12

3.3 Complexity of the random distribution compared . . . 13

3.4 Normalized complexity of the random distribution compared . . . 13

3.5 Shockwave-like behavior of time evolution . . . 14

3.6 Complexity of the 1 bit distribution compared. . . 14

3.7 Normalized complexity of the 1 bit distribution compared . . . 14

3.8 Rate of complexity for 1 bit distributions compared. . . 15

3.9 Complexity for fully aligned distributions compared . . . 15

3.10 Complexity for half aligned distributions compared . . . 16

3.11 Complexity for alternating distributions compared . . . 16

3.12 Size dependency for rate of complexity . . . 17

List of Tables

2.1 Sampled recurrence times compared to theoretical recurrence times . . . . 10

3.1 Rates of change for the 1 bit distribution . . . 15

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Abbreviations

CC Computational Complexity CGB Complexity Growth Bound CNOT Controlled NOT

QCC Quantum Computational Complexity

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Symbols

A Black hole action C Complexity

Eψ Expectation value of energy for a given state

G Gravitational constant H Hadamard gate

ˆ

H Hamiltonian operator

K Number of qubits in a system U Unitary operator

V Volume

φ Quantum state ψ Quantum state

ψref Quantum reference state

⊗ Tensor product

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Chapter 1

Introduction

In a letter written by John Michell to Henry Cavendish [1], Michell conjectured that there could be objects so big that the velocity one has to have to escape the gravity of the object exceeds the speed of light. Later, people like Simon Pierre Laplace and Karl Schwarzchild worked on predicting and defining what physics now calls black holes. Ever since then, and even more so after Hawking wrote a letter to Nature about the emission of particles by black holes [2], the massive objects have been a topic of discussion in the world of physics.

The fact that black holes seem to have this stream of radiation leads to a contradiction. The radiation, namely, should contain all the information about everything that ever fell in the black hole. This stems from the fact that present wave functions are uniquely determined in the future by their evolution operator. For a time-independent Hamilto-nian, this (unitary) operator would be e−i ˆHt. This evolution operator also has an inverse which should uniquely determine the past wave functions given its present form. Thus, while information is falling into a black hole, this information is also emitted out of the black hole. This would violate the no-cloning theorem stating that no unitary operator can clone a general quantum state as such:

U (|φi_A|ei_B) = eiα(φ,e)|φi_A|φi_B (1.1) It would violate this theorem since there would have to be some operation possible to clone the ingoing information to the outgoing information before it could be emitted. Several attempts were made at avoiding this violation such as suggesting that the clones could not be observed at the same time [3] or by allowing firewalls [4] to purge every par-ticle crossing the event horizon. However, these are currently not unanimously deemed satisfactory. The debate it sparks however may emerge into a larger understanding of the laws of physics around black holes [5].

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Introduction 2 The model for black holes suggested by Hayden and Preskill seems to allow for the possibility of an infalling observer to extract information from the Hawking radiation, provided the observer is able to perform an adequate nonlocal measurement [6]. This so-called ‘black hole information paradox’ lead Hayden and Harlow to investigate the matter using computational complexity (CC).

CC is the branch of theoretical computer science that focuses on determining the in-herent difficulty of solving a certain problem. Hayden and Harlow applied this technique to the paradox black holes seem to generate by looking at the quantum computational complexity (QCC) of decoding the Hawking Radiation. They found that for a wide variety of black holes it would take exponential time to decode the emitted information [7]. Exponential time allows for the black holes to evaporate long before any observer has a chance of decoding the information and quickly going across the event horizon to observe the clone. One might say the information is still there, even though it is not retrievable. Harlow and Hayden however state that, considering black hole evaporation, the operational constraints are important in understanding the structure of the Hilbert space [7].

So although this did not disprove the existence of firewalls per se, it did provide the insight that QCC may be a useful tool analysing black holes and maybe even physics problems in general. It motivated Leonard Susskind to start looking deeper into the QCC of black holes. Nowadays, he even states that it is essential to fully understanding the black hole horizons and their properties [8].

It is now clear how QCC found its way into physics in the first place and it is time to sketch the problem ahead for this specific project. A good place to start is how Susskind defines QCC. A quantum computation would be to start with a certain given quantum state and perform some unitary operation U on this system. The QCC of the operation U would be given by the minimum number of ‘simple’ unitary gates acting on qubits [8]. This statement may seem vague and it is. The next paragraph will deal with defining them one by one.

A gate, or a quantum gate, is the quantum version of the classic logic gate. They are represented by unitary matrices and act on a certain number of qubits. All quan-tum gates have the property of being reversible. One example of a quanquan-tum gate is the Hadamard gate: it maps |0i to |0i+|1i√

2 and |1i to |0i−|1i_√ 2 and is represented by H = √1 2 " 1 1 1 −1 # (1.2)

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Introduction 3 where |0i = " 1 0 # and |1i = " 0 1 #

. It can be represented in Pauli matrices:

H = √1

2(σz+ σx)

Other relevant quantum gates for this thesis are included in Appendix A.

A quantum gate is defined as a simple gate when it acts on a small number of qubits. Susskind suggests this small number to be two qubits. A computation is then defined by letting these gates apply on pairs of qubits in a certain manner to eventually represent the unitary operation U . When using quantum gates to look at black holes, two assumptions are made [6,9]:

1. Any unitary operator U can be represented with no more than eK number of quantum gates where K represents the number of qubits in the system.

2. The scrambling time of a system is given by K log K.

Thus, for every gate applied the complexity changes and statement 1 implies that the QCC is at its highest at eK. This is the same timescale as the classical recurrence time, the time it takes a classical system to return to its original state. There is also the quantum recurrence time, the time it takes a quantum system to return to its original state. This is given by eeK [10]. Thus, at the quantum recurrence time the QCC is back at its original value.

Brown et al. conjecture a bound exists with which this complexity can grow [10,11] namely:

d dtC(e

−i ˆHt_{|ψi) ≤} 2Eψ

π¯h (1.3) This complexity growth bound (CGB) is interesting since it sets a limit on how fast the complexity of a system can change. Uncharged black holes saturate this bound while most other black holes obey the bound [10]. However, there are problems with the CGB and one of them is: cat states. Cat states are superpositions of at least two states that are macroscopically different. An example would be:

approximately sin2(θ)E, and thus, a lot smaller than E. The rate of complexification however is given by E [10]. This makes cat states a violation of the CGB.

The current thesis will approach the subject of complexity from another perspective. Where Brown et al. approach the conjecture from a theoretical viewpoint, this thesis

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Introduction 4 will simulate simpler systems and make several assumptions. These assumptions will enable concrete simulations of comparable systems and their time evolution. Then, the complexity of systems can be numerically calculated, the behavior of systems can be analysed and whether this behavior obeys the proposed theoretical bound. Ideally, this simulation also sheds some light to the main discussion points in the Brown et al. paper, like the cat states. This introduces the main questions of this thesis:

1. What conditions and assumptions are needed to make the simulation concrete? 2. What complexity growth bounds are found?

3. How would this simulation hold up containing cat states?

which should combine in giving the answer to the main question: Do the simulated systems display complexity growth bounds comparable to the bound from the ‘complexity equals action’-conjecture?

The first question will handle all the conditions and assumptions that were made in order to make the calculations achievable for this thesis. The second question will treat the results of the simulation in-depth and look if any obvious CGBs can be found. As a last, a theoretical analysis of how this simulation would treat cat states is included. All in all, these questions hopefully lead up to what CGBs are present and in what way. These bounds can then be compared to the described behavior of the original bound.

This thesis will try and dive into concrete simulations of the bound and its vio-lations by using simpler quantum systems. This research will simulate the quantum systems like 1D Ising models without an external magnetic field present. As a discrete alternative to the time evolution operator, a unitary matrix will be used in the set-ups; one with a deterministic approach to the time evolution. Comparable to the original Ising model, nearest neighbor interaction will be used and simulated, allowing only for local interaction.

The Ising models lattice will be wrapped around in space as a circle to ignore pos-sible boundary effects by allowing the left edge to be the neighbour of the right edge. Also, multiple different systems to time evolve will be simulated, from simple distribu-tions of up and down states to aligned states with single ‘perturbation’-like unaligned states. A more in-depth analysis of the approach is offered further along in this thesis. Hopefully, the results this approach offers will be helpful to the community and provide the community with hints to extrapolate to black hole dynamics.

Furthermore, this text will first explain the theory and the methods used in simulat-ing the problem. Consequently, this section will discuss the actual simulation in-depth. Then, the results of the simulations will be presented and discussed as expected.

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Chapter 2

Theory & Methods

2.1 Theory

As mentioned before, the use of quantum computational complexity has been entered into the branch of black hole physics to resolve the problem Hawking radiation seems to propose [7]. However, QCC offers interesting perspectives in other branches of black hole physics as well [10]. One example of this is that black holes are observed to have grow-ing complexity internally long after local equilibrium has been reached [12]. Susskind proposed that complexity may shed some light on the matter [8]. Thus, the study of QCC cannot only hope to offer insight on matters where the complexity of a system or a problem is of interest but offers an entire new side of information about (quantum) systems.

In this thesis, this information will be used to investigate simpler systems that may have been discussed in pre-computational complexity times. The first step is to investi-gate how the CGB came to existence. The first duality between complexity and some other property of black holes was the complexity/volume duality. It related complexity to volume as such:

C ∼ V

G` (2.1)

where ` is a length scale appropriately chosen for the analysed situation [13]. This proposed relation had some unsatisfactory elements such as why the volume would play a role in the complexity of the system [10].

Here is where the ‘complexity equals action’-conjecture comes in, where volume is swapped by action. Brown et al. define the action as the Einstein-Hilbert action, the Maxwell electromagnetic action and the York-Gibbons-Hawking action1 [10]. They also

1_{A =} 1 16πG R Mp|g|(R − 2Λ) − 1 16π R Mp|g|FµνF µν +_8πG1 R ∂Mp|h|K 5

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Theory & Methods 6 define a state for which the complexity is calculated:

|ψ(tL, tR)i = e−i(HLtL+HRtR)|T F Di (2.2)

where |T F Di is the thermofield double state and the time evolution operator acts on two boundaries at times tLand tR. When considering the geometry of black holes, this

ultimately leads to the complexity being defined by Brown et al. as follows : C(|ψ(t_L, tR)i) =

AW

π¯h (2.3)

where π acts as a normalization factor [10].

Working with the conjecture of Seth Lloyd about the physical limits of computation [14] and the Margolus-Levitin bound which gives the minimum time for a quantum state to evolve to an orthogonal state, the paper conjectures the earlier mentioned complexity growth bound:

d dtC(e

−i ˆHt_{|ψi) ≤} 2Eψ

π¯h (2.4) There is now a full set of formulas to get into the method of this thesis’ simulation.

2.2 Assumptions, choices and conditions

The actual paper deals with complicated states to simulate the black holes, with difficult time evolution. There is not a lot known and simulating the problem would be hard. This simulation will be using a 1D Ising model to provide with a simpler alternative to the black hole quantum system. The Ising model has often proved to be a useful ap-proximation of reality in different branches of physics. The Ising model has earlier been used as a mathematical model of ferromagnetism. It consists of a lattice with discrete variables (spins) that can be in two states (−1 & +1). The simplified model only allows for local interaction by allowing the spins to only interact with their nearest neighbors (no diagonal interaction).

The use of this model could give its user a lot of insight and adaptability of the system. The Ising model is easily understood and is therefore a good start for this simulation. If future quantum complexity scientists start their experiments, they may need to know about the behavior of complexity growth under influence of different initial phases or constants. This easy to grasp model may provide them with this knowledge. The model also allows for easy resizability. The lattice will be circled unto itself to avoid boundary conditions. The spins will act as qubits in this simulation. The state of the spins will be |ψi. Thus, we have a |ψi for K qubits. The next thing we need is a universal set of quantum gates.

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Theory & Methods 7 A universal set of quantum gates is a set of quantum gates for which it has been proven that each quantum state can be achieved by the set. Ergo, it is necessary to choose more than one quantum gate (unless it is universal on its own) since one would like to be able to get to every quantum state possible.

The set that was chosen exists of the Hadamard gate, the π₈ phase shift gate and the CNOT gate. All of these gates are explained in Appendix A. Call this set the universal set for now. From this set, one operator was chosen to be the time evolution operator: the CNOT gate. It is a two qubit quantum gate and flips one bit when the other bit satisfies a certain condition. Although the operator seems trivial enough, its behavior is not predicted as trivially. Randomly flipping bits might be described as thermal fluctu-ations but flipping bits only when a certain condition is met, namely the controlled bit to be |1i could prove interesting along the way.

The last thing Brown et al. require to calculate the complexity is some reference state since the complexity is defined as the minimum number of gates needed for a sys-tem to get from some arbitrary state to the current state for which one would like to calculate the complexity [10,11]. It is possible to choose this reference state to be the initial state of the evolution or to be some arbitrary state. This thesis went along with two papers about Hamiltonian complexity [15,16] and chose to calculate the complexity from some arbitrary state.

The |000...000i state was chosen as the arbitrary state from which to start. Next, the assumption was made that by applying NOT gates (see Appendix A) to this state one by one until the current state was achieved the complexity could be calculated. It was mainly assumed to be either close to the minimum number of gates or was conse-quently wrong for each calculation of the complexity. The latter would still allow for an informative reading of the rate of change of the complexity. Sadly, it was out of scope for this thesis to find and prove the minimum number of gates needed to get to the state. If, for example, someone using the simulation would like to find the complexity of the state |10011i, it would start with |00000i, apply NOT operators to the first, fourth and fifth qubit to achieve the current state and give the state complexity 3. Then, when a time evolution step was performed and the current state would be, for example |10001i, the complexity would now be 2 and dC

dt would be -1.

This approach, though tractable, has one huge drawback. It does not allow for the complexity to become extremely large. The system can flip an arbitrary large number of times and the complexity could still be zero. In fact, it never becomes larger than K, if the system has K qubits. This is a large problem and shapes a serious concern for the conclusion of the thesis. Intuitively, one could see the simulation of superpositions have a C >> K. This is treated later on. Recall assumption 1 in Chapter 1, that stated that the QCC of a system is maximal at eK at the classical recurrence time. Since the universal set can form any unitary operator U , the universal set can also reach a QCC

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Theory & Methods 8 of eK, if we use only gates from the set. Later in this text, a thought experiment with superpositions will be treated.

To ultimately investigate the behavior of the systems under CNOT-propagation, five different initial phases were used. This section will go into all the different simu-lated scenarios. This full set of scenarios could provide with knowledge of the behavior of an Ising model under time evolution of a discrete operator. The CNOT operator is an operator that works on two qubits only and thus simulates the exchange of forces or effects between two qubits. When considering locality, the distance between the con-trolled qubit and the operated-on qubit was never bigger than one qubit. For the initial phases of the systems that displayed strong boundaries or massive initial alignment, the locality factor will play a massive role in the rate of complexity change.

1. Random distribution; example: |01001i 2. 1 |1i qubit in the middle; example: |00100i 3. Fully aligned; example: |11111i

4. Half aligned; example: |000111i 5. Alternating qubits; example: |01010i

Phase 1 had qubits in both |0i and |1i distributed throughout the entire array in a random way. This phase would simulate a system which went without any external forces and without preference to any (un)alignment. Phase 2 is completely filled with |0i qubits except for the middle qubit which was in the |1i state. The middle of a circle is, ofcourse, non existent but the middle of the 2D representation was used for best vi-sualisation. This phase simulates a completely aligned state with a small perturbation. This perturbation could allow for some interesting shockwave-like behavior.

Phase 3 is completely filled with |1i qubits. This phase simulates a totally aligned system. Phase 4 has the left half filled with |0i and the right half filled with |1i. This phase was included to investigate the behavior of the time evolution on a boundary be-tween alignments while the system has a complexity of approximately K₂ for K qubits. Phase 5 is maximally unaligned with alternating qubits in every direction of the array. This phase acts as a countersystem to phase 3. The influence of the alignment of a initial phase could prove interesting to investigate.

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Theory & Methods 9

2.2.1 Time evolution

Time evolution normally is a continuous process, while the application of quantum gates is discrete in its definition. This problem stayed unaddressed in the original discussion. In order to avoid this problem, the approach was inspired by the Metropolis Hastings algorithm. This algorithm is commonly used with Ising models to investigate phase transitions and magnetization. It simulates discrete time evolution by randomly sam-pling sites and applying some set of rules to those sites. Normally, this set of rules would be something like:

– If ∆E < 0, flip.

– If ∆E > 0, flip with probability relative to temperature

This allows for the system to tend to lower energy states but still simulate fluctuations towards higher energy states. In our simulation, this set of rules is exchanged for the chosen time evolution operator. In future research, operators allowing probabilities of flipping or operators displaying other behavior could be chosen to tweak this time evo-lution. This would allow for better correlation with reality.

2.2.2 Time steps

A time step in the simulation will be defined as a product of three unitary operators: U = U1URU2 where URperforms the CNOT operation on a random site. This was done

to avoid trivial and invariant behavior. Imagine an initial state:

01001011100101010001 (2.5) U1 will be applied first. It applies the CNOT operator on pairs of qubits where the odd

qubit is the operated on bit and the even qubit is the controlled bit. This is visualized for this thesis by an arrow above the qubits.

− →

0100−→−→10−→11−10→01−→−→01−→01−00→−01→ (2.6) UR will be applied next. This samples a random site and takes the right neighbour

as the controlled bit for the CNOT operation. This operator is where the operation is inspired by the Metropolis Hastings algorithm.

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Theory & Methods 10 even qubits as the operated on bits. Thus, it shifts the entire operation one position.

0−→10−→01−01→−11→−→00−→10−10→−→10−→001

←−−−−−−−−−−−−−−−−−− (2.7) One might intuitively ask why the random sampling is necessary in the operation. This operation was added to the process since U1U2 provides cyclical time evolution

quite fast. To give an idea of how fast: the simulation was run for different numbers of qubits and the time it took to get back to the initial phase was sampled.

Table 2.1: Sampled recurrence times compared to theoretical recurrence times

No. of qubits Recurrence time in time steps Classical rec. time

4 6 54 6 15 403 8 12 2980 10 17 22026 12 30 e12 14 62 e14

Since the quantum recurrence time is eeK, the sampled times proved to be nowhere close to both recurrence times and thus were seen as unwanted behavior. Therefore, the site was randomly sampled in between the two determined operators.

To look at the energy behavior of the operator, the operator was written in operator form. For the operator itself acting on two qubits, this operator form is:

(σz+ 1

2 ) ⊗ 1 − ( σz− 1

2 ) ⊗ σx (2.8) For U1 this becomes:

X i= odd (σ (i) z + 1 2 ) ⊗ 1 (i+1)_{− (}σ (i) z − 1 2 ) ⊗ σ (i+1) x (2.9) and for U2: X i= even (σ (i) z + 1 2 ) ⊗ 1 (i+1)_{− (}σ (i) z − 1 2 ) ⊗ σ (i+1) x (2.10)

Normally, one could find the Hamiltonian operator from the time evolution operator by using U = ei ˆHt. However, for this operator this seems nontrivial. For 2.8, one can change basis by diagonalizing the matrix, taking the logarithm of the diagonal values

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Theory & Methods 11 and changing basis back. This gives:

ˆ Ht = π 2        0 0 0 0 0 0 0 0 0 0 −1 1 0 0 1 −1        (2.11)

However, taking the logarithm of a summation of two tensor products is non-trivial. The resulting matrix of the tensor products will become 2K_{× 2}K _for K

2 pairs, becoming

non-trivial.

The Taylor series expansion of log(1 + z) = z − z₂2 +z₃3 + . . . , if one rewrites: log(1

2[1 + X

i

σ_zi + σi+1_x − σi_zσ_xi+1]) (2.12)

This expansion too, becomes non-trivial even for second order. For the first order, ˆHt ∝P

iσiz+ σxi − σizσ (i+1)

x , with higher orders becoming more

complicated quickly. The property of Pauli-matrices acting on different qubits keeps terms from dropping out.

This complicated ˆH without obvious ground states or invariant states was the reason for this thesis to not force a Hamiltonian operator to the system. This however meant, not looking for2.4per se, as the CNOT propagation showed no obvious energetic behavior. This made defining the expectation value of the energy of the system obscure. As such, this thesis will mainly focus on looking for possible other bounds using the chosen systems:

d dtC(e

−i ˆHt_{|ψi) ≤} α

π (2.13)

where α is some constant or property of the system and π is included to normalize the advancement of the action-phase [14].

2.2.3 Simulation

First of all, the coding was done in C++ and the actual code is included in Appendix B. The Ising model was recreated by an array and initialising this array with the phases. For the random distribution, a system time clock seed was used providing us with a pseudo-random distribution. The other phases were predetermined and constantly set. When both qubits were determined, the CNOT operator was applied on every odd qubit for U1 and every even qubit for U2. The complexity was calculated as described before,

by applying NOT gates from the reference state |000 . . . 000i until the final state is achieved. The rate was calculated between every timestep.

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Chapter 3

Results

All simulations were comparable, except for the initial phases and the number of qubits present in the array. This should allow for great comparison between all the variables. The simulations were all run a duration of 100 time steps. Between each step, the complexity of the current system was calculated and the rate of change was determined. These numbers were then outputted as a .txt file. This file, on its part, can then be visualized. All the different scenarios were run for different numbers of qubits, namely: 10, 20 and 50 qubits. Since the complexity is dependent on the size of the system, it is important to see if the behavior is present for all sizes.

3.1 Random

The state initialised with the random distribution, for example: |0100111010i.

Figure 3.1: The complexity over time of the random distributed 10

bits

Figure 3.2: The rate of change of the complexity per timestep of the random distributed 10 bits

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Results 13 The operation seems to tend towards a maximally unaligned state, where approx-imately half is one state, and one half is the other. Since the initial phase here usually starts with this state (aligned states are uncommon), nothing unusual happened here. The complexity varied between nine and two, the rate varied between minus four and four. Keep in mind, while reading the graphs that the first measurement of rate is an anomaly: it measures the rate of complexity from the reference state to the initial phase, instead of the rate of complexity between states.

Figure 3.3: The complexity over time of all the random

distribu-tions

Figure 3.4: The normalized complexity over time of all the

random distributions

Figure 3.3 displays the complexity for the random distribution for all three array sizes. As mentioned before, the complexity is dependent on the size and this graph displays this behavior. To compare the graphs, the data was normalized in figure3.4to belong to the same number of qubits. The complexity keeps changing around K₂ for K qubits. Average rate throughout the time evolution is approximately zero.

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Results 14

3.2 1 bit

The state initialised with a single |1i in the middle, for example: |0000010000i.

Figure 3.5: The single bit distribution showing the local behavior of the time evolution

Since all the operations act local - even the random sampling needs a |1i controlled bit - and all the controlled bits are on the right side, this simulation showed onesided shockwave-like behavior as displayed in figure 3.5.

Figure 3.6: The complexity over time of all the 1 bit distributions

Figure 3.7: The normalized complexity over time of all the 1

bit distributions

The shockwave-like behavior starts with low complexity and grows over time as the entire system is reached. Once again the complexity grows towards K₂ and saturates there. The normalized complexity visually displays a faster rising for the smallest array. This, however, is a property of the normalization of the complexity. In figure 3.8 no clear difference between the rates is shown.

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Results 15

Figure 3.8: The rate of complexity for all 1 bit distributions

The rate of change shows dependence on the size of the array:

Table 3.1: Rates of change for the 1

bit distribution

No. of qubits Max. dC_dt

10 5

20 8

50 12

3.3 Fully aligned

The state initialised with all |1i bits, for example: |1111111111i.

Figure 3.9: The complexity for all the different sizes of distributions with a completely aligned initial phase

Once again, the complexity tends towards K₂. This seems to be the preferred state of the time evolution. This time, the complexity drops since the initial phase is the furthest from the reference state as possible. It may show interesting to compare dC_dt in both directions of change. Also, the rate is once again dependent on number of qubits, or maybe more accurate: distance from K₂. If this distance is large, the rate is large,

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Results 16 until it comes closer to the preferred state, where the complexity finds an equilibrium and the rate starts ‘hugging’ zero.

3.4 Half aligned

The state initialised with 1 half |1i bits, for example: |0000011111i.

Figure 3.10: The complexity for all the different sizes of distributions with two un-aligned halves

The complexity shows similar behavior as for all phases. For the large number of qubits, it takes some time to find the ‘complexity equilibrium’, namely: approximately 25 time steps for the shockwave to reach the entire array.

3.5 Alternating

The state initialised with alternating bits, for example: |0101010101i.

Figure 3.11: The complexity for all the different sizes of alternating distributions

This distribution behaved comparable to the other initial phases and no other out-of-the-ordinary results were found.

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Results 17

3.6 Dependency on size

Apart from some difference in the beginning of the time evolution, the phases show no apparent distinction from each other. Whenever they reach apparent complexity equilibrium, they grow no more and the evolution apparently prefers a state with ap-proximately half of the bits one way, and half of the bits the other way, not necessarily in one order. No stationary states were found present around this equilibrium. The operator will always flip, whatever state it is in. However, there does seem to be a dependency of the rate of complexity on the size of the array. This may seem trivial: U1 and U2 transform the entire array, and thus, larger arrays provide for larger dC_dt but

what relation it has exactly may still prove interesting. Therefore, the simulation was run again (with the single bit distribution) for more array sizes.

Figure 3.12: Size dependency for rate of complexity, with a line of best fit.

3.7 Simulation superpositions

The first part of this chapter treated the results to investigate the bounds the simulation may have had in terms of rate of complexity. These results may give an answer to the second subquestion of this thesis. To take a look at the third question, a ‘thought sim-ulation’ was described below, as it was out of scope to actually simulate superpositions. This simulation may prove interesting as the CNOT time evolution does not allow for ever increasing complexity while a simulation of superpositions may.

Assume a small array: |00i. To achieve superposition using quantum gates, one of the other gates out of the chosen universal set shall be used: the Hadamard gate (described in Appendix A).

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Results 18 Putting each qubit in a superposition:

|00i → √1

2|(|0i + |1i)(|0i + |1i)i = 1

2(|00i + |01i + |10i + |11i) (3.1) Applying the CNOT operator where the first bit will be the controlled bit:

1

2(|00i + |01i + |10i + |11i) → 1

2(|00i + |01i + |11i + |10i) (3.2) Applying the CNOT operator on the other bit will give the same result. Thus, only applying CNOT operators on superpositions will have no effect as it will always flip two states into each other and end up with the same state.

Random sampling for an array of 2 qubits is just the same as either U1 and U2

and thus has no influence. Random sampling changes the behavior for 4 qubits and it is therefore interesting to also take a look at how this behaves.

(3.3) |0000i → 1

4(|0000i + |0001i + |0010i + |0011i + |0100i + |0101i + |0110i + |0111i + |1000i + |1001i + |1010i + |1011i + |1100i + |1101i + |1110i + |1111i)

Applying U1:

(3.4) → 1

Randomly choosing the second qubits and its neighbour to operate UR on:

(3.5) → 1

Applying U2:

(3.6) → 1

Once again, we end up with the same state and thus, the chosen time evolution does not allow for complexification anymore than the initial phase allows. However, if one chose a time evolution including Hadamard gates instead of CNOT operators, this may allow for complexification and is interesting for future research.

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Chapter 4

Conclusion

The minimum requirements one needs to define to calculate the complexity is a system, a universal set of quantum gates to time evolve the system and some reference state. For this particular simulation, the 1D Ising model was chosen as the system to evolve. The system was evolved with the CNOT operator from the universal set of quantum gates consisting of the CNOT gate, the Hadamard gate and the π₈ phase shift gate. The time evolution operator consisted of three different steps of applying the CNOT operator, inspired by the Metropolis Hastings algorithm: U1 applying on the odd bits,

UR applying one random bit to avoid cyclical behavior and U2 applies the operator on

all even bits.

The chosen reference state was the |000 . . . 000i state. This had a reason: it is the only state invariant under time evolution and therefore seemed a good state to take as a simple state. The next, maybe biggest assumption for this thesis, was calculating the complexity. In theory, one needs to find the minimum number of quantum gates to get from some |ψrefi to |ψi. However, finding and proving the minimum number of gates

was out of scope for this thesis and thus, an approach to calculating C was assumed to be close enough.

Now, the results of the simulation: for every initial phase the time evolution be-havior was very comparable. It tended from whatever initial phase to a state where approximately K₂ for K qubits was |0i and the other half was |1i. When the initial phase was already close to this state, the dC_dt became small and started varying around this ‘equilibrium of complexity’. However, no invariant states were found around this apparent equilibrium. This non-existence stems from both the random sampling URand

the array-wide evolution that U1U2 provides. The only invariant state was the reference

state, which was as far from the equilibrium as the system allowed.

Interestingly, the local nature of the time evolution gave some parts of the simula-tion where a clear rising of complexity towards saturasimula-tion point was visible. This rising

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Conclusion 20 was proportional to the size of the array and very similar when normalized (see fig. 3.7).

dC

dt ∝ Number of qubits (4.1) where some kind of bound on this rate is found by the local nature of the operator.

Interesting behavior was found when looking at initial phases far from equilibrium of complexity. Even when the shockwave had not yet had time to reach the entire system, the complexity had already settled around its K₂ point. The size dependency of the rate was naively sampled. Intuitively, this dependence exists -more qubits means more can change and thus the complexity can change quicker-, however, the sampling did show promise of a tighter bound.

To conclude the second question, one bound can be mentioned. When there are K qubits in the system, there are no unexpected interactions under the chosen time evolution. This means that at best, the rate of complexity can flip all qubits at the same time:

d dtC(e

−i ˆHt_{|ψi) ≤ K}

(4.2) In a naive reading, this bound is comparable to the conjectured bound where the expec-tation value of the energy of the system bounds the rate of complexity since the number of qubits in a system have some correlation to Eψ. However, the bound is also intuitive

and inherent to the simulation. Tighter bounds were not proven in this thesis, but as mentioned before, are suggested by findings that it takes approximately K₂ time steps to get to the ‘complexity equilibrium’ at K₂ and the sampling of the size dependency of the rate. Furthermore, the systems did not show any complexification after the system was ‘scrambled’ and the simulation is therefore not a good simulation of the complexification present in black holes. However, the complexification before local equilibrium was met, did show behavior proportional to the number of qubits in the system and the distance the current phase had from the phase with approximately K₂ bits |0i and the other half |1i. Thus, bounds are expected to include both K and the distance from the current state to the point of ‘complexity equilibrium’.

Last, a simple analysis of how this system and time evolution would behave with superpositions as initial phase showed that the time evolution was completely invariant. Therefore, it is not a good simulation of the behavior of cat states in black holes and cannot shed light on its complexity growth bound violation. Other time evolutions in future research may prove better simulations of reality for cat states.

All in all, these questions sum up to the answer of the lead question. The simula-tion, as it stands, does show some bounds comparable to the bounds in the ’complexity equals action’-conjecture but further research is needed to tighten and prove this bound.

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Chapter 5

Discussion

Next to the several options for the future research already mentioned in the conclusion, further research can focus on finding a time evolution with better energetic behavior. If the energy is clearly defined, one can simulate systems and see if they obey the conjectured bound. Using operators only flipping for a certain probability may help with this. Another thing future research can focus on is the calculation of the complexity. If the actual minimum of gates is found and proved, this may show for completely different numbers as were found in this thesis.

Last, to improve the simulation and have more comparable behavior to black holes, it is important to have a complexity that can become extremely large. For the current simulation, the complexity could never exceed K. Allowing the initial phase to consist entirely of superpositions provided with invariant states and thus, a rate of complexity of zero. A suggestion to allow for large complexity could be an initial phase consisting of both product states and superpositions. This initial phase may have both the variant behavior of the product states and the complexifying behavior that superpositions would have in our definition of calculating complexity. If this suggestion –or other initial phases– indeed allows for the complexity to exceed K, simulations would start being more similar to complexity behavior of black holes.

The starting point for this simulation was to provide future researchers with a database of knowledge about the simulation of the ‘complexity equals action’-conjecture. If a better time evolution and a good way to compute the energy and complexity for the Ising system were to be found, this simulation could very well prove to be useful for the calculations around this conjecture as it is a very tractable and easy to understand simulation of reality.

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Appendix A

Quantum gates

The following is a short summary of some well known quantum gates. It is meant as a short introduction for the unfamiliar reader to avoid having to investigate the subject matter outside this thesis.

Assume the vector representation of a single qubit to be

a |0i + b |1i → "

a b #

and that of two qubits to be

a |00i + b |01i + c |10i + d |11i →        a b c d       

A.1 Hadamard gate

The Hadamard gate maps

|0i → |0i + |1i√ 2 |1i → |0i − |1i√

2 It is represented in matrix notation by: √1

2

" 1 1 1 −1

#

and in Pauli matrices by:

1 √

2(σz+ σx)

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Quantum gates 23

A.2 Pauli-X gate

The Pauli-X gate maps

|0i → |1i |1i → |0i It is represented in matrix notation by:

" 0 1 1 0 #

and in Pauli matrices by: σx

A.3 Phase shift gate

The π₈ phase shift gate maps

|0i → |0i |1i → eiπ4 |1i

It is represented in matrix notation by: "

1 0 0 eiπ4

#

A.4 Controlled Not gate

The CNOT gate maps as follows:

|00i → |00i |01i → |01i |10i → |11i |11i → |10i

Hence, the controlled bit has to be |1i for the other bit to flip.

It is represented in matrix notation by:        1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0       

and in Pauli matrices by:

(σz+ 1

2 ) ⊗ 1 − ( σz− 1

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Appendix B

Simulations code

In this appendix, the code used for the simulation is included for the interested reader. The full code below will function for the random distribution. For the four other distri-butions, only the line-by-line changes to the code will be included.

B.1 Random distribution

1 # i n c l u d e < i o s t r e a m >

2 # i n c l u d e < cstdlib > // for r a n d () and s r a n d ()

3 # i n c l u d e < ctime > // for t i m e () 4 # i n c l u d e < cmath > // for f l o o r 5 # i n c l u d e < fstream > // for f i l e o u t p u t 6 7 u s i n g n a m e s p a c e std ; 8 9 int a r r a y S i z e =6;

10 int * a =new int[ a r r a y S i z e ];

11 int i ; 12 int o B i t =0; 13 int c B i t =0; 14 int p r e v C o m p l e x =0; 15 int c u r r e n t C o m p l e x =0; 16

17 // G e n e r a t e a r a n d o m n u m b e r b e t w e e n min and max ( i n c l u s i v e )

18 // A s s u m e s s r a n d () has a l r e a d y b e e n c a l l e d

19 int g e t R a n d o m N u m b e r (int min , int max )

20 {

21 s t a t i c c o n s t d o u b l e f r a c t i o n = 1 . 0 / ( s t a t i c _ c a s t <double>( R A N D _ M A X ) + 1 . 0 ) ;

22 // s t a t i c u s e d for e f f i c i e n c y , so we o n l y c a l c u l a t e t h i s v a l u e o n c e

23 // e v e n l y d i s t r i b u t e the r a n d o m n u m b e r a c r o s s our r a n g e

24 r e t u r n s t a t i c _ c a s t <int>( r a n d ()* f r a c t i o n *( max - min + 1 ) + min );

25 }

26

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Simulations code 25

27 f l o a t i n t e g e r R a n d (int min , int max , f l o a t l i m i t )

28 // P r o d u c e r a n d o m i n t e g e r s to f i l l a r r a y w i t h 29 { 30 f l o a t z ; 31 z = g e t R a n d o m N u m b e r ( min , max ); 32 if( z <= l i m i t ) 33 { 34 z =0; 35 } 36 e l s e 37 { 38 z =1; 39 } 40 r e t u r n z ; 41 } 42

43 f l o a t p l a c e R a n d (int min , int max ) // F i n d r a n d o m i n t e g e r s to s a m p l e s i t e s

44 { 45 f l o a t z ; 46 z = g e t R a n d o m N u m b e r ( min , max ); 47 f l o o r ( z ); 48 r e t u r n z ; 49 } 50 51 int a r r a y (int x ) 52 // I n i t i a t e a r r a y w i t h 0 and 1 e q u a l l y d i s t r i b u t e d . C h a n g e 0 . 4 9 to c h a n g e d i s t r i b u t i o n . 53 { 54 for ( i =0; i < x ; i ++) 55 { 56 a [ i ]= i n t e g e r R a n d (0 , 1 ,0. 49 ); 57 } 58 r e t u r n 0; 59 } 60 61 int a r r a y P r i n t e r (int x ) // P r i n t a r r a y 62 { 63 for ( i =0; i < x ; i ++) 64 { 65 c o u t < < a [ i ]; 66 } 67 r e t u r n 0; 68 } 69

70 int c o u n t (int x ) // C o u n t c o m p l e x i t y w i t h NOT g a t e s

71 { 72 int r =0; 73 74 for ( i =0; i < x ; i ++) 75 { 76 r = r + a [ i ]; 77 } 78 r e t u r n r ; 79 } 80

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Simulations code 26 82 { 83 if( a [ c o n t r o l ] = = 1 & & a [ o p t o r ] = = 1 ) 84 { 85 a [ o p t o r ] = 0 ; 86 } 87 e l s e if( a [ c o n t r o l ] = = 1 & & a [ o p t o r ] = = 0 ) 88 { 89 a [ o p t o r ] = 1 ; 90 } 91 } 92 93 v o i d s i t e S a m p l e r (int X o B i t ) // F i n d l o c a t i o n to s a m p l e 94 { 95 o B i t = X o B i t ; // G i v e o B i t l o c a t i o n 96 c B i t = o B i t +1; // C h e c k r i g h t n e i g h b o u r of o B i t 97 if( cBit > a r r a y S i z e -1) // A v o i d b o u n d a r y c o n d i t i o n s by a l l o w i n g s p a c e to c i r c l e 98 { 99 c B i t =0; 100 } 101 } 102 103 v o i d s i t e S a m p l e r R a n d o m () // F i n d l o c a t i o n to s a m p l e 104 { 105 o B i t = p l a c e R a n d (0 , a r r a y S i z e - 1 ) ; // -1 s i n c e a r r a y s t a r t s at 0 106 c B i t = o B i t +1; // C h e c k r i g h t n e i g h b o u r s i n c e l o c a l 107 if( cBit > a r r a y S i z e -1) // A v o i d b o u n d a r y c o n d i t i o n s by a l l o w i n g s p a c e to c i r c l e 108 { 109 c B i t =0; 110 } 111 } 112 113 int r a t e C o m p l e x i t y (int n e w C o m p l e x ) 114 { 115 p r e v C o m p l e x = c u r r e n t C o m p l e x ; 116 c u r r e n t C o m p l e x = n e w C o m p l e x ; 117 int r a t e = c u r r e n t C o m p l e x - p r e v C o m p l e x ; 118 r e t u r n r a t e ; 119 } 120 121 int m a i n () 122 { 123 s r a n d ( s t a t i c _ c a s t < u n s i g n e d int>( t i m e ( 0 ) ) ) ; 124 // set i n i t i a l s e e d v a l u e to s y s t e m c l o c k 125 a r r a y ( a r r a y S i z e ); // I n i t i a t e a r r a y to set s i z e 126 o f s t r e a m t e x t f i l e ; // M a k e t e x t f i l e v a r i a b l e 127 t e x t f i l e . o p e n (" r a n d o m _ o u t p u t . txt "); // O p e n f i l e 128 t e x t f i l e < < " C o m p l e x i t y : "; // P r i n t l i n e t i t l e to f i l e 129 t e x t f i l e < < " R a t e : " < < e n d l ; // P r i n t l i n e t i t l e to f i l e 130 for (int d =0; d < 2 0 ; d ++) // D e p t h of q u a n t u m c i r c u i t 131 { 132 a r r a y P r i n t e r ( a r r a y S i z e ); // P r i n t s a i d a r r a y 133 c o u t < < e n d l ; 134 t e x t f i l e < < c o u n t ( a r r a y S i z e ) < < " , "; // P r i n t c o m p l e x i t y to f i l e 135 t e x t f i l e < < r a t e C o m p l e x i t y ( c o u n t ( a r r a y S i z e )) < < e n d l ; 136 // P r i n t r a t e of c o m p l e x i t y to f i l e

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Simulations code 27

137 for (int w =0; w < a r r a y S i z e ; w ++)

138 // P e r f o r m for all odd b i t s U_1 , e v e n b i t s in c ++ s i n c e a r r a y s s t a r t at 0

139 {

140 if( w % 2 = = 0 )

141 {

142 s i t e S a m p l e r ( w ); // T h i s d e t e r m i n e s the o B i t and the c B i t

143 b i t F l i p p e r ( cBit , o B i t ); // T h i s p e r f o r m s the o p e r a t o r

144 }

145 }

146 for (int u =0; u <1; u ++) // P e r f o r m s a m p l i n g and f l i p t h e m a p p r o p r i a t e l y

147 {

148 s i t e S a m p l e r R a n d o m ();

149 // T h i s d e t e r m i n e s the o B i t and the cBit ,

150 // it a l l o w s for the s a m e bit to be s a m p l e d t w i c e

152 }

153 for (int v =0; v < a r r a y S i z e ; v ++)

154 // P e r f o r m for all e v e n b i t s U_2 , odd b i t s in c ++ s i n c e a r r a y s s t a r t at 0

155 {

156 if( v % 2 = = 1 )

157 {

158 s i t e S a m p l e r ( v ); // T h i s d e t e r m i n e s the o B i t and the c B i t

160 }

161 }

162 }

163 t e x t f i l e . c l o s e (); // C l o s e the b i t s t r e a m for the f i l e

164 r e t u r n 0;

165 }

B.2 1 bit in the middle

The function that initialises the array on line 51-59 is replaced by the following:

1 int a r r a y (int x ) 2 { 3 for ( i = 0; i < x ; i ++) 4 { 5 a [ i ] = 0; 6 } 7 int mid = f l o o r ( a r r a y S i z e / 2 ) ; 8 a [ mid ] = 1; 9 }

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Simulations code 28

B.3 Fully aligned

1 int a r r a y (int x ) 2 { 3 for ( i = 0; i < x ; i ++) 4 { 5 a [ i ] = 1; 6 } 7 }

B.4 Half aligned

1 int a r r a y (int x ) 2 { 3 for ( i = 0; i < x ; i ++) 4 { 5 if ( i <( a r r a y S i z e / 2 ) ) 6 a [ i ] = 0 ; 7 e l s e 8 a [ i ] = 1 ; 9 } 10 }

B.5 Alternating

1 int a r r a y (int x ) 2 { 3 for ( i = 0; i < x ; i ++) 4 { 5 a [ i ] = 1; 6 if ( i % 2 = = 0 ) 7 a [ i ] = 0 ; 8 } 9 }

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