Quantum Cellular Automata: Order andRelaxation

Diplomarbeit von

Alexander Kettler

8. Marz 2008

Hauptberichter: Prof. Dr. Gunter Mahler

Mitberichter: Prof. Dr. Ulrich Weiß

1. Institut fur Theoretische Physik

Universitat Stuttgart

Pfaffenwaldring 57, 70550 Stuttgart

Ehrenwortliche Erklarung

Ich erklare, daß ich diese Arbeit selbstandig verfaßt und keine anderen als die angegebe-nen Quellen und Hilfsmittel benutzt habe.

Stuttgart, 8. Marz 2008 Alexander Kettler

Contents

1. Introduction 11.1. Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Basic Concepts 32.1. Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1. Postulates and mathematical formalism . . . . . . . . . . . . . . . 32.1.2. 2-level systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3. Unitary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.4. Composed systems . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.5. Density operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.6. Networks of two-level systems . . . . . . . . . . . . . . . . . . . . 72.1.7. Calculating the time evolution . . . . . . . . . . . . . . . . . . . . 8

2.2. Classical Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . 92.3. Quantum Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1. Linear QCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2. Colored QCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.3. Margolus QCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3. Relaxation of MQCA with 2-qubit block partitioning 153.1. Trivial rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2. Rules with Uloc : {|01〉 , |10〉} → {|01〉 , |10〉} (excitation number conserv-

ing rules) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3. Rules with Uloc : {|00〉 , |11〉} → {|00〉 , |11〉} . . . . . . . . . . . . . . . . 293.4. Rules with Uloc : {|00〉 , |11〉} → {|00〉 , |11〉}, {|01〉 , |10〉} → {|01〉 , |10〉} . 333.5. Total mixing rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4. Relaxation of a two species LQCA 414.1. Trivial rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2. Non-propagating rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3. Rules with γ01 = γ10 = 0, γ00 6= 0, γ11 6= 0 . . . . . . . . . . . . . . . . . 464.4. Left propagating or right propagating rules . . . . . . . . . . . . . . . . . 494.5. left and right propagating rules . . . . . . . . . . . . . . . . . . . . . . . 514.6. Total mixing rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

v

Contents

4.6.1. MQCA equivalent rules . . . . . . . . . . . . . . . . . . . . . . . . 574.7. Summary and Comparison of Chapter 3 and 4 . . . . . . . . . . . . . . . 57

5. Relaxation control with a LQCA 595.0.1. Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.0.2. Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6. Simulating spin chain dynamics through QCA dynamics 656.1. Simulation of spin chains by MQCA . . . . . . . . . . . . . . . . . . . . . 65

7. Summary and outlook 677.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.2. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

A. Appendix 69A.1. Generators of SU(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69A.2. A note on the numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Bibliography 71

Danksagung 73

vi

1. Introduction

1.1. Historical Background

In the recent years quantum cellular automata (QCA) have attracted much attentionfrom the scientific community as a new model for quantum computation. QCA are thequantum extension to the concept of classical cellular automata (CCA), a model forparallel computing on a lattice of cells with discrete states, which was introduced in theearly 1950s by John von Neumann and Stanislav Ulam [20] and became widely knownwith the invention of the “Game of Life” (a 2-state, 2-dimensional cellular automaton)by John Conway in the 1970s. In 1983 Stephen Wolfram started to systematicallyinvestigate classical cellular automata [22], [9], [23] (to mention only a few publications.For a more complete listing see 1) and it soon became clear that some of them (amongothers the Game of Life) can emulate a universal Turing machine.With this in mind, it seems very promising to extend the principles of CCA into thequantum regime. An initial approach to this task was first proposed in 1982 by RichardFeynmann [5] and today there exists a wide range of different QCA models [21], [18],[3], [8], [11], [7], [12], some of them capable of universal quantum computing.

1.2. Motivation

Most research on QCA that has been done so far, has mainly been focused on quantuminformation processing aspects. However, it can be very interesting to look at problemsof quantum information processing from the view of quantum thermodynamics [6]. Thepurpose of this work will be to address some thermodynamic aspects of two differentmodels of QCA, the LQCA [8] and the MQCA [7]. In particular, it will be investigated,how the states of small subsystems of the automaton (consisting of a small number ofcells) will develop in time. Does one observe some kind of thermodynamic behaviouremerging from the dynamics that result from the application of the update rule of theautomaton? Can one expect a relaxation of the local subsystems into some local equi-librium state? And if so, under what conditions? Can one give a classification of theQCA with respect to this behaviour?Another interesting question, when it comes to quantum computing, is the control ofthe natural relaxation of coupled quantum systems [15]. It is therefore an interesting

1http://www.stephenwolfram.com/publications/articles/ca/

1

1. Introduction

question, to what extend this relaxation could be controlled by applying QCA updaterules on the system.

Classical Cellular Automaton (CCA)Quantum Cellular Automaton (QCA)Linear Quantum Cellular Automaton (LQCA)Margolus Quantum Cellular Automaton (MQCA)Colored Quantum Cellular Automaton (CQCA)

Table 1.1.: List of abbreviations used in this work for different Models of Cellular Au-tomata (CA). These models will be presented in short in the next chapter.

2

2. Basic Concepts

2.1. Quantum Mechanics

2.1.1. Postulates and mathematical formalism

The quantum mechanical description of physical systems can be based on 6 postulateswhich will be presented here in short, mainly following [2].

1st postulate : The state of a physical system is described by a state vector |ψ〉 of unitnorm, belonging to a Hilbert space H, which is a complex, linear, unitary, separable andcomplete vector space.

2nd postulate : For every observable A there exists a linear, hermitian operator Aacting in H. The eigenvectors of A form a complete orthonormal basis of H

3rd postulate : The only possible outcomes of a measurement of A are given by theeigenvalues ak of the corresponding operator A.

4th postulate : The outcome of the measurement is completely random with the propa-bility of obtaining the eigenvalue ak given by

w(ak) =

gk∑

i=1

|〈a(i)k |ψ〉|2 (2.1)

where gk denotes the degeneracy of ak and 〈a(i)k | denotes the eigenvectors associated with

ak. The expectation value of A is given by

〈A〉 =∑

k

akw(ak) = 〈ψ|A|ψ〉 (2.2)

5th postulate : After the measurement of A, |ψ〉 collapses into the projection onto the

3

2. Basic Concepts

subspace associated to the eigenvalue ak that has been observed:

|ψ〉 −→ Pk |ψ〉√

〈ψ|Pk|ψ〉(2.3)

Pk =

gk∑

i=1

|a(i)k 〉 〈a(i)

k | (2.4)

As one can see from (2.3), the measurements of two observables A and B will mutuallydisturb each other as long as their corresponding operators don’t have the same set ofeigenvectors (they commute). This effect ist described by the generalized Heisenberguncertainity relation:

∆A∆B ≥ 1

2|〈ψ|[A, B]−|ψ〉| (2.5)

6th postulate : The time evolution of the state vector |ψ〉 is given by

i~∂

∂t|ψ〉 = H(t) |ψ〉 , (2.6)

where H(t) denotes the Hamilton operator corresponding to the total energy of the sys-tem. (2.6) is also known as the (time dependent) “Schrodinger Equation”.

2.1.2. 2-level systems

A 2-level system is the simplest example for a quantum system, its Hilbertspace beingonly of dimension 2. Thus the state of the system can be described by a state vector|ψ〉 = c1 |1〉 + c0 |0〉 where |1〉 and |0〉 are the eigenvectors of a 2-dimensional hermitianoperator A with the eigenvalues a1 and a0. It is convenient to choose the eigenvectorsof one of the Pauli-matrices

σx =

(

0 11 0

)

σy =

(

0 −ii 0

)

σz =

(

1 00 −1

)

(2.7)

(usually σz) as a basis. For A=σz it is

|1〉 =

(

10

)

|0〉 =

(

01

)

(2.8)

a1 = 1 a0 = −1 (2.9)

The propability p1 of finding the system in the state |1〉 is given by

p1 = |〈1|ψ〉|2 = 〈ψ|P1|ψ〉 = |c1|2 P1 = |1〉 〈1| (2.10)

4

2.1. Quantum Mechanics

2.1.3. Unitary operators

An operator U is unitary, by definition, if its adjoint U † is equal to its inverse U−1:

U †U = U U † = 1 (2.11)

Unitary operators conserve the scalarproduct between two arbitrary vectors |ϕ〉,|ψ〉 (andtherefore also the norm of a vector):

〈Uϕ|Uψ〉 = 〈ϕ|U †U |ψ〉 = 〈ϕ|ψ〉 (2.12)

An example of a 2-dimensional unitary operator is the rotation matrix in the state spaceof a spin-1

2particle:

R(α, β, γ) =

(

e1

2i(α−β) cos γ

2e

1

2i(α+β) sin γ

2

−e− 1

2i(α+β) sin γ

2e−

1

2i(α−β) cos γ

2

)

(2.13)

Unitary operators can be constructed by using hermitian operators: V = eiA is unitary,if A is a hermitian operator (A† = A), because

V †V = e−iA†

eiA = e−iAeiA = 1 (2.14)

V V † = eiAe−iA†

= eiAe−iA = 1 (2.15)

The group of all n × n unitary matrices with the group operation being the matrixmultiplication is called the unitary group of degree n, U(n).The subgroup of U(n) consisting of all n × n unitary matrices with determinant 1 iscalled the special unitary group of degree n, SU(n).

2.1.4. Composed systems

A physical system (3) may be composed of two separate systems (1) and (2). In this casethe Hilbert space of the total system is given by the tensor product of the two partialsystems:

H(3) = H(1) ⊗H(2) (2.16)

and is spanned by the tensor product of the basis vectors |ϕ(1)i 〉 and |ϕ(2)

j 〉 of H(1) and

H(2):

|ϕ(3)ij 〉 = |ϕ(1)

i 〉 ⊗ |ϕ(2)j 〉 =: |ϕ(1)

i , ϕ(2)j 〉 . (2.17)

Only in special cases (e.g. if the two systems have not interacted), the total state vectorcan be written as a tensor product of a vector |χ(1)〉 out of H(1) and a vector |ζ(2)〉 outof H(2) (product state):

|ψ(3)〉 = |χ(1)〉 ⊗ |ζ(2)〉 (2.18)

5

2. Basic Concepts

otherwise it must be written in the general form (entangled state):

|ψ(3)〉 =∑

i,j

cij |ϕ(1)i 〉 ⊗ |ϕ(2)

j 〉 (2.19)

An operator A(1) acting only in (1) is written in the basis of (3) as

A(3) = A(1) ⊗ 1(2). (2.20)

2.1.5. Density operator

Often the system under consideration is only part of a composed larger system. Thus,the local state of the system is not perfectly well known but rather one has to deal with astatistical mixture of states |ψ1〉 , |ψ2〉 , . . . with propabilities p1, p2, . . . It is important tonote that this differs from the system being in the state |ψ〉 =

√p1 |ψ1〉+

√p2 |ψ2〉+ . . .

as this would describe a coherent state. To get a general description which can describeboth cases one has to introduce the density operator

ρ =∑

k

pk |ψk〉 〈ψk| (2.21)

The density operator is a hermitian operator (ρ† = ρ) with Tr[ρ] = 1, represented in anarbitrary basis {|ei〉} by a square matrix whose elements are given by

ρij = 〈ei|ρ|ej〉 . (2.22)

The expectation value for an observable A is given by

〈A〉 = Tr (Aρ). (2.23)

An important quantity of ρ is the purity

P = Tr[ρ2], (2.24)

which takes on its maximum value (P = 1) for pure states and its minimum value(P = 1

d) for a totally mixed state (ρij = 1

dδij) of dimension d. P is invariant under

unitary transformations that act only on the local system described by ρ.

Partial trace operation

To obtain the reduced density operator ρ(1) for a system (1) which is part of a globalsystem (1) + (2) with the density operator ρ, one has to perform a partial trace operationon system (2):

ρ(1) = Tr2 ρ =∑

i,i′

∑

k

〈ϕ(1)i , ϕ

(2)k | ρ |ϕ(1)

i′ , ϕ(2)k 〉 |ϕ(1)

i 〉 〈ϕ(1)i′ | (2.25)

6

2.1. Quantum Mechanics

As an example from quantum thermodynamics, consider a system in thermodynamicequilibrium in contact with a reservoir of inverse temperature β. The density matrix inthe energy eigenbasis {|n〉} of the system (En = H |n〉) has the following form [6]:

ρ =1

Z

∑

n

e−βEn |n〉 〈n| (2.26)

with the partition sum

Z =∑

n

e−βEn . (2.27)

2.1.6. Networks of two-level systems

All systems that will be dealt with in this work are composed systems, consisting of afinite number N of two-level systems (in the following often to be reffered to as “cells”).The Hilbert space for each single two-level system will be spanned by the eigenbasis ofthe σz-operator ({|1〉 , |0〉}). The state |ψ〉 of the total system is assumed to be pure(ρ = |ψ〉 〈ψ|) and will be written as

|ψ〉 =∑

s(1),...,s(N)

cs(1),...,s(N) |s(1)〉⊗|s(2)〉⊗· · ·⊗|s(N)〉 , s(i) ∈ {1, 0}, i ∈ {1, . . . , N}

(2.28)or in its short form:

|ψ〉 =111...1∑

s=0...000

cs |s〉 (2.29)

where s represents a N-digit binary number. The propability p1(i) of finding the two-levelsystem at position i in the excited state (|1〉) is given by

p1(i) = 〈ψ|P (i)1 |ψ〉 , (2.30)

with P1 as defined in eq. (2.10). Numbers in brackets (i, j, ...) at the upper right of anoperator indicate that the operator is to be applied on the local subsytem consisting ofthe two-level system(s) located at the position(s) indicated by i, j, . . .In analog, the reduced density matrix for a subsystem consisting of the two-level sys-tem(s) at position(s) (i, j, . . . ) will be denoted by ρ(i,j,...) and is obtained by tracing outall other two-level systems.An interesting quantity that measures the overall entanglement of all two-level systemswith each other is the multi particle entanglement measure defined in [10], which can bewritten in the form:

Q =1

N

N∑

i=1

2(1 − Tr[(ρ(i))2]) (2.31)

Q can be seen as a normalized measure of the sum of all purities. Q takes on its min-imum (Q=0), if all local states are completely pure (Tr[(ρ(i))2] = 1) and its maximum

7

2. Basic Concepts

(Q=1), if all local states are totally mixed (Tr[(ρ(i))2] = 12).

2.1.7. Calculating the time evolution

To calculate the time evolution of the state |ψ〉, one has to solve the time dependentSchrodinger Equation. In the following, some methods to do this will be sketched.

Exact solution

In the case of a time independent Hamilton operator, the Schrodinger Equation has thefollowing solution:

|ψ(t)〉 = U(t) |ψ(t = 0)〉 (2.32)

with the time evolution operator

U(t) = e−t i

~H . (2.33)

Thus, for not too big systems, |ψ(t)〉 can be calculated by diagonalizing H and calculatingthe exponential operator U(t).

Iterative approximation procedures

For bigger systems or systems with time dependent Hamiltonian exact diagonalizationworks only in very few special cases. Thus one needs approximation procedures. Thereare several methods to integrate the differential equation for |ψ〉 numerically, like theRunge-Kutta or the Fehlberg algorithm, which are more or less standard procedures.In the following will be presented another method, which was used for most of thecalculations done in this work.

Suzuki-Trotter Decomposition

The Suzuki-Trotter decomposition [17] is useful if exact diagonalization of the Hamiltonoperator H is not possible, but H can be split up into H = HA + HB, where HA and HB

can be diagonalized separately. The time evolution operator can then be approximatedby

exp[−t i~(HA + HB)] = lim

N→∞

(

exp[−∆ti

~(HA + HB)]

)N

= limN→∞

(

exp[−∆ti

~HA] · exp[−∆t

i

~HB]

)N

≈(

exp[−∆ti

~HA] · exp[−∆t

i

~HB]

)N

, ∆t =t

N(2.34)

8

2.2. Classical Cellular Automata

with an overall error of order 1/N . This can be improved by symmetrizing the incre-mental propagator

exp[−∆ti

~(HA + HB)] ≈ exp[−∆t

2

i

~HB] · exp[−∆t

2

i

~HA] · exp[−∆t

2

i

~HB] (2.35)

which results in an overall error of order 1/N2. One of the advantages of the Suzuki-Trotter method is that it conserves the norm (symplectic integrator).

2.2. Classical Cellular Automata

In the following, a definition of classical cellular automata will be given, according tovarious definitions found in literature:

Definition: A classical cellular automaton (CCA) consists of an (usually infinite) d-dimensional lattice of cells indexed by ~x ∈ Z

d, a finite neighborhood scheme {~ni} ⊂ Zd, a

local update rule f and a set of discrete cell states Σ. Each cell posesses a state S~x ∈ Σ.Time evolves in discrete timesteps. At each discrete timestep the state of each cell isupdated according to S~x(t+ 1) = f(S~x(t), {S~x+~ni

(t)}) which gives the new state of a cellas a function of its own state and the state of all cells in its neighborhood defined by{~ni}.

Usually, instead of using an infinite grid of cells, periodic boundary conditions are ap-plied. In this work, the grid will always be limited to a fixed size with periodic ornon-periodic boundaries.To illustrate this rather mathematical definition, an example of a one-dimensional CCAwill be given:

Example: Consider a one-dimensional CCA with possible cell states 0 (white/inactive)and 1 (black/active) and a neighborhood scheme that consists only of the nearest neigh-bors. Thus 256 different local update rules can be implemented. One possible rule couldlook like this:

Sx−1, Sx, Sx+1 111 110 101 100 011 010 001 000f(Sx, Sx−1, Sx+1) 1 0 0 1 0 1 1 0

This rule would be numbered as rule 150 according to Wolfram, reading the second lineas a binary number. By running the automaton with this rule on an initial state whereall cells except for one cell are in state 0, one obtains the pattern displayed in figure 2.1.

9

2. Basic Concepts

Figure 2.1.: Classical cellular automaton with rule 150

2.3. Quantum Cellular Automata

An intuitive way of introducing a quantum cellular automaton would be to identifythe lattice of cells with an array of qubits and the local update rule with a unitarytransformation on each qubit depending on the state of the qubits in its neigborhood.But when extending the concept of the classical cellular automaton into the quantumregime, one has to deal with the problem that a classical CA needs to memorize itscurrent state during the calculation of the subsequent state because the next value of acell depends on the values of the surrounding cells which will change their state duringthe calculation, that means one has to make a copy of the current state. This howeveris not possible for a quantum mechanical state due to the non-cloning theorem.As there exist several ways to circumvent this problem by introducing a partitioningscheme, many different models of QCA have been proposed, like the Watrous-Van-DamQCA [21], [18], [3], the Linear QCA [8], the Colored QCA [11], the Margolous QCA [7]or the Local Unitary QCA [12]. Some of them will be described in the following sections.

2.3.1. Linear QCA

This model was first proposed in 1993 by S. Lloyd [8] and is often referred to as SpinChain QCA. It consists of a one-dimensional chain of two-level systems (cells) withnearest neighbor interactions, partitioned into three different species, i.e. ABCABC . . .with different energy splittings for each species. The energy levels of each cell are shiftedas a function of the energy levels of its neighbors. This results in different resonantfrequencies ωA

00, ωA01, ω

A10, ω

A11 for A depending on whether C and B are in the state 0 and

0, 0 and 1, 1 and 0 or 1 and 1. By applying the adequate sequence of pulses, all cells of

10

2.3. Quantum Cellular Automata

a given species are updated in parallel with the local unitary update rule

Uloc = |0〉 〈0| ⊗ U00 ⊗ |0〉 〈0|+ (2.36)

|0〉 〈0| ⊗ U01 ⊗ |1〉 〈1|+|1〉 〈1| ⊗ U10 ⊗ |0〉 〈0|+|1〉 〈1| ⊗ U11 ⊗ |1〉 〈1|

which applies the unitary U00, U01, U10, U11 on a spin, depending on the state of its leftand right neighbors (see figure 2.2).

If one allows for different coupling strengths of the cell with its right and its left neigh-bor, respectively a partitioning into 2 different species i.e. ABAB . . . is sufficient. Theevolution of such an automaton with an even number of N cells and an initial state |ψ0〉can than be described as follows:

Let UA/B be the unitary, that updates all cells of the type A or B depending on thestate of their neighbors:

UA = U(1,2,3)loc U

(3,4,5)loc · · · U (N−2,N−1,N)

loc (2.37)

UB = U(2,3,4)loc U

(4,5,6)loc · · · U (N−1,N,1)

loc (2.38)

then the state ψ of the automaton after 2t steps is given by:

|ψ〉 = (UBUA)t |ψ0〉 (2.39)

Figure 2.3 illustrates the functioning of such a two-species linear QCA.

Figure 2.2.: Function of a 3-species linear QCA

11

2. Basic Concepts

Figure 2.3.: Function of a two-species linear QCA

2.3.2. Colored QCA

A generalization of Lloyd’s scheme for a partitioning into an arbitrary number ≥ 2 ofspecies (here: colors) for dimensions ≥ 1 has been introduced in [11] and is called ColoredQCA (CQCA). In a CQCA, each cell is assigned a color in a checkerboard fashion suchthat no two neighbors have the same color. At each time step only the cells of a certaincolor are updated with a unitary depending on their neighbor’s values. Neighbors of thesame color are not distinguishable in this context.

2.3.3. Margolus QCA

Figure 2.4.: Partitioning for an MQCA, as introduced by Margolus (a) and an examplefor a more general partitioning, following the definition by Schumacher andWerner (b).

A Margolus QCA (MQCA) as first introduced by N. Margolus [7] consists of a one-dimensional lattice of cells. In the first step all even ordered sites are grouped together

12

2.3. Quantum Cellular Automata

with their right neighbors and a unitary transformation is performed on each of thesepairs. Then the cells are regrouped, so that the even ordered sites are now groupedtogether with their left neighbors and a unitary transformation is performed on thesepairs (see figure 2.4a).Schumacher and Werner extended this concept to a more general form [13]. In theirversion, a Margolus scheme consists of 2 partitionings. Each partitioning dividies thelattice into finite, disjoint and uniformly arranged blocks, such that each block of onepartitioning overlaps with at least 2 blocks from the other partitioning.It can be shown that for every automata of the Margolus type there exists an automatonof the colored type that has exactly the same dynamics and vice versa [11].

13

2. Basic Concepts

14

3. Relaxation of MQCA with 2-qubitblock partitioning

Model

The model under consideration will be a Margolus cellular automaton where each cellconsists of a 2-level system. The partitioning consists of blocks containing 2 qubits asdepicted in figure 2.4a. The MQCA will have a finite number of cells N and periodicor non-periodic boundary conditions. As local update rule, one can therefore choose anarbitrary two-bit quantum gate represented by the unitary Uloc ∈ SU(4). An arbitraryunitary transformation out of SU(4) can be given by a 4 × 4 matrix

Uloc = eiG G =15∑

i=1

αiλi (3.1)

where αi are 15 real valued parameters and λi are the Gell-Mann matrices (see A.1).

Questions

The main questions, that will be investigated in this section are:

Does there exist some kind of stationary state, into which the automaton tends to relax(regardless of the initial state)?And if so, what does it look like?

General remarks

It is clear that a stationary state in the sense that the state vector of the total automa-ton |ψ〉 remains constant after some steps cannot exist due to the unitarity of the globalevolution. However if one looks at smaller subsystems, it is very likely that the systemevolves into some dominant region in Hilbert space, where the reduced density matricesof these subsystems remain approximately constant, what will be called in the followinga local stationary state.Of course, due to the unitarity of the global evolution, the automaton has to return to

15

3. Relaxation of MQCA with 2-qubit block partitioning

its original state at some point in time, but as the dimension of the Hilbert space growsexponentially with the size of the automaton, this will usually happen on a timescalethat is significantly longer than any observer could wait.The local update rule and the initial state of the automaton will determine the accessiblesubspace to which the evolution of the global state vector is constrained. As it turnsout, the shape of this subspace strongly affects the relaxation of the automaton. Dueto this, a classification of the local update rules depending on the accessible subspace isfeasible.

3.1. Trivial rules

Any local update rule Uloc that can be written as

Uloc = U1 ⊗ U2 = (U1 ⊗ 12)(12 ⊗ U2) U1, U2 ∈ SU(2) (3.2)

will only result in a rotation of the state of each cell regardless of the neighboring cells.Thus, no complex dynamics of the automaton are to be expected.

3.2. Rules with Uloc : {|01〉 , |10〉} → {|01〉 , |10〉}(excitation number conserving rules)

The excitation number of an arbitrary state |ψ〉 will be defined as

n =N∑

i=1

〈ψ|P (i)|1〉〈1||ψ〉 P|1〉〈1|= |1〉 〈1| . (3.3)

The subspace spanned by all basis states

|s(1)〉 ⊗ · · · ⊗ |s(N)〉 , s(1), . . . , s(N) ∈ {1, 0} (3.4)

with the same excitation number

n =N∑

i=1

δs(i),1 (3.5)

will be called “n-excitation subspace”. Its dimension is given by

dn =

(

Nn

)

. (3.6)

16

3.2. Rules with Uloc : {|01〉 , |10〉} → {|01〉 , |10〉} (excitation number conserving rules)

All local update rules which only transform between states belonging to the same n-excitation subspace will be called “excitation number conserving”. Those rules are givenby:

Uloc =

eiφ11 0 0 0

0 e1

2i(α−β) cos γ

2e

1

2i(α+β) sin γ

20

0 −e− 1

2i(α+β) sin γ

2e−

1

2i(α−β) cos γ

20

0 0 0 eiφ00

(3.7)

and transform only between the local states |01〉 and |10〉 (except for a phase shift of |11〉and |00〉). One can easily verify that the commutator between the operator for the totalexcitation number and Uloc applied on an arbitrary pair of cells (i, j) is always equal to0:

[N∑

k=1

P(k)|1〉〈1|, U

(i,j)loc ] = 0 (3.8)

Therefore, rules of this type conserve the total excitation number for any given state.

To give a first overview of the dynamics obtained by these rules, figures 3.1-3.5 showthe time evolution of the occupation propabilities 〈P (i)

|1〉〈1|〉 of periodic and nonperiodicMQCA with a fixed size of 20 cells and two different excitation number conserving rulesfor different initial states.

17

3.2. Rules with Uloc : {|01〉 , |10〉} → {|01〉 , |10〉} (excitation number conserving rules)

Obviously, there seems to exist a local stationary state, that the MQCA tends toreach, where not only the state of each cell remains approximately constant but alsohomogeneous along the array. As one can see, the strength of the relaxation is not thesame for all rules of this type, but depends also on the boundary conditions, the initialstate and on the phase factors φ00 and φ11 .

Dependence on the initial state

As one can see from the examples, the strength of the relaxation depends strongly on thethe initial state of the automaton. This is due to the fact that the initial state determinesthe dimension of the subspace which will be accessible for the global state vector duringthe evolution of the automaton. For the n-excitation subspace, this dimension is given

by the binomial coefficient

(

Nn

)

. The higher now the dimension, the more coefficients

cs(1),...,s(N) of the state vector change their values among each other during the evolutionand thus the more different frequencies are contained in the time evolution of each ofthese coefficients. Due to this, one should expect that the expectation value of a cell forbeing in the excited state,

〈ψ|P (k)|1〉〈1||ψ〉 =

∑

s(1),...,s(N)

|cs(1),...,s(N)|2δs(k),1 (3.9)

will get more and more constant, the higher the dimension of the subspace.

Properties of the local stationary state

If one additionally assumes that the time average in the local stationary state is thesame for all coefficients cs(1),...,s(N), one can deduce some more properties for smallersubsystems of size ≥ 1 in the local stationary state:Consider a small subsystem of the MQCA in the local stationary state, consisting of Mcells at positions pk (k ∈ {1, . . . ,M}).The diagonal elements of the reduced density matrix describing this subsystem are givenby

ρjj = 〈ψ|P (p1,...,pM )|j〉〈j| |ψ〉 =

1...1∑

s,r=0...0

c∗scr 〈s|P(p1,...,pM )|j〉〈j| |r〉 =

1...1∑

s=0...0

|cs|2M∏

k=1

δs(pk)=j(k) (3.10)

where j denotes an arbitrary M -bit string where the bits at positions (p1, . . . , pM) havethe values j(p1), . . . , j(pM). If the state of the total system lies entirely in the n-excitation

23

3. Relaxation of MQCA with 2-qubit block partitioning

subspace and under the assumptions made above, this can be written as:

ρjj =∑

s

|c|2M∏

k=1

δs(pk)=j(k) =

(

N −Mn−m

)

(

Nn

) m =M∑

k=1

δj(k),1 (3.11)

Thus the propability of finding the subsystem in any state, where m of the M cells arein the excited state, is given by the hypergeometric distribution

P (m) =

(

Mm

)(

N −Mn−m

)

(

Nn

) (3.12)

which can be approximated by the binomial distribution if M << N :

P (m) =

(

Mm

)

(n

N)m(

N − n

N)M−m (3.13)

Thus for M << N , the diagonal elements ρjj can be rewritten as

ρjj =1

Ze−βm 1

Z=(N − n

N

)M

, β = logN − n

n(3.14)

The offdiagonal elements of the reduced density matrix are correlation functions of theform

ρij = 〈ψ|P (p1,...,pM )|i〉〈j| |ψ〉 =

1...1∑

s,r=0...0

c∗scr 〈s|P(p1,...,pM )|i〉〈j| |r〉 =

1...1∑

s,r=0...0

c∗scr

M∏

k=1

δs(pk)=j(k)δr(pk)=j(k)

(3.15)over the coefficients cs and cr. The smaller a subsystem is chosen, the larger is thenumber of coefficients participating in this sum and thus, the smaller the value for ρij.

Thus, one would expect the reduced density matrix of a sufficiently small partialsystem to be a diagonal matrix with a canonical distribution of the diagonal elements,given by (3.14). Figure 3.6 illustrates this for a MQCA with 20 cells and an initial stateof n = 8 active cells. They show the diagonal elements of the reduced density matricesfor partial systems of different sizes.

24

3.2. Rules with Uloc : {|01〉 , |10〉} → {|01〉 , |10〉} (excitation number conserving rules)

Figure 3.6.: Diagonal elements of the reduced density matrix for a partial system ofcells at positions (8,9), (8,9,10) and (8,9,10,11) after 800 steps of a MQCAwith an initial state of n = 8 active cells. Update rule given by (3.7) withφ00 = φ11 = π

2, α = π

β= 0, γ = π

2.

25

3. Relaxation of MQCA with 2-qubit block partitioning

Figure 3.7.: Developing of the occupation propabilities of cell 1 and 15 for being in theexcited state for the first 150 steps of the automaton from figure 3.6. As onecan see, the relaxation of a single subsystem is quite good.

Entanglement in the local stationary state

According to the preceding considerations, one can expect the reduced density matrixof any single cell to be given by:

ρ(k) =

(

nN

00 1 − n

N

)

(3.16)

In this case, the multi-particle entanglement Q is given by

Qmax = 4(n

N− n2

N2) (3.17)

One can easily see, that this is the highest possible value for Q for a given excitationnumber n, if one maximizes Q for some general density matrices given by:

ρ(k) =

(

ak bkb∗k 1 − ak

)

(3.18)

under the constraintsN∑

k=1

ak!= n (3.19)

which gives (3.16) as solution. To illustrate this, figures 3.8-3.11 show the developing ofthe multi particle entanglement Q for the examples shown in figures 3.1-3.5.

26

3.2. Rules with Uloc : {|01〉 , |10〉} → {|01〉 , |10〉} (excitation number conserving rules)

Figure 3.8.: Evolution of the multi particle entanglement Q of the periodic MQCAshowed in fig. 3.1 (φ00 = φ11 = α = β = 0, γ = π

2). The dashed lines

show the maximum possible value for Q for a given excitation number n.The entanglement always tends to reach its maximum possible value, whichdepends on the number of excited cells in the initial state.

Figure 3.9.: Evolution of the multi particle entanglement Q of the periodic MQCAshowed in fig. 3.2 (φ00 = φ11 = π

2, α = β = 0, γ = π

2). The dashed

lines show the maximum possible value for Q for a given excitation numbern. The better quality of the relaxation for phase factors other than 0 canbe seen. The entanglement gets much closer to its maximum possible value.

27

3. Relaxation of MQCA with 2-qubit block partitioning

Figure 3.10.: Evolution of the multi particle entanglement Q of the nonperiodic MQCAshowed in fig. 3.3 (φ00 = φ11 = α = β = 0, γ = π

2). One can see that the

boundary conditions do not have a great influence on the developing of theentanglement.

Figure 3.11.: Evolution of the multi particle entanglement Q of the nonperiodic MQCAshowed in fig. 3.4 (φ00 = φ11 = π

2, α = β = 0, γ = π

2). Again, the different

phase factors result in a much better relaxation than in the preceding figure.

28

3.3. Rules with Uloc : {|00〉 , |11〉} → {|00〉 , |11〉}

Dependence on the phase factors φ00 and φ11

If one compares the results shown in figure 3.1 and 3.2, one can see that the relaxationof the MQCA is much better if the phase factors φ00 and φ11 are unequal to 0. As canbe seen in the following sections, the influence of such phase factors on the quality ofthe relaxation is not restricted to only this type of local update rules. Unfortunatelythe reason for this could not be discovered in this thesis. However it seems that it hassomething to do with the“scattering”of excitations amongst each other. This assumptionis based on the observation from the numerics, that for a non-periodic MQCA with aexcitation conserving rule with φ00 = φ11 = 0 and an initial state with n excited cells,the propagation of each single excitation can be calculated seperately from the others.But unfortunately, this could not yet be proven analytically.

3.3. Rules with Uloc : {|00〉 , |11〉} → {|00〉 , |11〉}These rules are given by

Uloc =

e1

2i(α−β) cos γ

20 0 e

1

2i(α+β) sin γ

2

0 eiφ10 0 00 0 eiφ01 0

−e− 1

2i(α+β) sin γ

20 0 e−

1

2i(α−β) cos γ

2

(3.20)

In this case Uloc transforms between basis states that differ in their excitation numberby multiples of two. But for a given excitation number, not all possible states withthe same excitation number can be reached. Thus by applying rules of this type, theevolution of the global state vector is again restricted to a subspace of the total Hilbertspace. But this time, the shape and dimension of this subspace cannot be easily givenin an analytical way. To give an impression of the basis states spanning the accessiblesubspace for a specific initial state, figure 3.12 shows an example of a MQCA with 8cells, together with the basis states for a local update rule given by

Uloc =

1√2

0 0 1√2

0 1 0 00 0 1 0

− 1√2

0 0 1√2

(3.21)

Figure 3.13 shows the dimension of the accessible subspace compared to the dimensionof the total Hilbert space depending on the size of a MQCA featuring such a local updaterule. From this, one can see, that the subspace dimension also scales exponentially withthe size of the system.Figures 3.14 and 3.15 show some examples of MQCA with a rule given by (3.20) anddifferent sets of parameters. One can see that this rule results in relaxational behaviour

29

3.4. Rules with Uloc : {|00〉 , |11〉} → {|00〉 , |11〉}, {|01〉 , |10〉} → {|01〉 , |10〉}

Figure 3.16.: Multi particle entanglement of the first two examples shown in figure 3.14a) and b). One can see that the entanglement gets much closer to itsmaximum possible value for phase factors φ01, φ10 6= 0.

3.4. Rules with Uloc : {|00〉 , |11〉} →{|00〉 , |11〉}, {|01〉 , |10〉} → {|01〉 , |10〉}

These rules are given by

Uloc =

e1

2i(α1−β1+δ) cos γ1

20 0 e

1

2i(α1+β1+δ) sin γ1

2

0 e1

2i(α2−β2) cos γ2

2e

1

2i(α2+β2) sin γ2

20

0 −e− 1

2i(α2+β2) sin γ2

2e−

1

2i(α2−β2) cos γ2

20

−e− 1

2i(α1+β1+δ) sin γ1

20 0 e−

1

2i(α1−β1+δ) cos γ1

2

(3.22)

and can be seen as a combination of the rules from the preceding two sections. In thiscase, the subspace to which the evolution of the global state vector is restricted to can beeasily described. Due to the part that transforms between |00〉 and |11〉, Uloc transformsbetween some basis states that differ in their excitation number by multiples of two. Inaddition, due to the part that transforms between |01〉 and |10〉, Uloc transforms nowalso between all possible basis states that belong to the same excitation number. Thus,one can see that Uloc can be split up in two parts, where the one part acts only in thesubspace spanned by all basis states with an even excitation number and the other partacts only in the subspace corresponding to an odd excitation number.Figure 3.17 shows examples of an MQCA featuring such a rule for different parametersets and initial states.

33

3.5. Total mixing rules

Figure 3.18.: Multi particle entanglement of the two first examples shown in figure 3.17b) and c)

The first three examples are a strong indicator, that there have to exist further sym-metries, that could not be found yet. As could already be seen in the previous sections,the phase factor δ seems to play an important role here, too.

3.5. Total mixing rules

All rules that do not fall under one of the cases already presented will act in the totalHilbert space of dimension d = 2N (eventually without the ground/all-excited state, thusd = 2N − 1).In this case, one would expect the strength of the relaxation to be more or less indepen-dent of the initial state and stronger than in the preceding cases.If, in analog to the assumptions made in the previous section, the coefficients of the statevector in the local equilibrium state would get totally “mixed”, one should expect thedensity matrix of a (not too small) partial system of dimension d then to be given by

1d

0. . .

0 1d

(3.23)

and the maximum value for the multi particle entanglement Q should be given by Q = 1.However, as it turns out, this is not always exactly fullfilled for all rules and initial states.Figures 3.19 and 3.22 show examples of MQCA for different initial states and 2 different

35

3.5. Total mixing rules

Figure 3.20.: Multi particle entanglement for the automaton showed in fig. 3.19. Dashedline: initial state with 1 excited cell. Solid line: initial state with 5 excitedcells. In both cases, the entanglement reaches the maximum possible valueof Q = 1 which means a perfectly mixed state.

Figure 3.21.: Mean value of the occupation propabilities for the excited state( 1

N

∑Ni=1〈P

(i)|1〉〈1|〉 = n

N) for the automaton showed in fig. 3.19. One can

see that there exists a slight dependence on the initial state which indi-cates that the assumption of a total mixing of the coefficients cannot becompletely correct.

37

3.5. Total mixing rules

Figure 3.23.: Multi particle entanglement for the automaton showed in fig. 3.22. Dashedline: initial state with 1 excited cell. Solid line: initial state with 5 excitedcells.

Figure 3.24.: Mean value of the occupation propabilities for the excited state( 1

N

∑Ni=1〈P

(i)|1〉〈1|〉 = n

N) for the automaton showed in fig. 3.22. Here the

dependence on the initial state is even stronger than in the previous exam-ple.

39

3. Relaxation of MQCA with 2-qubit block partitioning

40

4. Relaxation of a two species LQCA

Model

The model under consideration will be a Linear QCA with two different energy splittingsand different coupling strengths to the left and right neighbors. The LQCA will have a

finite number N of cells and periodic or non-periodic boundary conditions. At this point,it is assumed, that all states of the spin chain are long lived states, thus the “intrinsic”dynamics of the chain are slow compared to the time scale at which QCA dynamics takeplace. The local update rule of the QCA will be given by (2.36), with

U00 = R(α00, β00, γ00)

U01 = R(α01, β01, γ01)

U10 = R(α10, β10, γ10)

U11 = R(α11, β11, γ11) (4.1)

where R denotes a general rotation in the state space of a spin-12

system and will bedenoted by the set {α00 . . . γ11}.The main questions, that will be investigated are the same as in the previous chapter.Again, it is advantageous to define certain classes of update rules with special properties.

4.1. Trivial rules

Any local update rule Uloc that has

U00 = U01 = U10 = U11=V (4.2)

can be written asUloc = 12 ⊗ V ⊗ 12 (4.3)

and will only result in a rotation of the state of each cell regardless of the neighboringcells. Thus, no complex dynamics of the automaton are to be expected in this case.

41

4.2. Non-propagating rules

Dimension of the accessible subspace

To get an impression, of how the accessible subspace may look like in case a), consideran initial state, where all cells are inactive, except for the chain of cells between andincluding the positions a and b:

|ψ0〉 = |s(0)〉 · · · |s(N)〉 s(k) ={1 for a ≤ k ≤ b

0 otherwise(4.7)

By applying Uloc at an arbitrary 3-spin block one can create all kinds of states consistingof basisstates where one or more cells between a and b are flipped, but not on adjacentsites.These basis states span the accessible subspace in this case. To calculate its dimensiond, one therefore has to count all binary numbers of length n = b− a− 1 that don’t havetwo or more adjacent digits set to “0”. One can show [4] that this number is given by

d = F (n+ 2), (4.8)

where F (n) denotes the Fibonacci series, which is given by

F (n+ 2) = F (n) + F (n+ 1), n ≥ 1 and n(1) = n(2) = 1 (4.9)

and can be also be written in a closed form [1]

F (n) =1√5

([1 +√

5

2

]n

+[1 −

√5

2

]n)

(4.10)

Therefore, it should be possible for the automaton to evolve into a local stationarystate, if there exists a big enough area of the type described above, and therefore ahigher number of coefficients takes part in the evolution. One can see this behaviour inthe example shown in figure 4.2. For this type of rules of course, the local equilibriumstate cannot be homogenious across the lattice.

43

4.2. Non-propagating rules

Figure 4.3.: Multi particle entanglement for the LQCA shown in figure 4.2.

Figure 4.4.: Reduced density matrix of a partial subsystem consisting of cells 14-17 ofthe automaton shown in figure 4.2. States of the subsystem that have twoor more adjacent inactive cells cannot be found.

45

4. Relaxation of a two species LQCA

4.3. Rules with γ01 = γ10 = 0, γ00 6= 0, γ11 6= 0

In this case, excitation propagation is no longer restricted to special areas. However, thesubspace S of all global states that can be reached during the evolution of the automatonfor a given initial state is also a smaller part of the total Hilbert space, but not as easyto describe as in the preceding case. Consider an initial product state where all cells areinactive, except for the chain of cells between and including the positions j and l (l ≥ j):

|ψ0〉 = |s0〉 · · · |sN〉 sk ={1 for j ≤ k ≤ l

0 otherwise(4.11)

Because it is γ01 = γ10 = 0, one can only alter the state by applying Uloc on a block of 3cells whose outer left and outer right cells lie both inside the chain of active cells or bothoutside. By this, one creates states that are constituted of basisvectors with one cellinverted, that lies inside or outside the chain but not at the borders (at j − 1, j, l, l+ 1)except for chains of length 1. By repeating these considerations for the obtained basisvectors, one can find all states that span the subspace for a given initial state:If the initial state is a product state consisting of n chains of active cells, then these aregiven by all product states, where one or more of the n chains are extended or truncatedby multiples of two cells and/or have a cell inverted as described above.To illustrate this, figure 4.5 shows an example of a LQCA and the basis vectors thatspan the accessible subspace for the given initial state. The rule of the automaton isgiven by

U01 = U10 = 12

U00 = U11 =1√2

(

1 1−1 1

)

(4.12)

Figure 4.6 shows the dimension of the accessible subspace compared to the dimensionof the total Hilbert space depending on the size of the LQCA featuring such a localupdate rule. As one can see, the dimension grows exponentially with the size of theautomaton. One therefore should expect that the automaton tends to relax into a localstationary state if it is sufficiently large. Figure 4.7 shows an example of a larger LQCAthat confirms this expectation.

46

4. Relaxation of a two species LQCA

Figure 4.7.: Two examples of a LQCA with a rule given by (4.12) and non-periodic (a) orperiodic (b) boundary conditions. One can see that the boundary conditionsonly affect the relaxation of some few cells at the boundaries.

48

4. Relaxation of a two species LQCA

Figure 4.9.: Multi particle entanglement for the LQCA shown in figure 4.8. Here onecan also see, that the automaton does not reach any stationary state butrather shows some periodic behaviour.

Properties of the accessible subspace

Consider a LQCA with periodic boundary conditions, a left propagating rule and aninitial product state where all cells are inactive except for the cells at positions j and lwhich shall be separated by at least 1 cell (|l − j| > 1):

|ψ0〉 = |s0〉 · · · |sN〉 si = δi,j + δi,l |l − j| > 1 (4.16)

Because it is γ11 = γ10 = 0, one can easily see, that the cells at positions j + 1 and l+ 1always stay in the inactive state. From that follows with γ00 = γ10 = 0, that the cells atpositions j and l always stay in the active state. This means that one always stays inthe subspace S of all states, where the cells at positions (j, j + 1) and (k, k + 1) are inthe state |1〉 |0〉. In this case, the part U(S) of the global unitary U that acts in S canbe split up into two parts

U(S)(1,...,N) = U1(S)(j+1,...,l)U2(S)(l+1,...,N,1,...,j) (4.17)

Now the evolution of the periodic automaton can be described by the evolution of twoseparate non-periodic automata of size N1 = l− j− 1 and N2 = N −N1 and with initialstates where all cells are inactive except the cell at position 1. In analog, for initialproduct states with a higher number of excited cells, the automaton can be split up intoa higher number of non-periodic automata. One can easily see, that all states that canbe reached during the evolution of such a non-periodic automata of size Nm belong to

50

4.5. left and right propagating rules

the Nm-dimensional subspace spanned by the basis-vectors

|k〉 = |s1〉 . . . |sNm〉 k = 1, . . . , Nm − 1 si =

{ 1 for i ≤ k

0 otherwise(4.18)

Thus, the dynamics of a LQCA with a left or right propagating rule take place in severaldisjoint subspaces of maximum dimension d = N − 1. Due to this, one should expectthe relaxation into a local stationary state to be very weak.

4.5. left and right propagating rules

All rules that haveγ00 = γ11 = 0 (4.19)

andγ01 6= 0 and γ10 6= 0 (4.20)

will be called left and right propagating rules. Figure 4.11 and 4.10 show examples ofLQCA with a left and right propagating rule given by

U00 = U11 = 12

U01 = U10 =

(

1√2

1√2

− 1√2

1√2

)

(4.21)

for periodic and non-periodic boundary conditions and two different initial states. Fig-ure 4.12 shows the corresponding multi-particle entanglement.

As one can see from these two examples, the automaton tends to relax into a localstationary state. One can also see, that the relaxation depends on the initial state andon the boundary conditions. Again, this can be explained, if one looks at the underlyingsubspace structure.

Properties of the accessible subspace

Consider a LQCA of length N with non-periodic boundary conditions, a left and rightpropagating rule and a initial state drawn from the subspace spanned by all basis statesgiven by

|a, b〉 = |s(0)〉 · · · |s(N)〉 s(k) ={1 for a ≤ k < b

0 otherwise(4.22)

1 < a < b ≤ N

By applying Uloc at an arbitrary block of 3 cells, one can only create states that areconstituted of the old state and a basis state where one cell is flipped at the position a,a − 1, b or b + 1. Therefore one can easily confirm, that all states that can be reached

51

4. Relaxation of a two species LQCA

remain within this subspace. To generalize this, let Sn denote the subspace spanned byall basis states given by

|a1, b1, . . . , an, bn〉 = |s(0)〉 · · · |s(N)〉 s(k) ={1 for k ∈ [a1, b1 − 1] ∪ · · · ∪ [an, bn − 1]

0 otherwise(4.23)

1 < a1 < b1 < · · · < an < bn ≤ N

Then the global unitary U can be split up into

U = U(S1) + U(S2) + · · · + U(SN/2−1) (4.24)

where each U(Si) acts only in the subspace Sn.To calculate the dimension d(Sn) of the subspace Sn one has to count all possible combi-nations of a1, b1, . . . , an, bn. This is equivalent to counting all binary numbers of lengthN − 1 with exactly 2n bits set to “1”. Thus, the dimension of the subspace in thenon-periodic case is given by

d(Sn) =

(

N − 12n

)

(4.25)

For a periodic LQCA, the condition 1 < a < b ≤ N for eq. (4.22) has to be changed tobe 1 ≤ a < b ≤ N and one has to allow for rotations of these states along the chain:

|a, b, r〉 = |s(0)〉 · · · |s(N)〉 s(k) ={1 for a ≤ k + r < b

0 otherwise(4.26)

1 ≤ a < b ≤ N

Where k + r indicates periodicity ( k + r=(

(k + r − 1) mod N)

+ 1 ).From this, one might guess, that the subspace dimension in the case of n = 1 would be

given by d(S1) = N

(

N2

)

. But in fact this is not the case, as for a parametrization of

the form (4.26), the same basis vector may be described by multiple different parametersets {a, b, r}.By taking this into account, one finds the correct subspace dimension to be given by

d(S1) = 2

(

N2

)

and in the general case by

d(Sn) = 2

(

N2n

)

(4.27)

52

4.5. left and right propagating rules

a)

b)

Figure 4.12.: Multi particle entanglement for the LQCA shown in figure 4.11 (a) and 4.10(b). Solid Line: Initial state 1. Dashed line: Initial state 2. The relaxationin the periodic case is much better than in the nonperiodic case. This canbe explained by the observation that in the periodic case after some steps,all cells have the same occupation propabilities of ρ11 ≈ ρ00 ≈ 1

2while in

the nonperiodic case not. In addition, the number of participating basisvectors is higher as in the nonperiodic case.

55

4.7. Summary and Comparison of Chapter 3 and 4

Figure 4.14.: Multi particle entanglement for the LQCA shown in figure 4.13.

4.6.1. MQCA equivalent rules

A subclass of the total mixing rules are the MQCA equivalent rules. Any local updaterule Uloc that has

U00 = U01=V0 and U10 = U11=V1 (4.31)

can be written as

Uloc =(

|0〉 〈0| ⊗ V0 + |1〉 〈1| ⊗ V1

)

⊗ 12 (4.32)

This corresponds to a MQCA with the total mixing local update rule given by |0〉 〈0| ⊗V0 + |1〉 〈1| ⊗ V1. The same applies for rules with U00 = U10=W0, U01 = U11=W1, whichcan be written as

Uloc = 12 ⊗(

W0 ⊗ |0〉 〈0| + W1 ⊗ |1〉 〈1|)

(4.33)

4.7. Summary and Comparison of Chapter 3 and 4

It has been shown for quantum cellular automata of the margolus and the linear typethat nearly all local update rules lead to a relaxation into local stationary states.

The strength of this relaxation depends strongly on the dimension of the subspacewhich is accessible to the global state vector of the automaton during the evolution.Therefore, in some cases the strength of the relaxation depends also on the initial stateof the automaton.

57

4. Relaxation of a two species LQCA

The quality of the relaxation in the case of a MQCA has been found to be dependenton some relative phase factors of the local update rule. In the case of a LQCA, such adependence could not yet be found.

All examples have shown that the relaxation leads to an asymptotic increase of themulti-particle entanglement Q towards some maximum value which is never exceededduring the evolution. This means that the final state is a maximally mixed state withrespect to the accessible subspace.

58

5. Relaxation control with a LQCA

The model under consideration here will be a Linear QCA like in chapter 4 with afinite number of cells N and non-periodic boundary conditions. But this time it willbe assumed that the dynamics due to the coupling between the spins are on the sametimescale as the distance between the periodically interrupting control pulses of theLQCA. The coupling shall be a Forster type coupling and the total Hamiltonian of thechain shall be of the following form:

H =∆EA

2

(N−a)/2∑

k=1

σ(2k−1)z +

∆EB

2

(N+a)/2∑

k=1

σ(2k)z

+λAB

(N−a)/2∑

k=1

(σx ⊗ σx + σy ⊗ σy)(2k−1,2k) + λBA

(N+a)/2−1∑

k=1

(σx ⊗ σx + σy ⊗ σy)(2k,2k+1)

(5.1)

a ={0 for N even

1 for N odd

where ∆EA,∆EB denote the two different splitting energies and λAB, λBA denote thetwo different coupling constants to the left and right neighbors.The total evolution of the LQCA is now given by the “free” evolution of the chain,interrupted by periodic control pulses of the LQCA. According to [6], one can expectthat the free evolution will lead to a relaxation of smaller local subsystems into somelocal equilibrium state, and thus by applying the additional LQCA pulses, one can expectmuch more complex dynamics than in the case of a static spin chain.However, considering the results shown so far, one would expect that in most cases, thesystem will tend to reach some kind of local equilibrium state, anyway.If one wants to construct a quantum computer, an important question is, how one couldget rid of the unwanted free evolution resulting from the inevitable internal couplingbetween the individual qubits. It was shown by M.Stollsteimer and G.Mahler in [15], [16]how this can be done by applying specific pulse sequences acting either globally on allspins or selectively on some spins, depending on the type of interaction that one has todeal with. Especially for the case of only nearest neighbor coupling of the form

H(k,k+1)int = Jxσ

(k)x ⊗ σ(k+1)

x + Jyσ(k)y ⊗ σ(k+1)

y + Jzσ(k)z ⊗ σ(k+1)

z , (5.2)

they introduced a method which uses pulses that alternately act on all even numberedand then on all odd numbered qubits.

59

5. Relaxation control with a LQCA

In this chapter will be given some numerical examples of how the free evolution canbe altered or supressed in the formalism of a Linear QCA. This allows for example tosupress the free evolution for some initial states of the automaton, while other stateswill be not affected.It seems that there exists a field for investigations about the effect of all thinkable updaterules on the natural relaxation behaviour of a spin-1

2chain which could not be covered

completely in this thesis. So in this chapter can just be given some intriguing examplesfor LQCA with different update rules without the intention of a complete classification.

5.0.1. Example 1

An interesting type of rules are rules that have

γ00 = γ01 = γ10 = γ11 = 0 (5.3)

and were termed trivial in the case of slow intrinsic dynamics. But now the free dynamicsof the spin chain already induces a propagation of excitations along the chain and theserules can have a great effect on how this happens, while leaving the total excitationnumber of the state unchanged.This example shows, how rules of the form (5.3) can be used to control the strength andspeed of the relaxation of a spin-1

2chain. The local update rule shall have the following

form:

α01 = α10 = α11 = δ01 = δ10 = δ11 =π

2(5.4)

all others = 0.

This rule means a phase rotation of each spin, if its left and/or its right neighbor is inthe excited state.Figure 5.1 shows the dynamics of the occupation propabilities of such a LQCA and figure5.2 shows the corresponding multi particle entanglement. One can see that for the samerule, one can either accelerate or decelerate the relaxation depending on the distancebetween the QCA pulses, while in both cases, after some time the system seems to becloser to a local equilibrium state than in the case of free dynamics.

5.0.2. Example 2

We eventually consider another example of a rule of the form (5.3). This time, the localupdate rule shall have the following form:

δ01 = δ10 = −α01 = −α10 =π

2(5.5)

all others = 0

One can see from figure 5.3 that this leads to different speeds of relaxation dependingon the initial state.

60

5. Relaxation control with a LQCA

Figure 5.2.: Multi particle entanglement for the LQCA shown in figure 5.1. From this,one can see that the system with applied QCA pulses seems to be closer toa local equilibrium state than in the case of free dynamics.

62

5. Relaxation control with a LQCA

64

6. Simulating spin chain dynamicsthrough QCA dynamics

One of the possible applications where a quantum computer can outperform a classicalcomputer is the factoring of large numbers or the search in large databases. In 2001an example for an experimental NMR implementation of a quantum computer factoringthe number 15 by using the Shor-Algorithm [14] was given in [19]. This task requiredthe use of 7 qubits. One can expect that building a quantum computer that can factornumbers that cannot be factorized by classical computation will require the control overthousands of qubits. However, in 1981 R. Feynman sketched the possibility of using aquantum computer to simulate another quantum mechanical system [5]. Such a quantumsimulator consisting only of a few tens of qubits may be used for simulations of quantumsystems that would already be impossible to simulate using a classical computer. Thisis, why it is often claimed, that such a quantum simulator will be implemented far beforeany useful implementation of a quantum computer of the Shor-class.In the following will be shown, how QCA of the Margolus type can be used to simulatethe time-dependent behaviour of spin chains.

6.1. Simulation of spin chains by MQCA

In the following will be shown that any spin chain with only nearest neighbor couplingscan be simulated using a Margolus QCA. The Hamiltonian of such a spin chain shall beof the following form:

H =N∑

k=1

H(k)s +

N−1∑

k=1

H(k,k+1)c , (6.1)

where H(k)s corresponds to the splitting energy of the spin at position k and H

(k,k+1)c

corresponds to the coupling energy of the spins located at k,k+1.In order to calculate the time evolution of the chain using the Suzuki-Trotter method,

65

6. Simulating spin chain dynamics through QCA dynamics

one can split up the total Hamiltonian into H = HA + HB with

HA =

N/2∑

k=1

H(2k−1)s + H(2k)

s +H(2k−1,2k)c

HB =

N/2−1∑

k=1

H(2k,2k+1)c (6.2)

Now the time evolution can be calculated by

|ψ(t)〉 = |ψ(t = 0)〉 exp[−t i~(HA + HB)] (6.3)

= |ψ(t = 0)〉 lim∆t→0

(

exp[−∆ti

~HA] exp[−∆t

i

~HB]

) t

∆t (6.4)

= |ψ(t = 0)〉 lim∆t→0

(

UA(∆t)UB(∆t)) t

∆t (6.5)

Because both UA(∆t) = exp[−∆t i~HA] and UB(∆t) = exp[−∆t i

~HB] respectively are

only composed of operators acting on disjoint blocks of 2 spins, they can be written as

UA(∆t) =

N/2∏

k=1

U(∆t)(2k−1,2k)loc,A (6.6)

UB(∆t) =

N/2−1∏

k=1

U(∆t)(2k,2k+1)loc,B (6.7)

where

U(∆t)(2k−1,2k)loc,A = exp[−∆t

i

~

(

H(2k−1)s + H(2k)

s + H(2k−1,2k)c

)

] (6.8)

U(∆t)(2k,2k+1)loc,B = exp[−∆t

i

~

(

H(2k,2k+1)c

)

] (6.9)

are the unitaries acting on blocks of two spins at positions 2k − 1, 2k and 2k, 2k + 1.This corresponds to a MQCA as depicted in Fig. 2.4a with the local update rule

Uloc,A = exp[−∆ti

~

(

Hs ⊗ 1 + 1⊗ Hs + Hc

)

] (6.10)

applied to all blocks starting at odd positions and the local update rule

Uloc,A = exp[−∆ti

~Hc] (6.11)

applied to all blocks starting at even positions.

66

7. Summary and outlook

7.1. Summary

In this thesis the conditions, under which the cells of a Quantum Cellular Automatonrelax into local stationary states, due to the dynamics caused by the application of dif-ferent local update rules have been investigated for two different models of QuantumCellular Automata, namely for the Margolus Quantum Celular Automaton (MQCA)and the Linear Quantum Cellular Automaton (LQCA).

It has been shown, that the cells of the two investigated models of Quantum CellularAutomata will relax into local stationary states for most of the possible local updaterules if the size of the automaton is sufficiently large. It turned out, that the relaxationstrongly depends on the subspace that is accessible for the global state vector of theautomaton during its evolution. This subspace depends on the specific update rule thathas been chosen, as well as (for most of the local update rules) on the initial state of theautomaton. Therefore a classification of the different local update rules in terms of thesubspace structure that arises out of those rules has been given.

Furthermore there has been shown by two examples, that the natural relaxation ofa Forster coupled spin-1

2chain into equilibrium can to some extend be controlled by

applying LQCA update rules. In doing so, one can either slow down or accelerate therelaxation independently of the initial state or selectively slow down the relaxation forspecific initial states, while other initial states will relax more quickly.

In a short digression, there has also been shown the connection between a MargolusQCA and the simulation of spin chains via the Suzuki-Trotter Decomposition.

7.2. Outlook

A first step towards a classification of local update rules of two different QCA with re-spect to their ability to produce relaxation into local stationary states has been made inthis thesis. However there remain many open questions, as for example some subspacestructures are still lacking an analytical description. Another example would be thequestion whether the classification can be further extended with respect to other not

67

7. Summary and outlook

yet discovered symmetries (one example would be the strong dependence on the relativephases of some rules).Another field where there is much space for investigations is the control of relaxationusing LQCA rules. It seems that there exists a field for investigations about the effect ofall conceivable update rules on the natural relaxation behaviour of a spin-1

2chain, which

could not be covered completely in this thesis, as there was unfortunately not enoughtime left to systematically investigate this topic.One could also think about extending all these investigations into 2 or 3 dimensions,although the computational facilities for numerical simulations of such models will prob-ably very quickly reach their limits.

68

A. Appendix

A.1. Generators of SU(4)

Any unitary U drawn from the special unitary group SU(4) can be written as

U = eiG G =15∑

i=1

αiλi

where αi are 15 real valued parameters and λi denotes the Gell-Mann matrices given by

λ1 =

0 1 0 01 0 0 00 0 0 00 0 0 0

λ2 =

0 −i 0 0i 0 0 00 0 0 00 0 0 0

λ3 =

0 0 0 00 0 1 00 1 0 00 0 0 0

λ4 =

0 0 0 00 0 −i 00 i 0 00 0 0 0

λ5 =

0 0 0 00 0 0 10 0 1 00 0 0 0

λ6 =

0 0 0 00 0 0 −i0 0 i 00 0 0 0

λ7 =

0 0 1 00 0 0 01 0 0 00 0 0 0

λ8 =

0 0 −i 00 0 0 0i 0 0 00 0 0 0

λ9 =

0 0 0 00 0 0 10 0 0 00 1 0 0

λ10 =

0 0 0 00 0 0 −i0 0 0 00 i 0 0

λ11 =

0 0 0 10 0 0 00 0 0 01 0 0 0

λ12 =

0 0 0 −i0 0 0 00 0 0 0i 0 0 0

λ13 =

1 0 0 00 −1 0 00 0 0 00 0 0 0

λ14 =

0 0 0 00 1 0 00 0 −1 00 0 0 0

λ15 =

0 0 0 00 0 0 00 0 1 00 0 0 −1

A.2. A note on the numerics

All numerical simulations in this work have been performed using the library “libqn”which provides functions for simulating quantum registers consisting of two-level systemsby taking only the affected basis states into account. The library was written in C codeby the author during the making of this thesis and can be linked into Mathematica viathe MathLink interface. The time evolution of the spin-1

2chains was calculated using

the Suzuki-Trotter Decomposition.

69

A. Appendix

70

Bibliography

[1] Bronstein, Semendjajew, Musiol, and Muhlig. Taschenbuch der Mathematik. VerlagHarry Deutsch, 2001.

[2] C. Cohen-Tannoudji, B. Diu, and F. Laloe. Quantum Mechanics, volume 1. WileyInterscience, 1977.

[3] C. Durr, H. LeThanh, and M. Santha. A decision procedure for well-formed linearquantum cellular automata. Random Struct. Algorithms, 11(4):381–394, 1997.

[4] W. Feller. An Introduction to Probability Theory and Its Application, volume 1.Wiley, 3rd edition, 1968.

[5] R. Feynmann. Simulating physics with computers. Int. J. Theor. Phys., 21, 1982.

[6] J. Gemmer, M.Michel, and G.Mahler. Quantum Thermodynamics: Emergence ofthermodynamic behavior within composite quantum systems, volume 657. Springer,lecture notes in physics edition, 2004.

[7] H.Margolus and Norman. Parallel quantum computation. In W.H.Zurek, editor,Complexity, Entropy, and the Physics of Information, volume VIII. Addison-Wesley,1990.

[8] S. Lloyd. A potentially realizable quantum computer. Science, 261(5128):1569–1571, 1993.

[9] O. Martin, A. M. Odlyzko, and S. Wolfram. Algebraic properties of cellular au-tomata. Communications in Mathematical Physics, 93:219–258, June 1984.

[10] D. A. Meyer and N. R. Wallach. Global entanglement in multiparticle systems.Journal of Mathematical Physics, 43:4273–4278, Sept. 2002.

[11] C. A. Perez-Delgado and D. Cheung. Models of quantum cellular automata. ArXivQuantum Physics e-prints, Aug. 2005.

[12] C. A. Perez-Delgado and D. Cheung. Local unitary quantum cellular automata.Physical Review A (Atomic, Molecular, and Optical Physics), 76(3):032320, 2007.

[13] B. Schumacher and R. F. Werner. Reversible quantum cellular automata. ArXivQuantum Physics e-prints, May 2004.

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[14] P. W. Shor. Algorithms for quantum computation: Discrete logarithms and fac-toring. In IEEE Symposium on Foundations of Computer Science, pages 124–134,1994.

[15] M. Stollsteimer. Skalenverhalten und kontrolle von quanten-netzwerken. Master’sthesis, Universitat Stuttgart, 2000.

[16] M. Stollsteimer and G. Mahler. Suppression of arbitrary internal coupling in aquantum register. Phys. Rev. A, 64(5):052301, Oct 2001.

[17] H. Trotter. In Proc. Am. Math. Soc., 1959.

[18] W. van Dam. Quantum cellular automata. Master’s thesis, University of Nijmegen,1996.

[19] L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, andI. L. Chuang. Experimental realization of shor’s quantum factoring algorithm usingnuclear magnetic resonance. Nature, 414, 2001.

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72

Danksagung

An dieser Stelle seien einige Personen erwahnt, ohne die es mir nicht moglich gewesenware diese Arbeit anzufertigen. Mein besonderer Dank gilt...

... Prof. Dr. Gunter Mahler fur die Aufnahme in seine Arbeitsgruppe, die kompetenteBetreuung meiner Diplomarbeit und fur die selbstbestimmende Art des Arbeitens, dieer mir ermoglichte und die ich als sehr angenehm empfand.

... Prof. Dr. Ulrich Weiß fur die Ubernahme des Mitberichts dieser Arbeit.

... Prof. Dr. Wunner und dem gesamten 1. Institut fur Theoretische Physik fur dieangenehmen Arbeitsbedingungen.

... Markus Henrich, Thomas Jahnke, Georg Reuther, Florian Rempp, Harry Schmidt,Heiko Schroder, Jens Teifel, Pedro Vidal, Hendrik Weimer und Mohamed Youssef furdie freundliche Aufnahme in die Arbeitsgruppe, anregende und unterhaltsame Gespra-che (nicht immer nur fachlicher Natur ;) und die angenehme Atmosphare.

... Hendrik Weimer, Heiko Schroder und Florian Rempp fur Spiel und Spass in derMittagspause.

... nochmals Florian Rempp, der mit mir an der Uni das Zimmer teilte, sich von mirin den ersten Wochen geduldig diverse Locher in den Bauch fragen ließ und auch sonstfur anregende Diskussionen stets zu haben war.

... meiner Wohngemeinschaft bestehend aus meinem Bruder Jan Kettler, Daniel Mo-risson, Phillip Scholz, Johannes Hauser und Nicole Stricker fur die angenehmen Stundendaheim und den Jazz im Probenkeller :)

... Walter und Rita Stricker fur so manchen All-Inclusive-Urlaub am Wochenende.

... meinen Eltern Albrecht und Ute Kettler, die mich bis heute in allen Lebensberei-chen wundervoll unterstutzen und mir diesen Lebensweg ermoglicht haben.

... meiner Freundin Nicole Stricker, die immer fur mich da ist, an mich glaubt undmich mit ihrer Frohlichkeit verzaubert.

73

Danksagung

74