Geometric quantum computing with supersymmetric lattice models

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MSc Physics and Astronomy

Theoretical Physics

Master Thesis

Geometric quantum computing with

supersymmetric lattice models

by

Diko Hemminga

10680268 August 25, 2018 60 EC September 2017 - July 2018 Supervisor:

prof.dr. Kareljan Schoutens

Examiner: dr. Marcel Vonk

Institute for Theoretical Physics Amsterdam Institute of Physics

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Abstract

In this thesis we investigate quantum control of qubits defined by a super-symmetric lattice model. Periodic chains with length L = 3n produce two degenerate ground states, which can be interpreted as the computational states of a qubit. After the introduction of a staggering parameter in the lattice model we can choose a closed adiabatic path in this parameter space which results in a non-Abelian Berry phase. This unitary can be interpreted as a geometric quantum gate operation.

The triangle (L = 3) and hexagon (L = 6) lattice configurations both define a single qubit and allow us to investigate the construction of one-qubit quan-tum gates. We show that we can define the phase shift gate and rotation gate by a geometric procedure, which are sufficient for the construction of a general one-qubit gate.

In the bow tie lattice, defined as two connected triangles (qubits), we in-vestigate the construction of a non-trivial two-qubit quantum gate. We explore its non-trivial nature with the entanglement entropy as measure. We find that the procedure based on the non-Abelian Berry phase can pro-duce near-maximal entanglement entropy. While the non-trivial nature of the constructed two-qubit quantum gate is shown, the proof of equivalence to known non-trivial (universal) two-qubit quantum gates is not given in this work.

Title: Geometric quantum computing with supersymmetric lattice models Author: Diko Hemminga, diko.hemminga@casema.nl, 10680268

Supervisor: prof.dr. Kareljan Schoutens Examiner: dr. Marcel Vonk

Project conducted between September 2017 and July 2018 Date final version: August 25, 2018

Institute for Theoretical Physics Amsterdam University of Amsterdam

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Acknowledgements

I would like to express my gratitude to Kareljan Schoutens, for being willing to supervise my project, supplying me with suggestions and keeping me on the right track during the process. I would like to thank Marcel Vonk, for agreeing to be my second supervisor and helping with the mathematical picture. Many thanks are due to Herma, my mum, for the endless support and all the encouragements. Naturally also thanks are due to Haijo, my dad, for planting the seeds for my interest in physics when I was young and being there along the way. Lastly, I would like to thank Patrick, for being someone to talk to during lunch, and the other inhabitants of the Master Students room, for providing a nice place to work on this project.

Nederlandse populair wetenschappelijke samenvatting

In dit project werk ik een nieuw idee uit voor een quantumcomputer. De basiselementen van een quantumcomputer zijn qubits, die de wetten van de quantummechanica volgen. Dit in tegenstelling tot de bits in een klassieke computer, zoals je laptop of smartphone. De quantummechanica voegt nieuwe mogelijkheden toe, zoals superposities en verstrengeling van qubits. Daarvan gebruikmakend zal een quantumcomputer bepaalde rekentaken veel sneller kunnen uitvoeren. Het basiselement, de qubit, is een natuurkundig systeem. In mijn project gebruik ik een kleine keten van drie roosterpunten (een driehoekje). Er zijn regels waar een deeltje zich op het rooster mag bevinden. Deze keuze wordt ook wel een supersymmetrisch roostermodel genoemd. Het blijkt dat dit model twee mogelijke grondtoestanden voort-brengt, die we kunnen interpreteren als de toestanden van een qubit.

Om berekeningen te kunnen uitvoeren, moeten we de toestand van mijn ‘rooster-qubit’ gecontroleerd kunnen veranderen. Daarvoor introduceer ik een parameter in ons model, waarmee de grondtoestanden kunnen worden aangepast. Aangezien het veranderen van de parameter kan worden gezien als het volgen van een gesloten pad in de parameterruimte, gebruiken we concepten uit de meetkunde om het resultaat te beschrijven.

Met goede keuzes van de parameter kunnen we twee operaties (of quantum logische poorten) op een enkele qubit definiëren. Dit is zelfs genoeg om elke mogelijke operatie uit te voeren door de twee operaties te combineren. De volgende stap is het bekijken van een systeem van twee qubits. We koppelen twee driehoekjes tot een nieuw rooster dat lijkt op een vlinderdas. In dit systeem onderzoeken we of we de twee qubits kunnen verstrengelen, wat niet mogelijk is met operaties op maar één van de qubits tegelijk. Opnieuw gebruiken we een pad in de parameterruimte van het roostermodel. We vinden inderdaad een verstrengelende operatie op twee qubits, maar kunnen deze moeilijk vergelijken met operaties die bekend zijn uit de literatuur.

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List of Figures

2.1 Schematic view of the (empty) triangle and hexagon lattices, periodic chain of three and six sites, respectively. The edges correspond to neighboring sites in the lattice model. . . 6 2.2 Schematic view of the (empty) bowtie and trident lattices.

The edges correspond to neighboring sites in the lattice model. 6 2.3 The triangle lattice with the site-dependent staggering

para-meter ~λ. . . 11 2.4 Reduction of the staggering parameters on the hexagon lattice

using translational symmetry. . . 14 2.5 Reduction of the staggering parameters on the trident lattice

using rotational symmetry. . . 15 2.6 Reduction of the staggering parameters on the bow tie lattice

using mirror symmetry. . . 15

3.1 Representation of the pure-state space of a single qubit also known as the Bloch sphere. . . 18

4.1 The Bloch sphere S2 representing the state space of a single qubit. . . 55 4.2 The staggering parameter space for ~λ ∈ S2. . . 56 4.3 Schematic view of the map between the path in staggering

parameter space and the space of lattice states. . . 59 4.4 Schematic view of a conical path in a spherical parameter space. 62 4.5 Schematic representation of the tripod configuration (left)

and the Λ configuration (right) of a atomic level scheme. . . . 68

5.1 Schematic view of the |1i state on the triangle lattice, a pe-riodic chain of three sites. Empty sites are represented by white circles and occupied sites by black circles. The edges correspond to neighbouring sites in the lattice model. . . 72 5.2 The triangle lattice with staggering parameter ~λ included

(left) and the space of the staggering parameter S2 (right). . 74 5.3 Schematic view of the hexagon lattice, a periodic chain of six

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5.4 The hexagon lattice with the staggering parameter ~λ showing its reduction by translational symmetry. . . 82 5.5 Plots of a path on the sphere; triangular loop and a circle

around the equator. . . 88 5.6 Plots of a path on the sphere; a circle through both North

and South Pole and an orange-sliced-shaped loop. . . 89 5.7 Triangular path in the staggering parameter space . . . 95

6.1 Schematic view of the bow tie lattice configuration. . . 104 6.2 The bow tie lattice with the definition of the staggering

para-meter on this lattice. . . 107 6.3 Construction of the bow tie lattice by two triangle lattices and

therefore the creation of a two-qubit system by connecting two one-qubit systems. . . 117 6.4 The entanglement entropy as a function of dµb generated by

a geometric two-qubit quantum gate defined by path para-meters θb and φb. . . 127

6.5 The entanglement entropy as a function of θb (a,b) or φb(c,d)

generated by a geometric two-qubit quantum gate defined by path parameters θb, φb and µb. . . 128

6.6 The entanglement entropy S as a function of parameters θb

and φb for three values of the parameter dµb = 1, 10 and 1000. 130

6.7 The entanglement entropy as a function of dµb generated by

6.8 The entanglement entropy as a function of dµb generated by

7.1 Examples of longer lattice chains . . . 142 7.2 Two new lattice configurations as ansatz of two qubits (a) or

one qubit (b). . . 143

A.1 Two perspectives at the trident lattice configuration. . . 154 A.2 Tripod atomic level scheme as investigated by Leroux et al.

[29]. . . 159 A.3 Path in (φ1, φ2) phase parameter space as implemented by

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Chapter 1

Introduction and summary

In this thesis we will describe our research into geometric quantum com-putation based on supersymmetric lattice models. The elementary unit of quantum computation, the qubit, is defined in terms of the states of the lattice model. We use a special feature of the supersymmetric lattice model: For periodic chain with a length a multiple of three lattice sites the model produce two zero-energy degenerate ground states [16, 18]. This space de-fines the computational qubit space. We will introduce the supersymmetric lattice model in detail in chapter 2. The focus of this work is the investiga-tion of quantum control of the qubit defined in the supersymmetric lattice model. A general introduction to quantum computation will be given in 3. Quantum control is an essential component of an implementation of quan-tum computation in a physical system [12]. To this end, we introduce ex-plicit parameter dependence in the supersymmetric lattice model, by means of a staggering parameter on the lattice sites. This translates to parameter dependence in the lattice ground states and therefore in the qubit definition on the lattice. We consider a closed path in parameter space, transversed adiabatically. This action results in a geometric phase, known as the Berry phase, on the states in the ground state subspace. As the two ground states are degenerate, we find a matrix-valued non-Abelian Berry phase. We will see that the non-Abelian Berry phase can be interpreted in the context of gauge theory and fibre bundles [9, 43]. We will go into this mathematical structure in chapter 4.

The non-Abelian Berry phase on the lattice subspace after a closed adia-batic path naturally defines a unitary operation on the computational qubit space. In this way the non-Abelian Berry phase enables us to investigate quantum gate operations on the qubits after a geometric path in parameter space. This is definition of geometric or holonomic quantum computation. Our work can be of interest for the research to limit the influence of decoher-ence on quantum computation. As the size of contemporary experimental

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implementations of quantum computation are not suitable for fault-tolerant quantum computation yet [38], it is useful to investigate approaches which limit decoherence. To this end, we combine the supersymmetric lattice model with geometric quantum computation. The supersymmetric prop-erties produces protection of the ground states (the qubit states here) as long as supersymmetry is not broken. Geometric schemes are more resis-tant to small errors as the gate operation depends on a geometric quantity such as an enclosed surface (solid angle) of the path in parameter space. Geometric quantum computation can seen as an intermediate step towards topological quantum computation [46, 47].

We will implement the described method for geometric quantum computa-tion with a supersymmetric lattice model in the second part of this thesis. In chapter 5 we start with the explicit construction of the lattice ground states in the two simplest lattice yielding two degenerate ground states: the triangle and the hexagon lattices, periodic chains of three and six sites re-spectively1. We introduce two choices for the staggering parameter on the lattice and parametrize it by the standard spherical angles. This allows us to define a path in parameter space, with freedom in the surface enclosed by the path. The unitary matrices found by the non-Abelian Berry phase can be interpreted directly as the matrix form of known quantum gates. The two staggering choices result in the definition of the phase shift quantum gate and the rotation quantum gate. This set is sufficient for one-qubit quantum control as stated by quantum gate universality [36].

In chapter 6 we extend our discussion to a new lattice configuration. We attach two triangle lattices by an edge to create the bow tie lattice. The natural interpretation as two-qubit system allows for the investigation of a non-trivial two-qubit quantum gate. The additional parameter on the con-nection between the triangles acts as a coupling between the qubits defined on each of the triangles. We investigate the non-trivial nature of the uni-tary constructed by the non-Abelian Berry phase in the bow tie lattice. A non-trivial two-qubit quantum gate can not be decomposed into one-qubit quantum gates and is as such essential for quantum gate universality [36]. Its construction would enable us to perform any n-qubit operation, as the tri-angle lattice has been shown to be able to produce any one-qubit operation. The constructed two-qubit quantum gate unitary matrices are investigated by measuring the entanglement entropy of the final state after application of the quantum gate. As entanglement can not be produced by one-qubit oper-ations, it can give an indication for the wanted non-trivial behaviour of the two-qubit operations. We find that geometric paths can be chosen which produce a final state with near-maximal entanglement entropy. However,

1

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the dependence of the matrix form of the quantum gate on the parameters determining the geometric path is highly non-linear. This makes optimiza-tion of the geometric path to achieve the maximal entanglement entropy difficult. Although our analysis shows that the produced two-qubit quan-tum gate is non-trivial, we are not able to transform the found expression to any known universal two-qubit gate such as CNOT.

In conclusion, we show that for geometric quantum computation defined on the supersymmetric lattice we are able to construct a general one-qubit quantum gate and we are also able to find non-trivial two-qubit quantum gates. However, to be able to achieve quantum gate universality more re-search into the geometric two-qubit quantum gate is needed.

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Chapter 2

Supersymmetric lattice

models

This chapter about the supersymmetric lattice models is the first of the three theoretically oriented chapters. We will introduce the supersymmetric lattice model including staggering on the lattice sites. Then we will consider different supersymmetric Mk models and lattice symmetries.

The supersymmetric lattice model describes spin-less fermions on adjacent locations, called sites. We will first consider periodic chains, one-dimensional lattices. Each site has two neighbors. In particular, we will be investigating the periodic chain with three and six sites, named the triangle lattice and the hexagon lattice. These configurations are given in figure 2.1. Second, we will also consider other configurations. We combine two triangles by adding an edge between them to create a new lattice configuration, called the bow tie lattice. In the appendix we consider the trident, where three sites are only adjacent to a fourth site ‘in the middle’. These configurations are given in figure 2.2.

As is standard in condensed matter physics, the model is described by its Hamiltonian. The Hamiltonian is defined in terms of operators Q and Q†, which are called supersymmetric charges. The notation Q−= Q and Q+= Q† will be used in this text. The operators Q± change the number of fermions on the lattice by ±1. The Hamiltonian H is defined as the anti-commutator of the supersymmetric charges,

H = {Q, Q†} = {Q−, Q+} = Q+Q−+ Q−Q+. (2.1) The supersymmetric charges are nilpotent, they square to zero: (Q−)2 = 0 and (Q+)2 = 0 [54]. The result is that the following relations hold for the commutators with the Hamiltonian,

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Site 1 Site 2 Site 3 Site 1 Site 2 Site 3 Site 4 Site 5 Site 6

Figure 2.1: Schematic view of the (empty) triangle and hexagon lattices, periodic chain of three and six sites, respectively. The edges correspond to neighboring sites in the lattice model.

Site 1 Site 2 Site 3 Site 4 Site 5 Site 6 Site 1 Site 2 Site 3 Site 4

Figure 2.2: Schematic view of the (empty) bowtie and trident lattices. The edges correspond to neighboring sites in the lattice model.

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Another example of nilpotent operators in physics are the Grassmann vari-ables in the path integral formulation of fermions [2].

The spectrum of the Hamiltonian H is positive semi-definite. Eigenstates of this Hamiltonian are divided into doublet pairs and singlets. The doublets consist of two states with a difference in fermion number of one, but with the same energy (by eq. (2.2)). For example, a state |Ψi forms a doublet with its supersymmetric partner |χi = Q+|Ψi. As part of the definition we know that Q−|Ψi = 0 and Q+_{|χi = 0 must hold as annihilation properties.}

The ground states of a supersymmetric lattice model are the singlet states. By definition of a singlet state |Φi, it holds that Q+|Φi = 0 and Q−|Φi = 0. We find that its eigenvalue of the Hamiltonian, the energy, is equal to zero: H |Φi = {Q−, Q+} |Φi = 0 = 0 |Φi. A supersymmetric lattice model gives rise to singlet ground states with exactly zero energy. This special feature prompts us to use analytical calculations in this work and at the same time allows this.

In a more general formulation, the supersymmetric N = 2 algebra is formed by the set of operators {Q−, Q+, H, F }, where F is the fermion number operator. It is defined by F = PN

j=1c

†

jcj, effectively counting the

num-ber of fermions on the chain. The operators cj, c†_j are the standard

an-nihilation and creation operator, respectively. These satisfy the fermionic anti-commutation relations {ci, c†_j} = δij, {c_i†, c†_j} = 0 and {ci, cj} = 0 [18].

We can write the supersymmetric charge operators Q− and Q+ _{in terms}

of the fermionic operators [25]. We use also the expression P which is

defined by P= Y j next to i Pj = Y j next to i (1 − c†_jcj). (2.3)

The operator Pj = 1 − Fj = 1 − c†jcj is called a projector, as we can see

that P = P2_{. As F}

j is the fermion number operator on site j the projector

Pj returns 1-# number of fermions on site j. Therefore the expression for

P in eq. (2.3) requires all sites adjacent to site i to be empty to be

non-zero. In the case of the supersymmetric lattice model with nearest neighbor exclusion, later referred to as M1, the supersymmetric charge operators are

defined as Q =X i c†_iP= Q+, Q†= X i ciP= Q−. (2.4)

With these definitions we can rewrite the (M1) Hamiltonian completely in

terms of fermionic operators, the result is as given in Fendley et al. [18] is

H = {Q†, Q} (2.5) = X i X j next to i Pc†icjP<j>+ X i P. (2.6)

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On the periodic chain we can write out the sum over j, with index i to be understood with i mod N ,

H = N X i=1 h Pi−1 c†_ici+1+ c†i+1ci

Pi+2+ Pi−1Pi+1

i

, (2.7)

as presented in Fendley et al. [18].

2.1 Features of M

k

models

Mk models describe interacting spin-less fermions by an explicit N = 2

supersymmetric lattice model. These models are often considered on a one-dimensional lattice, such as the periodic chain, but can also be extended to higher dimensions. The index k refers to the used exclusion rule, which allows a group of at most k fermions on neighboring sites [19]. A model with nearest neighbor exclusion is therefore classified as M1, as it does not

allow two adjacent sites to be occupied. Equivalent terms used in literature are fermions with hard-core exclusion or ‘fat’ fermions. We can summarize the condition as PjPj+1 = 0, ∀j [16].

We need to extend the definition of the supersymmetric charges to the Mk

model. This will be done in terms of the constrained fermionic creation and annihilation operators d†_[a,b],j and d[a,b],j. The operator d†_[a,b],j, where

a, b = 1, . . . , k and b ≤ a, creates a fermion at lattice site j in such a way that a string of a particles is formed, with the newly created particle at position b in the string. For the M1 model we find the relation d

†

[1,1],j= Pj−1c † jPj+1.

The interpretation is that d†_j creates a fermion on position j if the two adjacent sites j − 1 and j + 1 are empty. The supersymmetric charge Q+ for a periodic chain of length L can now be written as

Q+= L X j=1 X a,b d†_[a,b],j, (2.8)

with Q− defined equivalently.

It is possible to generalize the definition of the supersymmetric charges Q± more [17]. For the Mk model we define a Hilbert space Hk. Now we can

define subspaces H(f )_k defined by the eigenvalue f of the fermion number operator F . In other words we partition the Hilbert space according to the number of fermions in the states. The supersymmetric charges Q± define maps from the subspace with fermion number f to the subspaces with fermion number f + 1, respectively f − 1, while preserving supersymmetry

Q−: H(f )_k → H(f −1)_k , (2.9)

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In other words the supersymmetric charges Q±define maps from the fermionic sector of the Hilbert space Hk to the bosonic sector and vice versa [54].

The ground states allow for a description by cohomology theory, which makes it a tool to compute the number of ground states. The ground states are the states |si that satisfy Q+|si = 0, but also cannot be written in a form |si = Q+|s0_{i for any state |s}0_{i. Therefore the ground states form}

the cohomology of operator Q+ and the number of ground states is the dimension of this space [16, 18].

2.2 Staggering

In the definition of the supersymmetric charges introduced above each lattice site is treated equally. By introducing staggering we can be more general. We construct an inhomogeneous lattice model where the probability ampli-tudes are explicitly site dependent. If we consider the simplest example of the M1 model on the periodic chain, we introduce staggering as follows. We

add a site-dependent parameter λj to our definition of the supersymmetric

charges. Note that λj = 1, ∀j reproduces the expression stated before [16].

The nilpotency of the supersymmetric charges is conserved after introduc-tion, Q− =PN j=1λjPj−1cjPj+1, (2.11) Q+ =PN j=1λ ∗ jPj−1c † jPj+1. (2.12)

For the general case of the Mk model we can extend the definition of

stag-gering parameter λj to λ[a,b],j. Directly from the definitions in eq. (2.8) and

eq. (2.12), as also presented in Fokkema [19], we can write

Q+= L X j=1 X a,b λ_[a,b],jd†_[a,b],j. (2.13)

If we consider the M2 model the set of λ[a,b],j can be simplified to two

parameters, λj and µj. We define

λ_[1,1,],j = λj, λ[2,1,],j = λjµj, λ[2,2,],j = λjµj−1. (2.14)

So we can interpret this as introducing the staggering parameter µj

specif-ically for cases with neighboring particles, which are allowed by the M2

model. Note that the index j − 1 is natively defined mod L in a chain of length L, but can mean any adjacent site. We can interpret the structure as λj to be defined on site j and µj on edges between sites.

Several staggering choices are described in literature, such as in [16, 19]. Dependent on the lattice configuration the staggering parameter is often chosen with a certain configuration too. For example, in Fokkema [19] the

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M2 model is considered with λ[1,1],j =

√

2λj and λ[2,1],j = λ[2,2],j = λj.

Additionally the λj are staggered with period two: λ2n+1 = λ1, λ2n = λ2,

where we could set one to 1 and the other to λ, to effectively consider the ratio of λi. Two limits are discussed: λ → 0 is the extreme staggering limit

and for λ = 1 the model is critical. An example of staggering with a period of three sites in a M1 supersymmetric model is λ3i= λ3i+2 = y, λ3i+1= 1, ∀i,

which is described in Fendley and Hagendorf [16].

2.2.1 Staggering in the triangle and hexagon lattices

In this thesis we will first consider two chains of length three and six, coined the triangle and the hexagon. The reason will be elaborated on below. This suggests to take the staggering parameter with period three too. Keeping in mind our wish to perform an adiabatic path through parameter space, we choose the vector describing the staggering parameters as the unit vector normal to the surface of the sphere at the point (θ, φ). The spherical coordi-nates (r, θ, φ) satisfy the ISO1 convention commonly used in physics; radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). Explicitly, our choice for the staggering parameter is the three-dimensional vector

~_{λ =}   sin θ cos φ sin θ sin φ cos θ   (2.15)

so we choose λ3n+1 = sin θ cos φ, λ3n+2 = sin θ sin φ and λ3n = cos θ. The

length

~_λ

is unity. We will give the point θ = 0, the North Pole on the sphere, a special function in our procedure. As ~λ

θ=0 = (0, 0, 1),

indepen-dently of φ, this is an ideal point to start and end our adiabatic path through parameter space. Another example is constant staggering, without site de-pendence, which can be achieved using θ = cos−1(√1

3) and φ = π

4. In that

case we find ~λ = √1

3(1, 1, 1). The triangle lattice with the added staggering

parameter ~λ = (λ1, λ2, λ3) is shown in figure 2.3.

Effectively the ‘staggering parameters’ are θ and φ. We could introduce a time dependence t in θ and φ, such that the adiabatic path in parametrized by t. We choose to keep θ and φ as independent variables, writing all derivatives and integrals in terms of these parameters. We will show that the staggering parameter ~λ will lead to a transformation matrix known as a rotation matrix, using the non-Abelian Berry phase, in chapter 5. It is useful to also think about a different, independent, choice for the staggering

1

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λ1 λ2 λ3

Figure 2.3: The triangle lattice with the site-dependent staggering para-meter ~λ.

parameter. Let us define

~_λ_Ω₌   0 − sin(θ/2) exp(iφ) cos(θ/2)  . (2.16)

We will show that this choice leads to a transformation matrix introducing a phase shift in section 5.5.

2.3 Features of different chain lengths

In literature it is described that different lattice configurations and super-symmetric lattice models allow for different numbers of supersuper-symmetric ground states [52, 17]. The main tool is the Witten index. Its general definition is

W = Tr[(−1)F exp{−βH}], (2.17) where F is the number of fermions, β is the inverse temperature and H denotes the Hamiltonian. Note that it can be thought of as a modification of the standard partition function, with the difference in the included factor (−1)F [2].

In a supersymmetric theory the expression for the Witten index is indepen-dent of β. The reason for this simplification is the following, which is a result of the level structure of a supersymmetric theory. The allowed states are arranged in doublets and singlets. The states in a doublet both have the same energy but a fermion number differing by one, therefore canceling each other in the calculation of the Witten index. The singlets are states with zero energy so also resulting in terms independent of β. So the expression can be written as

WS =

X

(−1)F, (2.18) where the sum goes over the possible states on the lattice and F denotes the number of fermions in each of those states. The Witten index defines a

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lower bound to the number of zero-energy ground states and is effectively equal to the number of bosonic ground states minus the number of fermionic ground states. In other words the absolute value |W | equals the number of zero-energy ground states [17], as in the cases considered here the ground states lie in a single sector. In our work we find the number of ground states using the Witten index; we write out all states allowed by the supersym-metric model and sum the number of states for each allowed fermion number with the suitable sign. The value of the Witten index is independent of all supersymmetry preserving deformations [54].

If we consider a M1 chain, a one-dimensional periodic lattice with hard-core

fermions, with a length of a multiple of three, the number of ground states is equal to two. By definition both have zero energy. This result is shown in Fendley and Hagendorf [16], Fendley et al. [18].

A argument for this feature is given by energy minimization. Consider the Hamiltonian in eq. (2.7): The first term allows fermions to hop to neighbor-ing sites (while maintainneighbor-ing nearest-neighbor exclusion). The second term favors adding more fermions, as long as they can be more than two sites apart. This can be interpreted as an additional repulsive potential [18]. So the energetically optimal lattice configuration consists of fermions three sites apart. In other words the number of fermions in the ground states is N for a periodic lattice chain of length 3N [16]. By taking every third site in the lattice chain as a sublattice, one can show that any periodic lattice chain of length 3N produces two supersymmetric ground states [18].

Number of ground states for the triangle and hexagon lattices

We can also calculate the number of ground states by using the Witten in-dex explicitly. In the triangle (figure 2.1) there exist one state with zero fermions, the empty state, which corresponds to the empty lattice without fermions. Moreover, there exist three states with one fermion, which already corresponds to the maximum occupation. The M1 condition, nearest

neigh-bor repulsion, forbids us to add another particle to the lattice. The Witten index (eq. (2.18)) is equal to W = 1 − 3 = −2. The supersymmetric M1

model on the triangle has two fermionic ground states. By looking at the structure of doublets and singlets, we find that the ground states are states with one fermion, one doublet is formed between the empty state and one of the one-fermion states.

We can also do the counting in the hexagon (figure 2.1): There is one state with zero fermions on the chain, the empty chain. Six states with one fermion, as the fermion can be one each of the vertices. There are nine states with two fermions; six states with the fermions two sites apart and three states with the fermion three sites apart. There are also two states

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with three fermions. The M1 condition results in half filling (three fermions

on six sites) as the maximal occupation. We can now find the Witten index W = 1 − 6 + 9 − 2 = +2 by eq. (2.18), so the supersymmetric M1 model on

the hexagon has two bosonic ground states. The structure of doublets gives that the ground states are two-fermion states. The ground states on the triangle lattice and the hexagon lattice are investigated in detail in Chapter 5.

Number of ground states for the trident and bow tie lattices In the trident lattice configuration (figure 2.2) we can also easily do the counting. We imposed the M2 model here. The possible states are an

empty state (zero fermions), four states with one fermion, six states with two fermions and one state with three fermions. Three fermions is the max-imum density for M2 on the trident lattice. The Witten index results in

W = 1 − 4 + 6 − 1 = +2. So the supersymmetric M2 model on the trident

has two bosonic ground states. The structure of doublets gives that the ground states are two-fermion states. The trident lattice is investigated in the appendix.

The bow tie lattice (figure 2.2) is constructed by connecting two triangles by one of their vertices. We will impose a combined M1+ M2 supersymmetric

model: In each of the triangles we allow at most one fermion (M1), but the

fermions may both be on the sites connected by the added edge between the triangles (M2). We can see in two ways that the ground states are states

with two fermions. The ground states of each of the triangles are one-fermion states and adding the extra edge (with the M2 condition) does not change

this. Therefore combined there are four ground states on the bow tie, with two fermions on the lattice. The number four is simply calculated as there are two possibilities on each of the two triangles. The second method is to again make use of the Witten index: There is one state with zero fermions (the empty state), six states with one fermion and nine states with two fermions. Therefore the Witten index is equal to W = 1 − 6 + 9 = 4. Two fermions on the bow tie is the maximum occupation. The ground states on the bow tie lattice are investigated in detail in Chapter 6.

2.4 Lattice symmetry

In our discussions and calculations for the different lattice configurations we will use lattice symmetries explicitly. We assume that the operator corre-sponding to each lattice symmetry commutes with the Hamiltonian. Peri-odic lattice chains can be equipped with translational symmetry. In the case we will consider this is a symmetry for translation over three sites, as we chose our staggering parameter with a period of three sites. The hexagon,

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λ1 λ2 λ3 λ4 λ5 λ6 λ1 λ2 λ3 λ1 λ2 λ3

Figure 2.4: Reduction of the staggering parameters on the hexagon lattice using translational symmetry.

the chain with six sites, is effectively staggered with staggering parameter ~

λsix= (sin θ cos φ, sin θ sin φ, cos θ, sin θ cos φ, sin θ sin φ, cos θ). (2.19)

The symmetry corresponds to a certain lattice operator, so the states on the lattice can be classified by their eigenvalue of this operator. On multi-ple occasions this will help to reduce the total Hilbert space to the Hilbert subspace which contains the ground states. It is then natural to take as the basis for our problem the set of eigenstates of this operator. For the transla-tion operator over three sites, denoted T3, we need that on a chain of length

3n T3 satisfies T3n = 1. In the triangle, n = 1, the translation operator is

trivial. If we look specifically at the hexagon, we can easily see that the translation operator T3 has eigenvalues Λ = ±1. The operator works on the

lattice states explicitly by T3|ii = |i + 3i, where i mod 6 (we use site index

6 instead of 0). The reduction of the staggering parameter on the hexagon lattice is shown in figure 2.4.

In the trident configuration, where three sites are all connected to a fourth one but not each other, we can use rotational symmetry. Starting from the set of staggering parameters λi, i = 1, . . . , 4; µi, i = 1, . . . , 3, the rotational

symmetry reduces this to λ, λ4, µ. The reason is that we have to identify

the three sites and their links to the fourth one under rotational symmetry. The reduction is also shown in figure 2.5.

In the bowtie lattice, constructed by connecting two triangles, we will intro-duce a mirroring operator. You can imagine the mirror placed in the middle of the connection between the triangles. While at first sight the problem is described by two sets of the spherical coordinates (θi, φi), i = 1, 2, the

mirror operator identifies the sets with each other. Additional to stagger-ing parameter ~λ on the sites of both triangles, we define a parameter µ on the connection between triangles. This is conform the definition of a M2

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µ1 µ2 µ3 λ1 λ2 λ3 λ4 µ µ µ λ λ λ λ4

Figure 2.5: Reduction of the staggering parameters on the trident lattice using rotational symmetry.

λ1 λ2 λ3 λ4 λ5 λ6 µ3 λ1 λ2 λ3 λ1 λ2 λ3 µ3

Figure 2.6: Reduction of the staggering parameters on the bow tie lattice using mirror symmetry.

model between the triangles as given in eq. (2.13) with the specific M2 choice

eq. (2.14).

2.5 Supersymmetry in particle physics

The theory of supersymmetry is most often associated with particle physics, so we can not neglect mentioning this. The theory of supersymmetric charges in a supersymmetric algebra is especially applicable. Supersym-metry (SUSY) describes a relation between fermions and bosons, the two classes of elementary particles with half-integer and integer spin, respec-tively. The theory states that each particle has an associated particle in the other class. This superpartner, with a spin differing by a half-integer, would be new and undiscovered particles. In a perfectly unbroken super-symmetry, pairs of superpartners would share the same mass and have the same internal quantum numbers except for their spin. As the particles but no superpartners are discovered, if supersymmetry exists, it will be a spon-taneously broken symmetry, which would allows superpartners to differ in mass. At this moment there is no evidence whether supersymmetry is cor-rect, but it is an attractive solution to some of the major concerns in particle physics. More information can be found in Martin [33].

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Chapter 3

Qubits and quantum

computing

This chapter about quantum computation is the second of three theoretically oriented chapters. In this chapter we will introduce concepts of quantum computation, such as the qubit, entanglement and quantum gates. We will also consider limiting decoherence and achieving quantum gate universality.

3.1 Qubits

The quantum bit, or qubit, is the fundamental unit in a quantum computer. A classical bit can take the values zero and one, or off and on. Important differences between a classical and a quantum mechanical system (or bit) are the concepts of superposition and entanglement. A quantum bit can also take values ‘in between’, that is, its state can be a superposition of zero and one. In the qubit the zero and one are written as quantum mechanical ‘ket’ states: |0i and |1i. A superposition is defined as follows

|Ψi = c₀|0i + c₁|1i , (3.1)

where the state is normalized by |c0|2 + |c1|2 = 1, c0, c1 ∈ C. We can

vi-sually imagine these states as a vector from the origin to the surface of the unit sphere. The sphere in this context is called the Bloch sphere, shown in figure 3.1, where |0i lies on the North Pole and |1i on the South Pole. In other words the Bloch sphere is a geometrical representation of the pure state space. The Bloch sphere is equivalent to the complex projective line, CP1. This definition will be explored further in chapter 4.

While the concept of superposition can be present in the state of a single qubit, to find the meaning of entanglement we will look at a two-qubit sys-tem. The general state of two qubits is a four-dimensional complex vector in

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|1i

|0i

Figure 3.1: Representation of the pure-state space of a single qubit also known as the Bloch sphere.

the two-qubit basis {|00i , |01i , |10i , |11i}. The standard notational simpli-fication used here is |iji = |ii ⊗ |ji. These states are generically entangled, such that they can not be written as a product between two one-qubit states. Let us look at two examples

|P i = √1

2(|00i + |01i) = |0i ⊗ 1 √

2(|0i + |1i), (3.2) |Ei = √1

2(|00i + |11i), (3.3) where |P i is a product state, as it is written as a product of one-qubit states, and |Ei is an entangled state. The state |Ei is also known as one of the two-qubit Bell states [36]. Entanglement will be discussed further in section 3.6.

A qubit can in principle be realized in any quantum mechanical two-level system, although we have to be able to define initialization, control and read out of the system [12]. This will be discussed in more detail in section 3.4. We will discuss now a couple of examples of physically realized qubits. Three simple physical systems which realize a qubit are a single electron spin, the ground state and an excited state of an atom and the polarization of a photon. The spin of an electron, ~S = 1/2, has two possible values for its projection with respect to a chosen z-axis: Sz = ±1/2. We can choose

these values as qubit states, for example |0i = |S = 1/2, Sz= −1/2i = |↓i

and |1i = |S = 1/2, Sz = 1/2i = |↑i. The photon can be measured to have

two possible polarizations, clockwise and counterclockwise, which can also be used as qubit states. The best hardware available today are qubit imple-mentations based on trapped-ions [4] and superconducting circuits [6]. This work is loosely related to the concept of the topological qubit [48]. We will work on geometrical (or holonomic) quantum computation, which can be seen as a intermediate step towards topological quantum computing. Our wish and expectation is that this concept is more tolerant to errors than quantum computation based on other concepts.

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Definition of a qubit with a supersymmetric lattice model

After discussing general definitions and implementations of the qubit, let us define our qubit on the supersymmetric lattice. The main point of our work is to investigate the supersymmetric lattice model in a configuration which produces two zero-energy ground states. This is directly related to our wish to define a qubit. Each of the degenerate ground states will correspond to a qubit state, |0i or |1i. In our work we let the lattice model depend on a staggering parameter, so we find also a dependence on parameters θ, φ in the explicit expression of the ground states. We choose to define the qubit states in terms of the coefficient vectors of the ground states on the North Pole, θ = 0. We will define quantum gates on these qubit states using a path through parameter space. It is natural to let these paths start and end on the North Pole, where we have made a clear interpretation of the ground states as qubit states.

3.2 Noisy Intermediate Scale Quantum

In the recent proceedings Preskill [38] introduces the terminology of Noisy Intermediate Scale Quantum (NISQ) technology. He describes NISQ tech-nology as a pivotal new era in quantum techtech-nology. ‘Intermediate scale’ refers to quantum computers of a size of fifty to a few hundred qubits which will be available in the next few years. Fifty qubits is a significant mile-stone, as they can outperform the most powerful (classical) supercomputers. ‘Noisy’ emphasizes imperfect control over the qubits, the noise will place se-rious limitations on quantum technology in the near future. Preskill does not expect NISQ to change the world by itself; it should be regarded as a step towards more powerful quantum technology. The NISQ era describes quantum computers with noisy gates unprotected by quantum error correc-tion. The error gate per gate will put a bound on the maximum circuit size and therefore the computational power of NISQ technology. Moreover, we need to prepare and measure qubits accurately. From an engineering point of view the manufacturing constraints of number of connections and reliability should be kept in mind.

In the near future NISQ quantum computers may be able to outperform classical computers. Its power is based on quantum complexity, with the underlying concept of quantum entanglement. Important is also quantum error correction, which will determine the scalability of quantum computa-tion.

Preskill gives three reasons why quantum computers may have capabilities surpassing a classical computer. Quantum algorithms have been discov-ered for problems that are believed to be hard for classical computers. The well-known examples are Shor’s factoring algorithm [42] and Grover’s search

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algorithm [22]. The second argument comes from complexity theory: ‘easy quantum states’ have superclassical properties. The possibly most persua-sive argument is that there is no known classical algorithm to simulate a quantum computer. For more details on quantum supremacy, see Harrow and Montanaro [23].

For a physicist the natural problem to look at is simulating a many-particle quantum system. In such a quantum system entanglement has profound consequences. Especially dynamics of highly-entangled many-particle sys-tems is a promising arena where quantum computers may have a significant advantage over classical computers. The size of the quantum mechanical Hilbert space grows exponentially with the number of particles. This fun-damental fact makes the calculations of large quantum mechanical systems difficult. Let us compare the simulation of a quantum mechanical system in more precise terms. Consider the memory requirements of a system of N spin-1/2 particles. Each state can be in two states; |↑i or |↓i, and each clas-sical bit can also be in two states: 0 and 1. So we can describe the system of N ‘classical’ spins by N bits, bi ∈ {0, 1}, i = 1, . . . , N . However, in the

quan-tum case we allow superpositions (linear combinations) of arrays of N spins, which means that we need 2N complex numbers, αj ∈ C, j = 1, . . . , 2N, to

describe the state of the system. Note the exponential scaling with the sys-tem size N . This is the simplest argument to research quantum computation, to simulate large quantum mechanical systems. As the quantum computer is itself quantum mechanical, it benefits from the same exponential scaling [36]. While an universal quantum computer has not been developed (yet), for the research into quantum many-body systems there has been progress in quantum simulation. The main idea is to use a quantum mechanical system over which you have a lot of control to simulate a Hamiltonian of a system you want to investigate. These controllable systems can be realized in lab setup, with for example laser-induced periodic potentials.

Quantum computing is hard as a result of a fundamental feature of the quantum world: we cannot observe a quantum system without disturbing the system. That means that we need to keep the system nearly perfectly isolated from the environment, if we want to use the system to store and process quantum information. At the same time, the qubits must strongly interact with each other to perform computation and we need to control the system from the outside. Eventually we also want to read out the result of the computation. It is very challenging to build a quantum system that satisfies all the criteria. The expectation is that eventually the quantum sys-tems can be protected using the principle of quantum error correction and then quantum computers can be scaled up. Entanglement lies at the basis of quantum error correction as it is possible to encode quantum information in a highly entangled state. Unfortunately, there is a significant overhead cost

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for quantum error correction; it requires many additional physical qubits. For more detail see Steane [49]. Clever design of noise-resilient algorithms may extend the computational power of NISQ devices.

Let us stress the fundamental difference of measuring a classical bit and a quantum bit. The classical bit takes values 0 and 1. If we measure the value of a bit equal to 0 we will find 0. A qubit can also be in superposition states, in general |Bi = c0|0i+c1|1i, where ci∈ C and |c0|2+|c1|2 = 1. This

requirement makes sure the probabilities sum to one (Born’s rule). However, measurement destroys the superposition state, measuring the same qubit again will always give the same result as measuring the first time. This is a result of the quantum mechanical nature of the qubit. At the other hand, this is also what is meant in popular science by saying that the qubit is in the two states 0 and 1 at the same time. From the point of view of error correction this nature is very inconvenient. In a classical system it is possible to measure a bit and put its value in another bit, effectively copying the value. The single bit error known as ‘bit-flip’ can then be corrected by comparing to the reference bit(s). In a quantum system, as it is impossible to measure the qubit without destroying its superposition value, it is not possible to clone the qubit. This is the premise of the research field of quantum error correction [36].

3.3 Decoherence

The concept of decoherence is very important to consider for quantum com-putation. Decoherence, or quantum noise, can be seen as loss of information from the system (the quantum computer) to the environment. The ideal pic-ture of a closed quantum system is not true in reality. The coherence, or phase relation between different states, can be lost in a decay process. The characteristic time is called the decoherence time. During the process the system becomes coupled to the environment and entanglement is created, in some unknown way, between the system and the environment. In other words, quantum information is shared or transferred to the environment.

Decoherence can be modeled as non-unitary dynamics of the system alone. The combined system with environment still evolves unitarily. The system undergoes a irreversible (non-unitary) transformation acting on its Hilbert space. The irreversibility underlines the fact that information is lost. Ex-amples include dissipative and dephasing contributions [30].

Decoherence is also studied in the context of quantum measurement and the transition from quantum to classical behavior. The existence of

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deco-herence poses a challenge for physical realizations of quantum computation. It is necessary to manage its influence as the quantum computer relies on preserving the interference and entanglement between quantum states to perform computation. A solution is presented in the next subsection called quantum error correction. Another approach to solving the decoherence problem is the theory of decoherence-free subspaces and subsystems. For a review see Lidar and Whaley [30].

One of the challenges of an experimental realization of the quantum com-puter is therefore its coupling to the environment. A quantum comcom-puter has to be well isolated to retain its quantum properties, but at the same time its qubits have to be accessible. The qubits may need to be manipulated to perform computation or read out the results. A realistic implementation must achieve a delicate balance in its coupling to the environment [36].

3.3.1 Quantum error correction

Quantum error correction is a method to perform quantum information pro-cessing in a reliable manner in the presence of noise. The goal of quantum error correction is to increase the fidelity with which quantum information is stored or communicated. Quantum-error-correcting codes work by encoding quantum states in a special way, to make them noise resilient. Fault-tolerant quantum computation concerns the protection of quantum information as it dynamically undergoes computation. The theory of fault-tolerant quantum computation produces the remarkable ‘threshold theorem’: provided that the noise in individual quantum gates is below a certain constant threshold it is possible to efficiently perform an arbitrarily large quantum computation [36].

The key idea of (classical) protection against noise is to encode the mes-sage by adding redundant information to the mesmes-sage. So if the encoded message is affected by noise, the redundancy allows to decode and thus re-cover the message. However, there are some important differences between the classical and quantum information related to error correction. Firstly, while creating redundancy we have to take the no-cloning theorem in mind. Secondly, the errors are continuous, instead of a classical (discrete) bit flip error. And thirdly, measurement destroys the quantum information, we can-not recover the quantum state in this way. In Nielsen and Chuang, Chapter 10 is shown that these differences do not prove fatal for quantum error cor-rection. The general approach to creating redundancy uses physical ancilla qubits used for measurement in addition to the physical code qubits storing the quantum information.

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3.3.2 Our approach to fight decoherence

Let us discuss the motivation of the approach to quantum computation pre-sented here, in the context of protection against quantum errors. We look into geometric quantum computation on the supersymmetric lattice. Both main ingredients are important here. We use the fact that the supersym-metry protects the computational states as motivation for using this lattice model. If the supersymmetry is unbroken, the system is robust with respect to dissipation and decoherence. The supersymmetry of a quantum system is called unbroken if the ground state energy is zero. In other words, the supercharges Q, Q†annihilate all ground states. So in this work we assume unbroken supersymmetry. Moreover, the dimension of the degenerate space of ground states is given by the Witten index and therefore has the property of topological invariance. More information can be found in Tomka et al. [50].

By definition topological invariant quantities can not be changed by small perturbations. This makes quantum computation based on topology more robust to errors, although the details depend on the experimental realiza-tion. This feature has incited research into a form of fault-tolerant quantum computation known as topological quantum computation. For more infor-mation, see Nayak et al. [35].

Our work presents a theoretical ansatz for geometric quantum computation. Geometric quantum computation inherits protection against errors from its geometric principles. It can be seen as an intermediate step towards topolog-ical quantum computation. In the next chapter we will introduce the non-Abelian Berry phase, which will allows us to construct geometric quantum gates. The non-Abelian Berry phase does not depend on the specifics of the path, only on its geometry such as the area spanned by a loop. This makes geometric phases robust to noise in the classical control parameters [48]. It is also flexible in the way a computation is implemented, as there many dif-ferent paths possible to find a certain value of the solid angle spanned [31]. Experimental implementations of geometric quantum computation will be shown in section 4.7. This literature also discusses the protection against errors that is provided by the geometric approach.

3.4 Quantum control

The work in this thesis will focus on the possibilities of quantum control of the qubits defined on a supersymmetric lattice. The basis of quantum control consists of a set of operations we are able to perform on the qubit. We will talk about quantum gates and their matrix representations in the next section. The main goal of this research is to find which quantum gates can be performed on the supersymmetric lattice based qubits. As will be

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introduced in chapters 5 and 6, we take a geometrical approach to define the quantum gate operations.

To be able to speak about quantum computation on a supersymmetric lat-tice, we have to satisfy a set of criteria, also known as DiVincenzo’s criteria. In his paper DiVincenzo [12] describes five requirements for the physical im-plementation of quantum computation and adds two additional requirements for quantum communication. This set of requirements can be summarized as follows

1. A scalable physical system with well-characterized qubits

2. The ability to initialize the state of the qubits to a simple fiducial state, such as |000 . . .i

3. Long relevant decoherence times, much longer than the gate operation time

4. A ‘universal’ set of quantum gates 5. A qubit-specific measurement capability

6. The ability to interconvert stationary and flying qubits

7. The ability faithfully to transmit flying qubits between specified loca-tions

DiVincenzo’s seven requirements are aimed towards experimental implemen-tations of quantum computation and quantum communication [12]. Our work is primarily aimed at presenting a theoretical ansatz which can be used for geometric quantum computation. The requirements stated can not be satisfied from a theoretical discussion alone, only a experimental imple-mentation will give definite answers of the possibilities. From our point of view we will discuss the relevant requirements. We will not touch upon quantum communication, described by requirements 6 and 7.

The first requirement

The supersymmetric model is used to provide qubits; the two degenerate ground states define the computational space. We know from the supersym-metric theory that other states have positive energy, such that we assume that in combination with the adiabatic approach the probability of populat-ing these states is small. The set of lattice states are also a finite set; it is convenient to consider a Hilbert space of finite size, for example to minimize decoherence [36]. We have to note that this system does not correspond directly to a physical system, which would mean an experimental imple-mentation. A first step towards the scalability of the supersymmetric model based qubits is investigated in the bow tie lattice. Two qubits, defined on each of the triangles, are coupled to form a two-qubit system. In principle this approach could be extended to coupling more triangles (qubits) in this manner.

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The second and third requirements

The specifics of the initialization of the qubit state are also dependent on the physical implementation. The construction of a simple initial state is vital for quantum computation. To extract all information from the result of a calculation we need to be able to repeat the process, with the same ini-tial state each time. In the approach presented in this work the staggering parameter enables us to initialize the lattice quantum states in the compu-tational subspace. However, this will not in general be a simple state, such as |0i, it could be some superposition of |0i and |1i. Therefore initialization on the supersymmetric lattice to a simple state will require a extra protocol. Note that a commonly used cooling approach [12, 36] may not work as the computational subspace coincides with the ground state manifold. We will continue the discussion in chapter 7. The requirement of long decoherence times will also naturally depend on the experimental implementation. Our approach is thought to provide some protection to decoherence by combina-tion of the supersymmetric lattice model and the adiabatic approach. We have considered this in section 3.3.2.

The fourth requirement

The fourth requirement we will consider in detail. The main focus of our work is defining a universal set of quantum gates for the supersymmetric lattice based qubits. We want to construct the necessary unitary opera-tions by a geometric protocol. Our approach will be to construct unitary operations by the non-Abelian Berry phase. More information on universal quantum control can be found in sections 3.5 and 3.7. We consider different choices for the staggering parameter in the supersymmetric lattice model on the triangle to find the rotation and phase shift quantum gates, as pre-sented in chapter 5. Moreover, we investigate the possibilities of creating entanglement with a two-qubit unitary operation in chapter 6. We note that in the system considered for two-qubit operations also the one-qubit opera-tions are still possible, if we address the staggering parameter in each of the triangles separately. In theory we have the capability to address any set of two adjacent qubits by means of turning on the parameter on the connection between the triangles. However, we have not investigated beyond the bow tie configuration.

The fifth requirement

The final (computation) requirement concerns a qubit-specific measurement capability, which is (again) dependent on the experimental implementation. You could imagine that this means that we translate the computational qubit states back to lattice states and measure the probability of finding a fermion on each of the sites.

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(3.4) Measurement is naturally the final element of quantum computation and denoted by the meter symbol shown in 3.4. This pictorial notation will also be used in the next section (section 3.5) to show the construction of quantum circuits by quantum gates. It corresponds to a projective measure-ment in the computational basis, whereas more general measuremeasure-ments can be represented by unitary transforms followed by projective measurements. A projective measurement corresponds to what is known as a collapse of the wave function [36]. A measurement with (near) perfect quantum efficiency can also be used for state preparation in a quantum computation imple-mentation [12]. Projective measurements are often difficult to implement. It is required that the coupling between the quantum and classical system is large and also switchable. Unwanted (projective) measurements can be considered as a decoherence processes. Measurement can be viewed as an interface between the quantum and the classical world, it is generically con-sidered to be an irreversible operation: Measurement is destroying quantum information and replacing it by classical information. Only in special con-structed cases, such quantum teleportation and quantum error-correction, this does not need to hold; in these instances the measurement result does not reveal information about the identity of the quantum state being mea-sured. Measurement can be thought of as a process of coupling one or more qubits to a classical system, such that after some time the state of the qubits is indicated by the state of the classical system [36].

An interesting and important observation is that the constraints on the physical realization of the quantum computer are opposing in general. A quantum computer has to be well-isolated in order to retain its quantum properties, that is reduce decoherence, but at the same time the system has to be accessible so that it can be manipulated to perform a computation or to read out the results. A realistic implementation must find a balance between these constraints [36].

3.5 Quantum gates

Quantum gates, or generally quantum operations, are the extension of classi-cal computer gates to the quantum computer. In classiclassi-cal computers, based on bits valued zero or one, the operations are called logic gates. Apart from the trivial identity operator, leaving its input unchanged, the only non-trivial one-bit operator is the negation gate, NOT, taking zero to one and one to zero. We will see that in the language of the quantum gates, the set of

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one-qubit operators is extensively increased. The standard two-bit opera-tors are AND and OR. Combining these two-bit operaopera-tors with the negation gate NOT, creates the two-bit operators NAND and NOR. Rounding out the set of two-bit gates are XOR and XNOR. It is shown that circuits of only NOR gates (Pierce, 1880) or alternatively only NAND gates (Sheffer, 1913) can reproduce the functions of all the two-bit gates. Consequently, these gates are called universal logic gates. We are interested in this notion of universal gates in the case of quantum gates: we will introduce the equivalent definition in section 3.7 and aim for defining operations on the lattice creating this set.

One-qubit gates

The one-qubit gates perform a operation on one qubit at the time. The state of a qubit can be represented by a point on the Bloch sphere, so the set of possible one-qubit operations can be viewed as rotations to another point on the Bloch sphere. Therefore quantum states and quantum gates are often defined in terms of a vector basis. A single qubit is represented as a vector

|Ψi = c0|0i + c1|1i ⇔ Ψ =

c0

c1

(3.5)

and a one-qubit gate by a 2 × 2 unitary matrix. In general a gate acting on k qubits is represented by a 2k× 2k _{unitary matrix in the vector basis of k}

qubits |Ψki. For example if k = 2 the two-qubit gate is defined in the basis

of

|Ψ_k=2i = c00|00i + c01|01i + c10|10i + c11|11i ⇔ Ψk=2=

    c00 c01 c10 c11     (3.6)

The set of possible operations is extensively increased with respect to the classical bit. The operation related to the classical NOT gate is now called the Pauli X gate, as it can be represented as the Pauli matrix σx eq. (3.7)

applied to the qubit vector [36].

The classical negation operation is embedded as |0i = (1, 0) is changed to |1i = (0, 1) and vice versa. However, the quantum gate can be applied to any quantum state |Ψi = c0|0i+c1|1i, with the action X |Ψi = c0|1i+c1|0i.

The Pauli z-matrix defines the Pauli Z gate, which add a minus sign to |1i but leaves |0i unchanged. This gate is also called the phase-flip gate and has no classical analog. Its matrix form equals the Pauli z-matrix eq. (3.7). The third Pauli gate is Y, defined by the Pauli matrix σy 3.7, which by the

properties of the Pauli matrices can also be written as Y = iYZ. The Pauli gates NOT = X, Y and Z are shown in quantum circuit notation in 3.8.

σx= 0 1 1 0 σy = 0 −i i 0 σz = 1 0 0 −1 (3.7)

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X Y Z (3.8) An important gate in quantum computation algorithms is the Hadamard gate H. It is particularly known for its function in creating entangled states. Entanglement is the most important quantum mechanical concept for quan-tum computing. The Hadamard gate can be represented in the standard |0i , |1i one-qubit basis as

H = √1 2 1 1 1 −1 . (3.9) H (3.10) We can see that the Hadamard gate is related to the Pauli gates by H =

1 √

2(X + Z). The application of this gate creates a superposition state, special

to quantum computing compared to classical computing. The superposition is a linear combination of the classical zero and one states.

H |0i = √1

2(|0i + |1i), H |1i = 1 √

2(|0i − |1i) (3.11)

In the Bloch sphere picture the vector in moved from the poles to the equa-tor by the Hadamard gate [36]. The simple quantum circuit notation is shown in 3.10.

A more general one-qubit gate is the phase shift gate, which constitutes a family of gates parameterized by α. Its matrix definition is equal to

Phaseα =

1 0 0 eiα

(3.12)

where α is called the phase shift.

Phase(α) = Rα (3.13)

Common examples are taking α = π/4 and α = π/2, which are known as T and S. The phase shift α = π reproduces the Pauli Z gate, which is therefore also known as the phase flip gate. The quantum circuit notation is shown in 3.13.

The rotation quantum gate also constitutes a family of one-qubit quantum gates, parametrized by β. Its matrix representation in the one-qubit basis is equal to the rotation matrix in two-dimensions over an angle β

Rotationβ =

cos β − sin β sin β cos β

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Rot(β)

(3.15) For example the angle choice β = π₂ leads to the iY, the Pauli y-matrix multiplied by the imaginary unit i. It quantum circuit representation is given in 3.15.

Two-qubit gates

Two-qubit gates perform an action on two qubits. This is often represented in the two-qubit basis by a 4×4 unitary matrix. In quantum circuit notation the gate is extended to two qubits, as shown in 3.16.

|0i U2

|0i

(3.16) A simple example is the SWAP two-qubit gate, which swaps the values of two qubits. Its matrix representation in the 00, 01, 10, 11 basis is equal to

SWAP =     1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1     (3.17)

An important family of two-qubit gates are the controlled gates. The basic function is applying a operation on the second qubit if the first qubit equals |1i and leaving the second unchanged if the first equals |0i. This opera-tion can for example the Pauli gates X(= NOT), Y, Z. The general form of a controlled gate CU where U is the ‘controlled’ operator is given by

CU =     1 0 0 0 0 1 0 0 0 0 0 0 U     (3.18)

The CNOT quantum gate, where U = X, is generally used to create entan-glement between qubits as discussed in the next section 3.6. The controlled gate CU is given in quantum circuit notation in 3.19 and the special notation for the well-known CNOT quantum gate is shown in 3.20.

• U

Geometric quantum computing with supersymmetric lattice models

MSc Physics and Astronomy

Master Thesis

Geometric quantum computing with

supersymmetric lattice models

Diko Hemminga

Contents

List of Figures

Chapter 1

Introduction and summary

Chapter 2

Supersymmetric lattice

models

2.1

Features of M

models

2.2

Staggering

2.3

Features of different chain lengths

2.4

Lattice symmetry

2.5

Supersymmetry in particle physics

Chapter 3

Qubits and quantum

computing

3.1

Qubits

3.2

Noisy Intermediate Scale Quantum

3.3

Decoherence

3.4

Quantum control

3.5

Quantum gates