Hypergraph states: their entanglement and resource properties

for quantum computation

Henry Costello1

Project supervisor: Dr Dan Browne1,2

Degree Course: MSci Theoretical Physics

1 Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, UK

2 Centre for Doctoral Training in Delivering Quantum Technologies, University College London, Gower Street, London, WC1E 6BT, UK

1

LONDON’S GLOBAL UNIVERSITY

Abstract

The threshold theorem states that, if the total physical error rate in a quantum circuit is below a certain

threshold value, the amount of error that accumulates in the circuit is tolerable and can be corrected

for by error-correcting codes. It has been shown that entanglement purification protocols can be

implemented to reduce the noise in a quantum system below this threshold value. Using such a

protocol, a fault-tolerant quantum gate could be generated which, with the addition of operations from

the Clifford group, would be sufficient for a fault-tolerant universal quantum computer.

Here a protocol is presented which uses two identical copies of some noisy three qubit state as an

input and produces one pure three qubit hypergraph state as an output. The protocol is tested for three

different noise models: a dephasing channel, a depolarizing channel, and a global white noise model.

The protocol is first of all run just once, purifying qubit 1 then 2 then 3 in that order. It is then run

three times on the noisy three qubit state in several different orders. Finally, the number of iterations

of the protocol is also varied. These parameters are changed so that the optimal order and number of

iterations can be found to allow for the best performance of the purification protocol to hypergraph

states.

It is demonstrated that the optimal order of purifying the qubits of some noisy three qubit state is the

order 1,2,3 – 3,1,2 – 2,3,1 (where i={1,2,3} denotes the qubit which is purified). Once this is

established, the optimal number of iterations of this order is shown to be four iterations. It is

demonstrated that, using this order and number of iterations, the hypergraph purification protocol can

2

be used to obtain pure hypergraph states from an initial fidelity of: F>0.730 for noise introduced by

the dephasing channel, F>0.641 for noise introduced by the depolarising channel, and F>0.651 for

noise introduced by the global white noise model.

The results of this purification protocol are compared with previous known results for purifying to

three qubit hypergraph states and it is shown that this purification protocol is the most effective

protocol for purifying to such states.

Contents

Abstract……………………………………………………………………………………...…………3

Quantum gate notation………………………………………………………………….…………….7

1 Introduction……………………………………………………………………...……………8

1.1 Noise in a quantum system……………………………………………………………8

1.2 Fault-tolerance and the error threshold……………………………………………..…9

1.3 Universal quantum computation…………………………………………..…………10

2 Entanglement purification protocols…………………………………………………….....11

2.1 Introduction to entanglement………………………………………………………...12

2.2 Quantifying entanglement……………………………………………………………13

2.3 Entanglement purification protocol for bipartite states……………………………...14

3 Hypergraph states……………………………………………………………………...……16

3.1 Graphs………………………………………………………………………………..16

3.2 Multi-qubit graph states…………………………………………………………...…18

3.3 Generalisation to hypergraph states………………………………………………….20

3

3.4 The three qubit hypergraph state…………………………………………………….22

4 Entanglement purification protocol to hypergraph states………………………………..22

4.1 Motivation for developing a hypergraph purification protocol……………………...23

4.2 Noise models…………………………………………………………………………24

4.2.1 The dephasing channel……………………………………………………...25

4.2.2 The depolarising channel…………………………………………………...26

4.2.3 The global white noise model……………………………………………….26

4.2.4 Comparison of the different noise models…………………………………..27

4.3 The hypergraph purification protocol………………………………………………..28

4.3.1 Steps involved in the hypergraph purification protocol…………………….28

4.3.2 The purification protocol for one qubit……………………………………..29

5 Results for the hypergraph purification protocol…………………………………………31

5.1 Performance of the protocol after one iteration………………...……………………31

5.1.1 The fidelity of the output state after the purification of one qubit…………..32

5.1.2 The fidelity of the output state after the purification of all three qubits…….34

5.2 Purification to completely pure hypergraph states…………………………………..37

5.2.1 Achieving a fidelity of one for the output of the protocol…………………...37

5.2.2 Improving the performance of the purification protocol……………………38

5.2.3 Optimising the purification protocol………………………………………..41

6 Conclusion…………………………………………………………………………………...42

6.1 Comparison with the purification protocol to LMESs……………………………….42

4

6.2 Comment on purifying to an output state with a fidelity of one……………………..44

6.3 Generalising the hypergraph purification protocol…………………………………..45

References…………………………………………………………………………………………….47

Appendix A…………………………………………………………………………………………...51

Appendix B…………………………………………………………………………………………...52

Appendix C…………………………………………………………………………………………...54

Appendix D…………………………………………………………………………………………...56

Appendix E…………………………………………………………………………………………...58

Appendix F…………………………………………………………………………………………...60

Appendix G…………………………………………………………………………………………...62

Appendix H…………………………………………………………………………………………...63

Appendix I……………………………………………………………………………………………64

Appendix J……………………………………………………………………………………………65

5

Identity operator I=[1 00 1]

Pauli operators σ 1=X=[0 11 0]

σ 2=Y=[0 −ii 0 ]

σ 3=Z=[1 00 −1]

Hadamard operator H= 1√2 [1 1

1 −1]

Phase operator S=[1 00 i ]

π/8 operator T=[1 00 e iπ /4 ]

Controlled NOT operator CNOT =[1 0 0 00 1 0 00 0 0 10 0 1 0 ]

Controlled Z operator CZ=[1 0 0 00 1 0 00 0 1 00 0 0 −1]

Toffoli operator CCNOT=[1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0

]

Quantum gate notation

6

1 Introduction

1.1 Noise in a quantum system

The principles of quantum mechanics have proved to be both a blessing and a curse for the

development of the field of quantum computation. While the properties of quantum states have

allowed for such successful discoveries as Shor’s algorithm for integer factorisation [1] or Grover’s

search algorithm [2], the continuous nature of errors on these states makes noise in a quantum

computation very troublesome.

A quantum computer, like all information processing systems, is very vulnerable to noise and this

needs to be accounted for in order for the output of the quantum computation to be of any use. A

similar approach to that which is taken with classical error-correcting techniques might be used,

where an input is embedded in several copies of itself in order to introduce a level of redundancy into

the input. However, the no-cloning theorem forbids the duplication of quantum states [3], so this

method cannot be used for quantum error-correcting techniques. Furthermore, classical error-

correcting techniques require a measurement of the output state in order to determine what kind of

error needs to be corrected for. However, a measurement on a quantum state collapses the state into a

single eigenstate, so this cannot be included in a quantum error-correcting technique either.

Fortunately, despite these difficulties, there have been many great advances in the development of

effective quantum error-correcting codes. The first breakthrough came with the Shor code, which can

correct for arbitrary errors on a single qubit by encoding one logical qubit in nine physical qubits [4].

This was followed by the Steane code which does the same only with seven physical qubits [5]. These

types of code were generalised with the invention of the CSS codes, named for their inventors

Calderbank, Shor and Steane [6], which can correct for arbitrary errors on a single qubit by encoding

one logical qubit in an arbitrary number of physical qubits (although it has been shown that the

quantum Hamming bound limits the number of qubits needed to encode a single qubit, in order to

7

correct for arbitrary errors on the qubit, to a minimum of five [7]). The ultimate success of quantum

error-correcting codes came with their generalisation by Gottesman using stabilizer codes [8]. A

stabilizer code is defined by the +1-eigenstate of the subgroup S of the Pauli group on n qubits Gn

such that −I∉S , where Gn is the group generated by the operators

G1= {± I , ± iI ,± X , ±iX ,± Y , ±iY ,± Z ,±iZ }applied to each of the n qubits [6]. The dimensions of

this space are 2ksuch that k logical qubits can be encoded into nphysical qubits, although there is still

the limitation that five physical qubits are needed to encode one logical qubit in order to correct for

arbitrary errors on the one qubit.

1.2 Fault-tolerance and the error threshold

While stabilizer codes have made error-correction in a quantum computation both feasible and

relatively straightforward, they alone may be insufficient to perform quantum computations of

arbitrary length without the level of noise in the system rendering any output useless. In order for

quantum error-correcting codes to be able to remove the errors on a quantum computation, it is

necessary that such a computation be fault-tolerant. If the components of a quantum circuit are fault-

tolerant, the spread of errors is small enough that they can be corrected for by error-correcting codes.

If the system is not fault-tolerant, however, errors will spread at such a rate that they cannot be

corrected for and the output of the system will be unusable [9]. Developing fault-tolerant components

to a quantum circuit is therefore of great importance.

To simplify this task, a key number has been defined which is known as the error threshold. The

threshold theorem states that if the physical error rate in a quantum circuit is below a certain threshold

valueγ, the circuit can be transformed into one which is fault-tolerant and quantum error-correcting

codes can handle the level of noise in the system [10]. This implies that the physical error rate on each

component of the circuit must be below a certain value as well. If the physical error rate is above γ

then the level of noise in the system will be too much for error-correcting codes to cope with.

Unfortunately, current estimates of this value vary from being on the order of 10−6 [11] to 10−4 [12],

8

which is far too low to feasibly implement a fault-tolerant quantum computation with current

technological capabilities.

A very exciting step towards the ability to perform quantum computations with manageable levels of

noise has come with the development of entanglement purification protocols. These protocols involve

the transformation of N copies of some noisy state ρ to a smaller number of pure states [13, 14]. It

has been shown by Bennett et al. [15] that entanglement purification protocols are equivalent to a

quantum error-correcting code for a noisy channel and in some cases can actually perform better than

any error-correcting code. This means that entanglement purification could be a very useful tool in

constructing fault-tolerant quantum circuits, and much work has been done with different protocols

towards achieving this. In fact, entanglement purification protocols have proved to be so successful

that they can be used in the construction of a fault-tolerant universal quantum computer.

1.3 Universal quantum computation

In order to construct a fault-tolerant universal quantum computer, all of its constituent gates must be

individually fault-tolerant. A small number of quantum gates are known to be implementable fault-

tolerantly without modification [16], including the gates in the Clifford group C where C is the group

generated by the operatorsCg={H , S , CNOT } [6]. However, if a quantum computer were

constructed using only the following constituents then such a computer, although being fault-tolerant,

could also be efficiently simulated on a classical computer:

Qubits prepared in the computational basis states

Quantum gates from C

Measurements in the computational basis

This is known as the Gottesman-Knill theorem [17], the result of which is that these components

alone are not sufficient for a fault-tolerant universal quantum computer. However, this quantum

computer model is very simple so it is of interest to see if there is a simple modification to the model

that can be made in order to achieve universal quantum computation. It is known that the addition of

one gate which is not in C is enough to achieve universality. For instance, a quantum computer using

9

the gates from C with the addition of a π/8 phase gate is sufficient for universality [18]. The three

qubit Toffoli gate CCNOT is also known to be universal [19] and so a quantum computer constructed

using gates from C plus CCNOT would also be universal.

This is where entanglement purification protocols have proved to be useful. Although only one

additional gate is needed to construct a universal quantum computer, this additional gate must also be

fault-tolerant in order for such a computer to be fault-tolerant. Entanglement purification protocols

could provide a method by which a fault-tolerant gate could be constructed. Bravyi and Kitaev have

shown how purification protocols can be used to produce so-called “magic” states which, when

combined with gates fromC, are sufficient for universal quantum computation [20]. Their “magic”

state distillation protocol purifies these “magic” states from several copies of some noisy mixed stateρ

, and so such a computer would also be implementable fault-tolerantly.

This report presents a purification protocol based upon the class of multiparticle entangled states

known as hypergraph states, specifically three qubit hypergraph states. Hypergraph states form a

subset of an even larger class of multiparticle entangled states called the locally maximally

entangleable states (LMESs). As such, the approach taken is similar to the purification to LMESs

protocol presented by Carle et al. [21]. However, while the protocol presented by Carle et al. requires

the creation of local maximally entangled two qubit states in addition to the input state to be purified,

this protocol only uses measurements on the input state and a pre-prepared ancilla state. A purification

protocol to hypergraph states is presented as these states occur naturally in the analysis of important

quantum algorithms such as the Deutsch-Josza algorithm [22] and the Grover algorithm [23]. An

effective method of obtaining pure hypergraph states would be an important contribution to the field

of fault-tolerant quantum computation.

2 Entanglement purification protocols

Before presenting the hypergraph purification protocol, an overview of entanglement purification

protocols in general will be considered. First, the concept of entanglement will be explained, followed

10

by a more thorough quantitative analysis of both pure state and mixed state entanglement. The

entanglement purification protocol for bipartite states [14] will be presented as a useful example to

demonstrate how an entanglement purification protocol can produce output states with an improved

fidelity.

2.1 Introduction to entanglement

The curious characteristic trait of quantum mechanics known as entanglement has been near the

forefront of all discussion of quantum theory since it was first postulated by Schrodinger in 1935 [24],

and the importance of entanglement to quantum computation cannot be overstated. The concept of

entanglement is easily definable for pure quantum states; specifically, a pure quantum state is said to

be entangled across n systems if it cannot be expressed as a tensor product of states in those systems

⨂n|ψ n ⟩ [25]. For example, the state |ψ ⟩of a composite system of the two states|ψ1 ⟩=a1|0 ⟩+b1|1 ⟩and

|ψ2 ⟩=a2|0 ⟩+b2|1 ⟩which can be described by their tensor product is not entangled:

|ψ ⟩=|ψ1 ψ2 ⟩=(a1|0 ⟩+b1∨1⟩¿(a2|0 ⟩+b2∨1⟩)

¿a1 a2|00 ⟩+a1 b2|01 ⟩+b1 a2|10 ⟩+b1b2∨11⟩. (1)

Now consider the quantum system described by a state with only the first and last terms or second and

third terms of Eq.(1):

|Φ ⟩=a1 a2|00 ⟩+b1 b2∨11⟩, (2)

|Ψ ⟩=a1b2|01 ⟩+b1 a2|10 ⟩. (3)

If the state ¿Φ⟩ could be factorised to a form like Eq.(1) then comparing Eq.(2) and Eq.(1) reveals

that |Φ ⟩ → a1b2=b1a2=0 which implies that some of {an , bn} are equal to zero. However, this

cannot be the case as comparing Eq.(2) and Eq.(1) also gives |Φ ⟩ → a1 a2=b1b2=1 so none of {an ,

bn} can be equal to zero. Therefore, the state ¿Φ⟩ cannot be decomposable as the tensor product of

states of its constituent parts and so can be described as a maximally entangled state (corollary |Ψ ⟩ is

11

also a maximally entangled state following a similar argument). A very important set of states in

quantum computing (and for entanglement purification protocols in particular) which take this form

are the Bell states:

|Φ±⟩= 1√2

|00 ⟩ ± 1√2

∨11⟩,

(4)

|Ψ ± ⟩= 1√2

|01 ⟩ ± 1√2

|10 ⟩. (5)

Entanglement is a very useful property in quantum computation and is the key component of many

processes in quantum information theory, with the most well-known being quantum teleportation [26]

and super-dense coding [27]. In fact, it has been shown that for any quantum algorithm acting on pure

states to offer an exponential speed-up over classical computation there must be some degree of

multipartite entanglement present, though the same is not necessarily true of mixed states [28].

2.2 Quantifying entanglement

The distinction between entangled and unentangled states for pure states is well defined and as such

the entanglement of pure states is easily quantifiable. For some entangled pure state δ of a system

composed of subsystems A andB, its entanglement is measured by its entanglement entropy [29]

E (δ )=S ( ρA )=S( ρB), (6)

where S ( ρi )=−Tr (ρi ln ρi) is the von Neumann entropy of system i[30] and ρA=TrB|δ ⟩ ⟨δ| is the

reduced density matrix of subsystem A (similarly ρB=Tr A|δ ⟩ ⟨δ| is the reduced density matrix of

subsystem B). An unentangled state can be expressed as a product state and so its entanglement

entropy will be equal to zero as expected for a measure of entanglement.

The concept of entanglement for mixed states is not as well defined as it was for pure states. Mixed

states have been described which appear to be unentangled but still exhibit subtle nonlocal properties

[31] which would indicate some level of entanglement. However, the presence of noise in quantum

12

systems means that mixed states occur in quantum computations with a much higher frequency than

pure states. As such, a useful quantitative theory of mixed state entanglement is arguably of even

greater importance than for pure state entanglement. Many measures of entanglement for mixed states

have been introduced [32], the simplest of which is the entanglement of formation. In order to

demonstrate this, consider a mixed state

ρ=∑i

p i|ψ i ⟩ ⟨ψ i|. (7)

The entanglement of formation of the mixed state ρ is then defined as the minimum average

entanglement entropy of the pure states ¿ψ i ⟩ [33],

E ( ρ )=min∑i

p i E (ψ i ). (8)

2.3 Entanglement purification protocol for bipartite states

The basic principle of an entanglement purification protocol is that some amount of noisy states are

purified to a smaller amount of less noisy states. The motivation behind the protocols comes from the

fact that all input states to a quantum computation will be subjected to noise as they are transmitted

through noisy channels, an inevitable component in any quantum computer model. The amount of

noise on initial input states that a protocol can handle, as well as how quickly the fidelity of the output

states of the protocol approaches one, are good reference points to gauge the success and overall

efficiency of an entanglement purification protocol. For example, when Bennett et al. introduced the

entanglement purification protocols for bipartite states [14], they demonstrated that the yield of pure

Bell state singletsD ( M )>0if¿whereM is the mixed state from which the pure states are distilled [14].

In order to explain how an entanglement purification protocol can produce output states with an

improved fidelity, this purification protocol for bipartite states will be considered. The basics of this

purification protocol will be similar to the protocol for purifying to hypergraph states, as will be

demonstrated. The following steps are taken from the paper [14] by Bennett et al.

13

Two observers, Alice and Bob, who share pairs of some two-particle mixed state M , can increase the

fidelityF=⟨ Ψ−¿|M|Ψ −¿ ⟩¿¿ of M with local operations and two-way classical communication. First of

all, Alice and Bob perform a random bilateral rotation (or “twirl” [34]) on each shared pair in order to

transform the general state M into the form of a Werner state [31] which has the same fidelity as M :

W F=F ¿. (9)

The operator σ 2 is then applied to each of the two pairs. This operator maps |Ψ ± ⟩ ↔∨Φ∓⟩ and so

serves to transform the pairs from mostly ¿Ψ−¿ ⟩¿ Werner states to mostly ¿Φ+¿⟩¿ Werner states. A

bilateral controlled-NOT operation is then applied between the two impure ¿Φ+¿⟩¿ states. The

bilateral controlled-NOT operation is analogous to the CNOT operation on two qubits held by one

observer, only now operating between pairs of qubits held by two observers. If Alice holds qubits one

and three and Bob holds qubits two and four then, under the operation of the bilateral controlled-

NOT, qubit three is flipped if and only if qubit one is in the state ¿1⟩ and qubit four is flipped if and

only if qubit two is in the state ¿1⟩.

After the bilateral controlled-NOT operation has been applied, the target pair is locally measured

along the z axis and, if the target pair’s spins are parallel along the z axis (i.e. the target pair is in one

of the states |Φ±⟩= 1√2

|↑ ↑ ⟩ ± 1√2

|↓↓ ⟩, a completely equivalent definition to Eq.(4)), the

corresponding source pair is kept. If the spins are not parallel along the z axis then the source pair is

discarded. It is here that two-way classical communication is required, as Alice and Bob need to know

the result of one another’s measurement. If the source pair is kept, the operator σ 2 is then applied to

the state to convert it back to a mostly ¿Ψ−¿ ⟩¿ Werner state.

The resulting final Werner pair will have a fidelity F ' satisfying the recurrence relation

F '=F2+ 1

9(1−F )2

F2+ 23

F (1−F )+ 59(1−F)2

,

(10)

14

which gives F '>F for F> 12 , and this can clearly be seen in Figure 1. It was this success with the

entanglement purification protocol for bipartite states that has inspired the development of many more

purification protocols for many different states, including the protocol presented in this report.

3 Hypergraph states

Hypergraph states are a generalisation of graph states, a very useful class of multi-qubit state that can

be conveniently described by a mathematical graph. The mathematics of graph theory will be briefly

considered in order to give a solid basis with which to introduce graph states. The concept of a graph

state will then be generalised to hypergraph states, and finally the three qubit hypergraph state to be

used in the hypergraph purification protocol will be introduced.

3.1 Graphs

Describing a multi-qubit state using the mathematics of graph theory allows the depiction of the state

in a diagrammatic form, which can be very useful to conceptualise the dynamics of such a state. A

15

Figure 1. Plot of the initial fidelity F against final fidelity F ' for the bipartite entanglement purification protocol presented in [14]. The shaded area shows the difference between the two fidelities with the blue line being the final fidelity F ' and the

purple line being the initial fidelityF . It can be seen that F '>F for F> 12

and that F '=F forF=12

. It can also be seen

that the protocol reduces the fidelity of the final state for some initial fidelities; specifically F '<F for14<F<1

2. The

Initial fidelity, F

Final fidelity, F’

graph is completely described by the pair of setsG= (V , E ) [35]. Here V is the set of vertices of the

graph such that V= {1 , 2 ,…,n } where n is the total number of vertices in the graph. E is the set of

edges of the graph where an edge can join any two and only two vertices. Thus the elements of E

form two-element subsets of V [35] such thatE⊆ {{i , j }, i , j∈V ,i ≠ j }.

Graph theory defines many properties of graphs that will prove useful when the extension is made to

multi-qubit graph states. The order of a graph is defined as the number of vertices in the graph [35]

and is denoted by ¿G∨¿ (the number of edges in the graph is called the size of the graph and is

denoted by||G||). Graphs can be directed or undirected according to whether or not the set of edges E

is given by ordered pairs {a , b } where the order implies that vertices a and b are connected by a

directed edge from a tob [36]. The degree of a vertex is defined as the number of edges which are

connected to that vertex [37]. Two vertices x , y of G are adjacent if the edge {x , y }∈G [35] (note

that a specific vertex or edge can be referred to as an element of the graph ( v∈G ,e∈G ), or as an

element of the set of vertices ( v∈V ) or set of edges (e∈E) respectively). If the vertices x , y are

adjacent then they can be referred to as neighbours and the set of all vertices adjacent to x is referred

to as the neighbourhood ofx. The colourability of a graph can then be defined such that a graph G is

k -colourable if k colours can be assigned to each vertex so that neighbours have different colours

[37]. Finally, a simple graph is defined as a graph which contains no loops (where one edge starts and

ends at the same vertex) or multiple edges (where more than one edge joins the same two vertices)

[37]. For the most part only simple, undirected graphs will be considered for the remainder of this

report.

The usefulness of graph theory stems from the fact that these graphs can easily be depicted

diagrammatically. Figure 2 shows a typical mathematical graph in diagrammatic form and this can be

used to explicitly illustrate the different properties of graphs. The graph G1 in Figure 2 is of order

¿7∨¿ and size||5||. Vertex 5 is of degree three, the vertices V 2={1,2} are of degree two, the vertices

V 1={3,4,7 } are of degree one, and vertex six is of degree zero. Vertex 1 is neighbours with vertices

{2,5 }, vertex 2 is also neighbours with vertex5, and vertex 5 is also neighbours with vertex7.

16

Vertices{3,4 } are also neighbours while vertex 6 has no neighbours. As such, it is clear that G1 is a

three-colourable graph. It can also be seen that G1 is a simple, undirected graph as it contains no loops

or multiple or directed edges.

Figure 2. The graph G1 with set of vertices V= {1,2,3,4,5,6,7 } and set of edges

E={{1,2 } , {1,5 }, {2,5 } , {3,4 } , {5,7 }}. This graph is taken from [35]. G1 is a three-colourable, simple, undirected graph

of order ¿7∨¿ and size ||5||.

3.2 Multi-qubit graph states

The mathematics of graph theory can be used to describe the multi-qubit class of states known as

graph states with the simple definition that the qubits of the state are represented by the vertices of a

graph and that interactions between these qubits are represented by the edges of the graph [38]. To

obtain a graph state, start with n qubits prepared in an initial state vector

|Ψ ⟩=¿¿, (11)

where ¿. An interaction pattern is then applied which is represented by the graph G where adjacent

qubits i and j interact according to the unitaryU ij=e−i φ ij H ij [36]. Here φ ij denotes the coupling

strength of the interaction and H ij represents the interaction Hamiltonian. The only graphs which this

project is concerned with are simple, undirected graphs and as such the unitaries must be symmetric

U ij=U ji so that one direction is not preferred in the interaction. As well as this, there will be no

17

1

2

3

6

4

5 7

ordering of the edges in G and as such all of the unitaries of a graph state must commute

[U ij ,U jk ]=0 [36].

It has been shown that these conditions limit the choice of interaction in the graph state to one

described by the Ising interaction pattern U ijI (φ ij)=e−i φ ij Hij

I

(whereH ijI=σ 3

i σ 3j), along with additional

local z rotations [36]. In the graph states used for this project, all of the qubits will interact with the

same interaction which means that the interaction coupling strength must be set to one specific value.

Noting that local z rotations can be included in the interaction pattern without going against the

restrictions imposed on the unitaries, and setting φ ij=π , gives

U ij=e−iπ H ij (12)

with

H ij=14 (I ij−σ3

i −σ3j+H ij

I ), (13)

to finally give the interaction pattern with which the graph states to be used in this project will be

defined:

U ij=e−i π

4 ( I−σ 3i −σ3

j )U ij

I ( π4 ). (14)

The reasoning behind this choice of interaction pattern, as well as the choice of initial state vector, can

now be explained. The unitary U ij is equal to theCZ operation between qubits i and j . This operation

applies the Z operator to qubit j (the target qubit) if qubit i (the controlled qubit) is ¿1⟩, and

otherwise leaves it unchanged. With this in mind, it is clear thatU ij=U ij† , the implication of which is

that U ij will create or delete the edge {i , j } in G depending on whether or not the edge is already in G

[36]. More importantly, these choices ensure that U ij ¿+⟩i ¿+⟩ j is a maximally entangled state. This

means that the graph state described by such a unitary can be used for the implementation of an

entanglement purification protocol.

18

The concept of a graph state can now be formally defined. The n qubit graph state ¿G ⟩ which

corresponds to the graph G=(V ,E) is the pure state with state vector [36]

|G ⟩= ∏{i , j }∈ E

CZ ij¿¿¿. (15)

As an explicit example consider the six qubit graph state

|G6 ⟩=CZ 23CZ 36CZ35 CZ45 ¿¿. (16)

This graph state is represented diagrammatically in Figure 3a), along with the corresponding quantum

circuit that could be used to construct such a state in Figure 3b).

19

Diagram a) depicts the diagrammatic representation of the graph state G6 in Eq.(16). A quantum circuit that creates this entangled state is depicted in b).

a) b)

3.3 Generalisation to hypergraph states

The extension to hypergraph states is very simple and shall now be explained. Hypergraph states can

be considered as a generalisation of graph states and thus all of the concepts introduced for graph

states also apply to hypergraph states. However, while only two qubits in a graph state can be

involved in one interaction, and therefore an edge in the corresponding graph can only join two

vertices, in a hypergraph state any number of qubits can be involved in the interactions. Thus a

hypergraph can have edges which join more than one vertex, also known as hyperedges.

Before formally defining a hypergraph state, some additional terminology shall be introduced. The

cardinality of a hyperedge is the number of vertices within this edge, denoted byk [39]. A hyperedge

of cardinality k can be referred to as a k-hyperedge. A hypergraph is referred to as uniform of

cardinality k if all of the hyperedges in the hypergraph are k -hyperedges. Such a hypergraph can also

be called a k-uniform hypergraph [39].

Formally a hypergraph H=(V , E) is completely described by the set of vertices V= {1 ,2 ,…,n }

where n is the total number of vertices in the hypergraph, and the set of edges E connecting the

vertices [39]. The n qubit hypergraph state ¿ H ⟩ which corresponds to the graph H=(V , E) is then

the pure state with state vector

|H ⟩=∏k=1

n

∏{i1 , i2 ,… ,ik }∈E

Ck−1 Z i1 ,i2 , …,ik¿+⟩⊗n, (17)

where {i1 ,i2 , …,ik }∈ E means that k vertices are connected by a k-hyperedge [22]. The operator CZ

is not sufficient to describe interactions between multiple qubits, so for this the C k−1 Zi1 , i2 ,… ,ik operator

is introduced in Eq.(17). This operator is the same as the CZ operator except that it is now required

that multiple control qubits all be in the state ¿1⟩ in order for the Z operator to be applied to the

target qubit. If k=1 then the Z operator alone is applied to the single qubit. The first product in Eq.

(17) is to account for hyperedges of different cardinalities.

20

Illustrating this with an example shall make both Eq.(17) more understandable and the differences

between hypergraph states and graph states more obvious. Consider the seven qubit hypergraph state

|H 7 ⟩=C Z12C Z13CC Z234 CC Z456C Z56 Z7CCCCCC Z1234567 ¿¿.

(18)

This is a non-uniform hypergraph state with one one-hyperedge, three two-hyperedges, two three-

hyperedges, and one seven-hyperedge. The hypergraph which corresponds to ¿ H 7⟩ is depicted in

Figure 4a) with a quantum circuit that could be used to construct ¿ H 7 ⟩ depicted in Figure 4b).

21

Diagram a) depicts the hypergraph corresponding to the hypergraph state |H 7 ⟩ in Eq.(18). Here the yellow line depicts a one-hyperedge, the blue lines depict two-hyperedges, the red lines depict three-hyperedges, and the green line depicts a seven-hyperedge. Diagram b) gives a quantum circuit which could be used to create¿ H 7 ⟩.

a) b)

3.4 The three qubit hypergraph state

Having introduced the mathematics of hypergraph states, the specific state which shall be used in the

hypergraph purification protocol can be introduced:

|H ⟩=CC Z123 ¿¿. (19)

This is a simple, undirected, three-colourable, three-uniform, three qubit hypergraph state of order

¿3∨¿ and size||1||. This state corresponds to a hypergraph with vertices V={1,2,3 } and one three-

hyperedge connecting all three verticesE={{1,2,3 }}. For a state of the form of Eq.(19), the qubits of

such a state will be labelled in the same fashion as the vertices of the corresponding hypergraph (the

qubits of the state |H A ⟩=CC Zabc ¿¿, for example, would be labelled qubit a, qubit b, and qubit c).

Note that this state can be rewritten in an equivalent form in terms of the states ¿0 ⟩ and ¿1⟩

|H ⟩= 1√8

(|000 ⟩+|001 ⟩+|010 ⟩+|011⟩+|100 ⟩+|101 ⟩+|110 ⟩−¿111⟩), (20)

and the corresponding hypergraph can be represented diagrammatically as in Figure 5a), where again

Figure 5b) shows the quantum circuit that can be used to create this state.

4 Entanglement purification protocol to hypergraph states

Now that the mathematical basis of hypergraph states has been established, and considering the

analysis of the bipartite entanglement purification protocol in section 2.3, the entanglement

22

Figure 5. The three qubit hypergraph state from Eq.(19) is depicted diagrammatically in a) and a quantum circuit to generate this state is depicted in b).

a) b)

purification protocol to hypergraph states can now be described in detail. First of all, the noise models

which shall be used to test the efficiency of the protocol will be introduced and then the specific steps

of the protocol itself will be described. The density matrix formalism will be used for this purification

protocol, with the density matrix for the pure three qubit hypergraph state of Eq.(19) being

ρH=|H ⟩ ⟨ H|=CC Z123¿¿. (21)

4.1 Motivation for developing a hypergraph purification protocol

The motivation for developing an effective entanglement purification protocol to hypergraph states

comes from the successful purification protocols which have been developed for other multipartite

states. A purification protocol for the three qubit W-state

|W ⟩= 1√3

(|100 ⟩+|010 ⟩+¿100⟩) (22)

has been introduced [40], demonstrating the capabilities of purification protocols for three qubit

entangled states. Furthermore, and arguably of more importance to the purification protocol presented

here, several successful protocols have been developed for graph states [41 – 46]. However, the most

important motivational factor for the development of an entanglement purification protocol to

hypergraph states is the demonstration (by Carle et al.) of a purification protocol to locally maximally

entangleable states [21].

Recall, from section 3.2, that the coupling strength of the interactions between qubits in graph states

(and in hypergraph states) was set to π (i.e. φ ij=π ) from Eq.(12) onwards. Locally maximally

entangleable states (LMESs) are those states where φ ij can take any value from φ ij=0 to φ ij=2 π [47]

so these states represent a generalisation of hypergraph states (which are themselves a generalisation

of graph states). Therefore, a similar approach to the purification protocol to LMESs is taken when

constructing this purification protocol to hypergraph states. Both protocols prepare identical copies of

some noisy state ρ and apply local operations and measurements on the noisy state in order to purify

it. However, the purification protocol for LMESs also requires the local creation of maximally

23

entangled two qubit states. The purification protocol presented in this report does not require this

additional step; the preparation of the noisy states and the application of local operations and

measurements are sufficient for purifying the noisy states, as will be demonstrated. In addition to

being simpler to implement than the purification to LMESs, a purification protocol which is

specifically for hypergraph states is also of interest as these states appear in important quantum

algorithms, such as the Deutsch-Josza algorithm [22] and the Grover algorithm [23]. A protocol to

purify hypergraph states is, therefore, vital for the fault-tolerant application of these algorithms.

4.2 Noise models

Noise can be introduced to a quantum computer in many different ways. In fact, all of the systems

which constitute a functional quantum computer are vulnerable to noise in one way or another.

Quantum states can decohere when they are transported through channels, or the channels themselves

can be noisy and hence the output will be noisy. Quantum gates may have a certain amount of noise

which means that noise will be introduced to the states upon which these gates act. Noise can even be

introduced when the output of a quantum circuit is measured. An effective method for simulating all

of these different forms of noise in a quantum circuit is to use models constructed from applying the

Pauli operatorsσ i to the pure input state, also known as Pauli channels. Here two such Pauli channels

are introduced: the dephasing channel, and the depolarising channel. A third model will also be

introduced which is a much more straightforward way of modelling the noise in a system: the global

white noise model.

The efficiency of an entanglement purification protocol is most easily tested by exposing an input

state to noise and then seeing for what level of noise the purification protocol can still distil a pure

state from the noisy input state. As such, the noise models in this protocol are defined for some error

parameter p which represents the amount of error introduced to the state. The purification protocol

will be tested for a range of different values of parameter p and the results in terms of the success of

the entanglement purification protocol shall be discussed.

24

4.2.1 The dephasing channel

The dephasing channel that shall be used in this protocol takes the form [21]

Edephasing ( ρ )=(1+ p2 )

3

ρ+ ∑i∈ {qi ,q j ,q k}

(1+ p2 )

2

( 1−p2 )( Z i . ρ . Z i )+¿

∑ij∈ {{qi ,q j} , {qi , qk} , {q j , qk} }

( 1+ p2 )( 1−p

2 )2

( Z ij . ρ .Z ij)+( 1−p2 )

3

(Zqiq j qk. ρ . Zqi q j qk ), (23)

where ρ is the density matrix for some pure state. Formally the Z operation acting on the left of the

density matrix in each term should be the Hermitian conjugateZ† . However, the Pauli operators are

Hermitian hence Z†=Z and the unconjugated Z is used for simplicity (taking the Hermitian conjugate

also unnecessarily increases the computation time of the Mathematica program for this protocol

(Appendix C-D)).

The first term of Eq.(23) represents the probability of no noise being introduced to the input state. It

can be seen that if p=1 (i.e. there is zero probability of noise being introduced to the input state) then

Edephasing ( ρ )=ρ as expected. The second term of Eq.(23) represents the probability of noise being

introduced to just one of the qubits {q i , q j , qk } in ρ. Noise is applied to each qubit individually and so

the summation over the qubits must be taken. This also means that p represents the error on one qubit

and so the coefficient of the second term has one factor of ( 1+p2 ) and two factors of( 1−p

2 ) (( 1+p2 )

for the qubit upon which noise has been applied and ( 1−p2 ) for each qubit where no noise has been

applied). The third term of Eq.(23) follows from a similar argument, only this term represents the

probability of noise being introduced to two of the qubits {q i , q j , qk } in ρ . Thus the summation is

over the set { {qi , q j } , {qi , qk } , {q j , qk }} and the coefficient of the third term has two factors of( 1+p2 )

25

and one factor of ( 1−p2 ). The last term of Eq.(23) represents the probability of noise being

introduced to all three of the qubits {q i , q j , qk } in ρ. It can be seen that if p=0 then

Edephasing ( ρ )=18(ρ+Z i ρ Z i) (where i is the sum of all qubit combinations) which is the maximally

phase damped three qubit state.

4.2.2 The depolarising channel

The second noise model that will be used to simulate noise for this purification protocol is the

depolarising channel

Edepolarising ( ρ )=(1+3 p4 )

3

ρ+ ∑i∈ {qi ,q j ,qk }

( 1+3 p4 )

2

( 1−p4 )( X i . ρ . X i+Y i . ρ . Y i+Z i . ρ . Z i )+ ∑

ij∈ {{qi ,q j } , {q i, qk }, {q j , qk} }(1+3 p

4 )( 1−p4 )

2

( X i j . ρ . X ij+Y ij . ρ .Y ij+Zij . ρ . Z ij )+( 1−p4 )

3

( Xqi q j qk. ρ. Xq iq j qk

+Y q iq j qk. ρ .Y qi q j qk

+Zqi q jq k. ρ . Zqi q j qk )

. (24)

The terms in Eq.(24) arise from a similar argument to the one that was used to explain the terms in

Eq.(23). However, for the depolarising channel all three Pauli operators are applied to a qubit to

simulate noise on the qubit rather than just Z. Again the operators acting on the left of the density

matrix should be Hermitian conjugated but, as was the case withZ, X†=X and Y †=Y so the

unconjugated operators are used for simplicity. The coefficients of the depolarising channel are

chosen such that, if p=1 then Edepolarising ( ρ )=ρ as expected, and if p=0 then

Edepolarising ( ρ )= 164

(ρ+ X i ρ X i+Y i ρ Y i+Z i ρ Z i) which is just the maximally mixed state for three

qubits I8 . Apart from these differences, the depolarising channel is identical to the dephasing channel

and so the arguments in section 4.2.1 apply here as well.

4.2.3 The global white noise model

26

The final noise model that will be introduced is a much simpler model called the global white noise

channel

Eglobal noise ( ρ )=( p ) ρ+(1−p ) I2n , (25)

where n is the number of qubits in the input state corresponding to the density matrix ρ. Thus for this

purification protocol on the three qubit hypergraph state ¿ H ⟩ the global white noise channel will be

Eglobal n oise ( ρ )= ( p ) ρ+(1−p) I8 . (26)

It is clear that this is a much simpler noise model than the dephasing channel and the depolarising

channel and its function is equally as simple. For p=1 the channel will leave the pure input state

unchanged and for p=0 the channel will replace the pure input state with the stateI2n which is the

maximally mixed state for n qubits. Thus the global white noise channel acts in a similar way to the

depolarising channel but, as it is linear in p rather than cubic, the noise from this channel should be

easier to deal with for the hypergraph purification protocol.

4.2.4 Comparison of the different noise models

Now that the different noise models which will be used to test this purification protocol have been

introduced, a comparison between them can be discussed. This will be useful when analysing the

effectiveness of the hypergraph purification protocol.

27

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Figure 6. Plot of the fidelity, F for an input state exposed to noise against the noise parameter, p. Here the blue line shows the effect upon the fidelity of the input state for the dephasing channel in Eq.(23), the red line shows the effect upon the fidelity of the input state for the depolarising channel Eq.(24), and the green line shows the effect upon fidelity of the input state for the global white noise model Eq.(26). The plot range for p goes from 0 for the maximal amount of noise introduced on the input state to 1 for the minimal amount of noise introduced on the input state. The Mathematica code for this plot can be found in Appendix B.

It can be seen from Figure 6 that the most destructive noise model is the depolarising channel, as the

fidelity of states exposed to this channel drops off rapidly for a decrease in the parameter p. The

dephasing channel is less destructive than the depolarising channel but is still fairly destructive. Again

the fidelity of states exposed to this channel drops off rapidly for a decrease in the error parameter p .

Figure 6 shows that the global white noise model is similarly as noisy as the dephasing channel but,

observing that Eq.(26) is linear in p while Eq.(24) is cubic in p, the fidelity drops off at a more

gradual rate for the global white noise model.

4.3 The hypergraph purification protocol

Now that the noise models which will be used to test this protocol have been introduced and analysed,

the protocol itself can be introduced.

4.3.1 Steps involved in the hypergraph purification protocol

28

Fidelity, F

Parameter, p

The first step of the protocol is to take the density matrix ρH (see Eq.(21)) and subject it to noise with

one of the noise models Eqs.(23,24,26) for some value of p. An identical copy of this state is then

prepared

|H ' ⟩=CC Z1' 2'3'¿¿, (27)

with density matrix

ρH '=|H ' ⟩ ⟨ H '|, (28)

and subjected to the same noise model with the same p. The convention will be taken that the

protocol is used to purify the state ¿ H ⟩ (see Eq.(19)) using the state ¿ H ' ⟩ as an ancilla state. The

state of the total system upon which operations and measurements will be applied can then be thought

of as a six qubit so –called “superstate”

|H tot ⟩=CC Z123 CC Z1'2' 3'¿¿. (29)

In order to purify the whole three qubit state¿ H ⟩, each of the qubits1,2, and 3 must be purified in

turn. Each qubit is purified by applying local operations between the qubits 1 and1 ', the qubits2 and

2 ', and the qubits3 and3 '. Each of the qubits of ¿ H ' ⟩ (i.e. qubits1 ',2 ', and 3 ') are then measured in

the X -basis and only those instances where the outcome +1 is obtained are kept.

It can be seen from these steps that the entanglement purification protocol to hypergraph states

follows the same format as the entanglement purification protocol for bipartite states explained in

sections 2.3. Both apply local operations between qubits of identical copies of some noisy state,

measure in some basis and then keep only the outcomes where a specific value is obtained. However,

the entanglement purification protocol to hypergraph states requires more operations than the

purification protocol for bipartite states due to the nature of multipartite entanglement. The specifics

of these operations, as well as the reason why they are required, will be explained by considering the

purification protocol for one qubit.

4.3.2 The purification protocol for one qubit

29

The purification for one qubit will now be explained by considering the purification of qubit1. The

specific steps of this protocol are as follows:

1. Apply a CNOT operation between qubit 1 and qubit 1 ' and then apply the projector

X+¿=¿¿ (30)

on qubit 1 '.

2. Apply the projector

PE=|00 ⟩ ⟨ 00|+|11 ⟩ ⟨11| (31)

between qubit 2 and qubit 2 '. Then apply the projector X+¿ ¿ on qubit 2 '.

3. The projector PE is then applied between qubit 3 and qubit 3 ' and the projector X+¿ ¿ is

applied on qubit 3 ' in the same fashion as the second step.

The applications of the operator CNOT and the projector PE serve to map some information about

the state ¿ H ⟩ onto the state|H ' ⟩. The projector X+¿ ¿ is then applied in order to measure the state

¿ H ' ⟩ in the X basis, which reveals this information. Thus the state ¿ H ⟩ is purified in a similar way

to the purification of the state M in section 2.3; information about the first copy of some noisy state is

revealed by measuring the second copy of the state which serves to increase the fidelity of the first

copy. The reason why each of these steps is necessary to purify ¿ H ⟩ will now be explained in detail.

The CNOT operation applied between qubits 1 and 1 ' followed by the application of the projector

X+¿ ¿ on qubit 1 ' has the following effect on the state |H tot ⟩:

CNO T 11' CC Z123 CC Z1' 2'3'¿¿. (32)

The ¿+⟩ state is the eigenstate of the Pauli X operator with eigenvalue+1. Therefore, applying the

projector X+¿ ¿ on qubit 1' is equivalent to a measurement of that qubit in the X -basis and keeping

only those outcomes where +1 is the result of the measurement. Applying the CNOT operation

between qubit 1 and qubit 1 ', followed by measuring qubit 1 ' in the X -basis and keeping only the +1

outcomes, serves to purify qubit 1.

30

However, Eq.(32) reveals that the CNOT operation also entangles the states ¿ H ⟩ and|H ' ⟩, evident

from the additional CC Z1' 23 operator between qubit 1' of ¿ H ' ⟩ and qubits 2 and 3 of¿ H ⟩. Figure 7

shows this extra entanglement using the diagrammatic representation of hypergraphs introduced in

section 3.3. This explains why the projector PE is required, rather than just the CNOT operation and

measurements in some basis, as was the case for the bipartite purification protocol. The projectorPE is

used to break the entanglement of qubit 1 ' with qubits 2 and 3. It is in breaking this entanglement that

the protocol for purifying to LMESs [21] requires the additional local creation of a maximally

entangled two qubit state. The application of the projector PE and subsequent measurements in the X -

basis achieve the same aim for purifying to hypergraph states and so the creation of this additional

state is not necessary.

After the second and third steps of the protocol, ¿ H tot ⟩ will be in the state

CC Z123CC Z1' 2'3' CC Z1' 2'3' ¿¿, (33)

as the operation CCZ is Hermitian so CC Z1' 2' 3' CC Z1' 2'3'=CC Z1' 2' 3' CC Z1' 2' 3'†=I 1'2' 3'. Recalling that

the state ¿ H tot ⟩ is actually a superstate composed of the states ¿ H ⟩ and¿ H ' ⟩, it can now be seen

that |H ⟩=CC Z123 ¿¿ where qubit 1 has been purified, while|H ' ⟩=I 1' 2'3' ¿¿. The purification of qubit

1 thus disentangles the state ¿ H ' ⟩ and so a new copy of the noisy state must be prepared in order to

31

Figure 7. Diagrammatic representation of the hypergraphs in Eq.(32), which represents the effect of the CNOT operation between the first qubits of the states ¿ H ⟩ and ¿ H ' ⟩. The operation has entangled qubit 1' of ¿ H ' ⟩ with qubits 2 and

3 of ¿ H ⟩, represented by the red hyperedge.

purify another qubit. For the sake of simplicity, this new copy will also be referred to as ¿ H ' ⟩ where

¿ H ' ⟩ is redefined as

|H ' ⟩=CC Z1' 2'3'¿¿ (34)

as in Eq.(27). The state ¿ H ' ⟩ must, therefore, be thought of as an ancilla state used for the

purification of one qubit of the state ¿ H ⟩ before being discarded.

5 Results for the hypergraph purification protocol

Now that the entanglement purification protocol for hypergraph states has been presented and

explained in detail, the effect of this protocol on the fidelity of some noisy input state can be assessed.

The fidelity of the output state after one iteration of the purification protocol will be presented to

begin with. The capabilities of the protocol will then be tested by repeatedly applying the protocol to

some noisy input state to try and achieve a fidelity of one for the output state.

5.1 Performance of the protocol after one iteration

To test this protocol, noise is introduced to the pure input state ρH with the three noise models from

Eqs.(23,24,26). Thus the input state to the purification protocol becomesEdephasing (ρH),

Edepolarising (ρH), or Eglobalnoise(ρH). The purification protocol shall be tested for each of these noisy

input states with the error parameter ranging from p=0 top=1. The fidelity that is of interest to this

protocol is the fidelity of the output state of the protocol, defined as

Υ i(E)

(35)

(where i denotes the order in which the qubits have been purified by the protocol and E denotes the

noise model), and the pure state ρH (see Eq.(21)). The fidelity between these two states is defined as

32

F=Tr ( ρH Υ i ( E ) ρH ),

(36)

but using the fact that ρH=|H ⟩ ⟨ H|, along with the invariance of the trace under cyclic permutations,

this can be rewritten in a simpler form

F=Tr ( ρH Υ i ( E ) ρH )=Tr (|H ⟩ ⟨ H|Υ i ( E )|H ⟩ ⟨ H|)=Tr ( ⟨ H|Υ i ( E )|H ⟩ ⟨ H|H ⟩ )

¿ ⟨ H|Υ i ( E )|H ⟩ ⟨ H|H ⟩=⟨ H|Υ i ( E )|H ⟩,

(37)

where the fact that |H ⟩ is normalised, and that the trace of a number is just the number, have also

been used.

5.1.1 The fidelity of the output state after the purification of one qubit

Using this definition of the fidelity, the performance of the purification protocol can be presented.

Figure 8 shows the effect on fidelity of the purification protocol on just one qubit for the dephasing

channel. From this graph it can be seen that the purification protocol on just one qubit has actually

decreased the fidelity of the final state.

The same can be seen for the effect on fidelity of the purification protocol on just one qubit for the

depolarising channel and global white noise model in Figure 9. For the global white noise model, the

fidelity of the output state Υ 1(Eglobalnoise) is actually slightly higher than the fidelity of the input state

Eglobalnoise(ρH) for high p, but this is only a very slight increase and the purification protocol should

be able to do much better than this.

33

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Figure 8. Plot of the fidelity, F against the parameter p for the dephasing channel. The dark blue line shows the fidelity of

the stateΥ 1(Edephasing) and the light blue line shows the fidelity of the state Edephasing ( ρH ). The Mathematica code for this plot can be found in Appendix E.

The reason why purifying just one qubit decreases the fidelity of ¿ H ⟩ can be explained by

considering the specific steps of the protocol described in section 4.3.2. Recall that the projector PE is

applied between qubits 2 and 2 ' and between qubits 3 and3 '. While this step is necessary to

disentangle qubit 1 ' from qubits 2 and 3, it also increases the amount of noise on qubits 2 and3. The

result of this is that, while the purification protocol on one qubit reduces the noise on that qubit, it

increases the noise on the other two qubits and so, if only one qubit is purified, the overall fidelity of

the state ¿ H ⟩ has decreased. This demonstrates why it is necessary to purify all three qubits in turn in

order to purify the whole state¿ H ⟩.

It

should be noted here that the graphs in Figure 8 and Figure 9 are

for a purification of qubit 1 of¿ H ⟩. However, as the same amount

of noise is added to each qubit for all three of the noise models, if

only one qubit is purified then it should not matter which qubit this

34

Fidelity, F

Parameter, p

Fidelity, F

Parameter, p

Fidelity, F

a) b)

Figure 9. Graph a) shows the plot of the fidelity, F against the error parameter, p for the depolarising channel. The dark red line shows the fidelity of the stateΥ 1(Edepolarising) and the light red line shows the fidelity of the state Edepolarising (ρH). Graph b) shows the plot of F against p for the global white noise model. The dark green line shows the fidelity of the stateΥ 1(Eglobalnoise) and the light green line shows the fidelity of the state Eglobalnoise(ρH). The

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

is. Figure 10 demonstrates that this is indeed the case and the purification of all three qubits can now

be presented.

35

Figure 10. Plots of the fidelity, the qubit that has been purified is qubitindicates that the qubit that has been purified is qubitdepolarising channel, and the graphs in c) are for the global white noise model. The Appendix E.

Fidelity, F

Fidelity, F Fidelity, F

Fidelity, F Fidelity, F

Fidelity, F Fidelity, F

Fidelity, F

Parameter, p

Parameter, p

Parameter, pParameter, pParameter, p

Parameter, pParameter, p

Figure 9. Graph a) shows the plot of the fidelity, F against the error parameter, p for the depolarising channel. The dark red line shows the fidelity of the stateΥ 1(Edepolarising) and the light red line shows the fidelity of the state Edepolarising (ρH). Graph b) shows the plot of F against p for the global white noise model. The dark green line shows the fidelity of the stateΥ 1(Eglobalnoise) and the light green line shows the fidelity of the state Eglobalnoise(ρH). The

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Fidelity, F Fidelity, F

a)

c)

b)

5.1.2 The fidelity of the output state after the purification of all three qubits

Figure 11 demonstrates the performance of the purification protocol when all three qubits of |H ⟩ have

been purified. The qubits of ¿ H ⟩ are purified in the order qubit1, then qubit2, and then qubit3. The

noise model used for this plot is the dephasing channel.

Figure 11. Plot of the fidelity, F against the error parameter, p for an input state exposed to noise from the dephasing channel and purified with the purification protocol applied to all three qubits in the order 1, 2, 3. The light blue line shows

the fidelity of the noisy input stateEdephasing ( ρH ). The dark blue line shows the fidelity of the output state

Υ 123 ( Edephasing) and the dark blue shaded area demonstrates the values of p for which the purification protocol improves the fidelity. The Mathematica code for this plot can be found in Appendix F.

It can be seen from Figure 11 that the purification protocol successfully produces output states with an

improved fidelity from input states subjected to noise with the dephasing channel. It can be calculated

numerically that the purification protocol can improve fidelity for p>0.789, which means that the

purification protocol can reduce the noise on a state if the initial fidelity of the state is

⟨ H|Edephasing ( ρH )|H ⟩>0.716. (38)

Figure 12 demonstrates the performance of the purification protocol when all three qubits of |H ⟩ have

been purified, but this time for the depolarising channel. The qubits of ¿ H ⟩ are again purified in the

order qubit1, qubit2, and then qubit3. It can be seen from Figure 12 that the purification protocol

performs better for states subjected to noise from the depolarising channel than the dephasing channel.

36

Fidelity, F

Parameter, p

Explicitly, the value of p for which the purification protocol can improve the fidelity of states

subjected to noise by the depolarising channel is p>0.760, and so the purification protocol can

reduce the noise on such a state if the initial fidelity is

⟨ H|Edepolarising ( ρH )|H ⟩>0.587. (39)

Figure 12. Plot of the fidelity, F against the error parameter, p for an input state exposed to noise from the depolarising channel and purified with the purification protocol applied to all three qubits in the order 1, 2, 3. The light red line shows

the fidelity of the noisy input stateEdepolarising ( ρH ). The dark red line shows the fidelity of the output state

Υ 123 ( Edepolarising) and the dark red shaded area demonstrates the values of p for which the purification protocol improves the fidelity. The Mathematica code for this plot can be found in Appendix F.

Recall from Figure 6 that the global white noise model is a much simpler noise model than the

dephasing and depolarising channels, and so it is expected that the purification protocol would be able

to handle a higher level of noise for this model than for the dephasing and depolarising channels.

Figure 13 shows that this is indeed the case; with the same purification order as for Figure 11 and

Figure 12, the value of p for which the protocol can improve the fidelity of states subjected to noise

by the global white noise model is p>0.561. This is significantly lower than for the depolarising

channel and dephasing channel, as expected. However, the fidelity of an input state for which the

purification protocol can reduce the noise is

⟨ H|Eglobalnoise ( ρH )|H ⟩>0.616 (40)

37

Parameter, p

Fidelity, F

calculated using the fact that the protocol can handle p>0.561. This is due to the fact that the fidelity

of states subjected to noise from the global white noise model drops off less rapidly for decreasing p

than for states subjected to noise from the depolarising channel.

From this analysis, it can be seen that, after one iteration, the entanglement purification protocol for

hypergraph states performs best for a state subjected to noise by the depolarising channel. However,

none of the plots in Figures 11, 12 or 13 reach a fidelity of one for any value of p. This implies that

more than one iteration of the protocol is required to obtain completely pure states from noisy input

states.

Figure 13. Plot of the fidelity, F against the error parameter, p for an input state exposed to noise from the global white noise model and purified with the purification protocol applied to all three qubits in the order 1, 2, 3. The light green line

shows the fidelity of the noisy input stateEglobalnoise ( ρH ). The dark green line shows the fidelity of the output state

Υ 123 ( Eglobalnoise ) and the dark green shaded area demonstrates the values of p for which the purification protocol improves the fidelity. The Mathematica code for this plot can be found in Appendix F.

5.2 Purification to completely pure hypergraph states

5.2.1 Achieving a fidelity of one for the output of the protocol

The ultimate aim of any entanglement purification protocol is to be able to purify to completely pure

output states from input states which have some amount of noise. This means that, in order to

consider the entanglement purification protocol for hypergraph states a successful purification

protocol, it needs to be demonstrated that

38

Parameter, p

Fidelity, F

⟨ H|Υ i ( E )|H ⟩=1

(41)

for any of the noise models with some value of p>0. In order to achieve this aim, the purification

protocol will be applied repeatedly to some noisy input state to try and attain an output state with a

fidelity of F=1.

For the first attempt, the order of the purification protocol used in the previous section is retained and

simply repeated three times. Recalling from section 3.4 that the convention is taken that the qubits of

a hypergraph state are labelled as the vertices of the corresponding hypergraph (so the qubits for the

state to be purified are {1,2,3}), the qubits are purified in the order 1,2,3 – 1,2,3 – 1,2,3 (denoted O1

). The performance of the purification protocol for this order is demonstrated in Figure 14. It can be

seen in Figure 14 that, after three repetitions of the purification protocol, a completely pure state can

be obtained from a noisy input state. This is true for all three of the noise models used to test the

protocol.

5.2.2 Improving the performance of the

purification protocol

39

Fidelity, F

Fidelity, F Fidelity, F

Parameter, p

Parameter, p

Figure 14. Plots of fidelity, F against error parameter, p for the hypergraph purification protocol applied repeatedly to some noisy input state. The purification protocol was applied to the qubits in the orderO1. Here plot a) is for the dephasing channel, plot b) is for the depolarising channel, and plot c) is for the global white noise channel. The Mathematica code for these plots can be found in Appendix G.

a) b)

c)

Fidelity, F

Fidelity, F

b)a)

Now that it has been demonstrated that the purification protocol can be used to obtain completely pure

hypergraph states from some noisy input state, it is of interest to explore whether or not the

performance of the protocol can be improved by varying both the order in which the qubits are

purified and the number of iterations of the protocol.

Rather than applying the purification protocol to all three qubits before repeating it, each qubit could

be purified multiple times before moving on to the purification of the next qubit. In order to see

whether or not this would improve the performance of the purification protocol, the qubits of some

noisy input state will be purified in the order1,1,1 –2,2,2 – 3,3,3 (denoted O2), and this order will be

tested for all three noise models.

Figure 15 demonstrates that the performance of the protocol when the qubits are purified in the order

O2 is actually worse than when the qubits are purified in the orderO1. In order to explain this, recall

from section 5.1.1 that when one qubit is purified, the noise on the other two qubits is increased.

Therefore, when the qubits are purified in the orderO2, the repeated purification of qubit 1 causes a

40

Figure 15. Plots of fidelity, F against error parameter, p for the hypergraph purification protocol applied repeatedly to some noisy input state. The graphs compare the performance of the purification protocol for the qubits purified in the order O1 (light colour lines), and in the order O2 (dark colour lines). It can be seen that the protocol performs better for the first order than for the second order, and the shaded light colour areas represent the range of error for which the first order outperforms the second order. Again, a) is the dephasing channel, b) is the depolarising channel, and c) is the global white noise model. The Mathematica code for these plots can be found in Appendix H,

Fidelity, F Fidelity, FFidelity, F

Parameter, p

Parameter, p

a) b)

c) Fidelity, F

Parameter, p

build-up of noise on the other two qubits (and similarly for qubit 2 and qubit3). The overall effect of

this is that the performance of the protocol is worsened when the qubits are purified in this order.

It is reasonable to assume, then, that the most efficient order in which to purify the qubits would be

the one which minimises this build-up of noise. A good candidate is the order 1,2,3 – 3,1,2 – 2,3,1

(denoted O3 ¿. For this order, the build-up of noise on qubit 3 from purifying qubits 1 and 2 is

counteracted by purifying qubit 3 first on the next iteration of the protocol. The build-up of noise on

qubit 2 from purifying qubits 1 and 3 in the second iteration is then counteracted by purifying qubit 2

first in the final iteration.

Figure 16 demonstrates the performance of the

purification protocol when the qubits are

purified in the order O3. As was expected, this order outperforms the order O1 for all three of the

noise models.

5.2.3 Optimising the purification protocol

Now that the optimal order in which to purify the qubits has been found, the next step is to see what

threshold of p is required such that a completely pure output state can still be obtained. To this end,

41

Figure 16. Plots of fidelity, F against error parameter, p for the hypergraph purification protocol applied repeatedly to some noisy state. Here the dark colour lines show the performance of the protocol when qubits are purified in the order O3 and the light colour

lines show the performance of the protocol when qubits are purified in the order O1. The dark shaded areas demonstrate the range

of p for which the protocol performs better with order O3 than O1. As usual, a) is the dephasing channel, b) is the depolarising channel, and c) is the global white noise model. The Mathematica code for these plots can be found in Appendix I.

a)

Fidelity, F Fidelity, FFidelity, F

Parameter, p

Parameter, p

b)c)

the order O3 will be repeatedly applied to noisy input states in order to determine how many

repetitions are needed to optimise the purification protocol.

Here the convention is introduced that, if an order is applied multiple times, this new order is denoted

Oin (42)

where i is the original order and n is the number of times it is applied to a noisy state. Figure 17

demonstrates the performance of the protocol when the qubits are purified in the order O34. While less

than four applications of order O3 results in a slightly worse performance for the protocol, any more

than four applications of order O3 does not improve the performance of the protocol and so O34 is the

optimal order as it optimises performance while minimising computational time. For this order, the

threshold p for which the protocol can produce completely pure output states is: p>0.801 for the

dephasing channel, p>0.798 for the depolarising channel, and p>0.601 for the global white noise

model. In terms of the initial state fidelities, this means that:

42

Figure 17. Plots of fidelity, F against error parameter, p for the optimised purification protocol applied to some noisy state. The optimal order and number of repetitions is order O3 applied four times to the noisy state (i.e. order

O34). Graph a) is the dephasing channel, b) is the depolarising channel, and c) is the global white noise model. The

Mathematica code for these plots can be found in Appendix J.

a)

Fidelity, FFidelity, F

Fidelity, F

Parameter, p

Parameter, p

b)

c)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

For the dephasing channel, a completely pure hypergraph state,

⟨ H|Υ O 34 ( Edephasing )|H ⟩=1, (43)

can be obtained from an initial fidelity of

⟨ H|Edephasing|H ⟩>0.730. (44)

For the depolarising channel, a completely pure hypergraph state,

⟨ H|Υ O 34 ( Edepolarising )|H ⟩=1, (45)

can be obtained from an initial fidelity of

⟨ H|Edepolarising|H ⟩>0.641. (46)

For the global white noise model, a completely pure hypergraph state,

⟨ H|Υ O 34 ( Eglobalnoise )|H ⟩=1, (47)

can be obtained from an initial fidelity of

⟨ H|Eglobalnoise|H ⟩>0.651. (48)

6 Conclusion

6.1 Comparison with purification protocol to LMESs

In this report, an entanglement purification protocol has been presented which successfully purifies to

hypergraph states from noisy three qubit input states. In order to determine whether this protocol is of

any practical relevance, the performance of the protocol needs to be compared to other purification

protocols. In the same paper that Carle et al. present the purification to locally maximally

entangleable states, they give numerical results for the performance of their purification protocol for

states in the form of Eq.(19) [21].

43

Figure 18 demonstrates the value of p for which the purification protocol to LMESs can obtain pure

states from input states subjected to noise from the dephasing channel and the depolarising channel.

Of interest to this report are the results for a three qubit input state (note also that, in Figure 18, linear

pattern refers to states of the form ∏i=2

n−1

U i−1 ,i ,i+1¿¿¿ which, for three qubits, is just ¿ H ⟩ (see Eq.

(19))). Figure 18 shows that, for a three qubit input state, the purification protocol to LMESs can

obtain pure states from noisy input states for p>0.8 for the dephasing channel, and for p>0.8 for the

depolarising channel. Recall that the hypergraph purification protocol can obtain pure states from

noisy input states with p>0.801 for the dephasing channel, and with p>0.798 for the depolarising

channel. The hypergraph purification protocol, therefore, performs just as well as the purification

protocol to LMESs for the dephasing channel, and performs slightly better than the purification

protocol to LMESs for the depolarising channel.

Figure 18 also demonstrates that, for three qubits, the purification protocol to LMESs can obtain pure

states from input states subjected to noise by the global white noise model for an initial fidelity

F>0.65. The hypergraph purification protocol can obtain pure states from input states subjected to

44

Figure 18. This figure is taken from the paper by Carle et al. in which they present a purification protocol to LMESs [21]. This figure shows the value of noise parameter, p for which the protocol can produce pure states for the depolarising channel and the dephasing channel, as well as the minimal fidelity, F for which the protocol can produce pure states for the global white noise model.

Parameter, p

noise by this noise model for p>0.601 which corresponds to an initial fidelity of F>0.651.

Therefore, the hypergraph purification protocol also performs just as well as the purification protocol

to LMESs for the global white noise model.

6.2 Comment on purifying to an output state with a fidelity of one

Note that a clarification of what is meant by obtaining an output state with a fidelity of one is

required. Purifying to an output state with a fidelity of one cannot be achieved by this, or any other

[15], entanglement purification protocol. Rather, purifying to a pure state means that the fidelity of the

output state can be brought arbitrarily close to one in the limit of infinitely many iterations of the

protocol [21]. To demonstrate this, note that the actual fidelity of states purified from an input state

exposed to noise from the dephasing channel with p=0.802 (i.e. p>0.801 to a precision of three

decimal places) is ⟨ H|Υ O 34 ( Edephasing )|H ⟩=0.99998.

Therefore, it must also be clarified why the order O34 was designated the optimal order for the

hypergraph purification protocol. While it was stated in section 5.2.3 that more than four iterations of

the order O3 does not improve the performance of the protocol, this is not strictly true. More than four

iterations will improve the performance of the protocol; it has already been mentioned that infinitely

many iterations are required in order to achieve an output state with a fidelity of one. However, this

has very little practical relevance and a precision of three decimal places is taken as standard instead.

To this precision, the order O34 produces output states with a fidelity of one for the minimal initial

state fidelity. As such, this order is referred to as optimal, although more iterations of the protocol

would be required to achieve output states with a fidelity of one to a finer precision.

6.3 Generalising the hypergraph purification protocol

45

This report has presented a purification protocol to hypergraph states that can obtain pure states from

some noisy input states for a variety of noise models. The inspiration for this protocol comes from the

success of the entanglement purification protocol for bipartite states presented by Bennett et al. [14]

(see section 2.3). The basic principles of the bipartite purification protocol are the basis for the

hypergraph purification protocol; two copies of some noisy state are used to obtain information about

the first copy which is then revealed by measuring the second copy, thereby purifying the first copy.

Hypergraph states are mathematically more complex than bipartite states, however, and, as such, it

has been demonstrated that a more complex purification protocol is required in order to purify to

hypergraph states.

It has been demonstrated that the purification protocol to hypergraph states outperforms the

purification protocol to LMESs for noisy three qubit states. This suggests that a generalisation of the

hypergraph purification protocol to many qubits could be more successful than the purification

protocol to LMESs for a wide variety of noisy input states.

In order to explore this possibility, the generalisation to multiple qubits must be made. Before this,

however, the case of a non-uniform three qubit hypergraph state might be considered. This relaxes the

condition in the equation

|H ⟩=∏k=1

3

∏{i1 , i2 ,… ,ik }∈E

Ck−1 Z i1 ,i2 , …,ik¿+⟩⊗3 (49)

that k=3 which leads to the three-uniform hypergraph state |H ⟩=CCZ123¿¿. As such, one-

hyperedges and two-hyperedges are now allowed between the qubits1, 2, and 3. A hypergraph state

of this form should require no modification to the hypergraph purification protocol.

After generalising the hypergraph purification protocol to the case of a non-uniform three qubit

hypergraph state, the next logical step is the generalisation to multiple qubits. This means that the

restriction n=3 in Eq.(17) is also relaxed, which means that a purification protocol to hypergraph

states of the form

46

|H ⟩=∏k=1

n

∏{i1 , i2 ,… ,ik }∈E

Ck−1 Z i1 ,i2 , …,ik¿+⟩⊗n (50)

is required. For hypergraph states of this form, the hypergraph purification protocol will need to be

modified. Specifically, recall from section 4.3.2 that, in step two (and step three) of the purification

protocol, the projector PE was applied between qubits 2 and 2 ' (and qubits 3 and 3’) and then the

projector X+¿ ¿ was applied to qubit 2 ' (and qubit 3 '). This step was required to counteract the extra

entanglement introduced between the two copies of the noisy input state by the CNOT operation (see

section 4.3.2). Therefore, when the generalisation to multiple qubit hypergraph states is made, these

projectors will need to be applied between more qubits as the CNOT operator may now introduce

entanglement between more than just three qubits.

The importance of the purification protocol to hypergraph states stems from the fact that these states

form a large class of states which include and generalise the stabiliser states and graph states [21],

both of which are of great importance and interest to quantum computation. Therefore, this

purification protocol provides a way to reduce the noise in a quantum system for many important

quantum computational processes. As such, the purification protocol to hypergraph states shows much

promise for the construction of a fault-tolerant quantum system, thereby paving the way for a

successful theoretical model of a fault-tolerant universal quantum computer.

47

References

[1] P.W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, Proc. 35nd Annual

Symposium on Foundations of Computer Science (Shafi Goldwasser, ed.), IEEE Computer Society Press, 124-

134 (1994)

[2] L.K. Grover, A fast quantum mechanical algorithm for database search, arXiv:quant-ph/9605043 (1996)

[3] W.K. Wootters and W.H. Zurek, A single quantum cannot be cloned, Nature 299, 802 – 803,

doi:10.1038/299802a0 (1982)

[4] P.W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493(R)

(1995)

[5] A. Steane, Multiple particle interference and quantum error correction, Proc.Roy.Soc.Lond. A452 2551

(1996)

[6] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information (10th Anniversary Edition),

Cambridge University Press (2010)

[7] P.K. Sarvepalli and A. Klappenecker, Degenerate quantum codes and the quantum Hamming bound, Phys.

Rev. A 81, 032318 (2010)

[8] D. Gottesman, Stabilizer codes and quantum error correction, arXiv:quant-ph/9705052 (1997)

[9] D. Gottesman, An introduction to quantum error correction and fault-tolerant quantum computation,

arXiv:0904.2557 (2009)

[10] D. Aharonov and M. Ben-Or, Fault-tolerant quantum computation with constant error,

arXiv:quant-ph/9611025 (1996)

[11] E. Knill, R. Laflamme, and W. Zurek, Resilient quantum computation, Science 279, 342 – 345 (1998)

[12] A. Steane, Active stabilization, quantum computation, and quantum state synthesis, Phys. Rev. Lett. 78,

2252 (1997)

[13] C.H. Bennett, H.J. Bernstein, S. Popescu, and B. Schumacher, Concentrating Partial Entanglement by

Local Operations, Phys. Rev. A53, 2046-2052 (1996)

48

[14] C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, and W.K. Wootters, Purification of

Noisy Entanglement and Faithful Teleportation via Noisy Channels, Phys. Rev. Lett. 76, 722-725 (1996)

[15] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, Mixed State Entanglement and Quantum

Error Correction, Phys. Rev. A54,3824-3851 (1996)

[16] D. Gottesman, A theory of fault-tolerant quantum computation, Phys. Rev. A 57, 127 (1998)

[17] D. Gottesman, The Heisenberg representation of quantum computers, arXiv:quant-ph/9807006 (1999)

[18] K. Matsumoto and K. Amano, Representation of quantum circuits with Clifford and π/8 gates,

arXiv:0806.3834 [quant-ph] (2008)

[19] Y. Shi, Both Toffoli and controlled-NOT need little help to do universal quantum computation,

arXiv:quant-ph/0205115 (2002)

[20] S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas ,

Phys. Rev. A 71, 022316 (2005)

[21] T. Carle, B. Kraus, W. Dür, and J.I. de Vincente, Purification to locally maximally entangleable states,

Phys. Rev. A 87, 012328 (2013)

[22] M. Rossi, M. Huber, D. Bruß, and C. Macchiavello, Quantum hypergraph states, New J. Phys. 15, 113022

(2013)

[23] M. Rossi, D. Bruß, and C. Macchiavelo, Hypergraph states in Grover’s quantum search algorithm, Phys.

Scr. T 160, 014036 (2014)

[24] E. Schrodinger, Discussion of Probability Relations between Separated Systems, Mathematical

Proceedings of the Cambridge Philosophical Society 31, 555-563 (1935)

[25] V. Vedral, Introduction to Quantum Information Science, Oxford University Press (2006)

[26] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an Unknown

Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels, Phys. Rev. Lett. 70, 1895 (1993)

[27] C.H. Bennett and S.J. Wiesner, Communication via one- and two-particle operators on Einstein-Podolsky-

Rosen states, Phys. Rev. Lett 69, 2881 (1992)

49

[28] R. Jozsa and N. Linden, On the role of entanglement in quantum computational speed-up,

arXiv:quant-ph/0201143 (2002)

[29] P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406:P06002

(2004)

[30] I. Bengtsson and K. Zyczkowski, Geometry of quantum states: An introduction to quantum entanglement,

Cambridge University Press (2007)

[31] R.F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable

model, Phys. Rev. A 40, 4277 (1989)

[32] M.B. Plenio and S. Virmani, An introduction to entanglement measures, Quant. Inf. Comput. 7, 1-51

(2007)

[33] W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80, 2245-

2248 (1998)

[34] M.S. Anwar, L. Xiao, A.J. Short, J.A. Jones, D. Blazina, S.B. Duckett, H.A. Carteret, Practical

implementations of twirl operations, Phys. Rev. A 71, 032327 (2005)

[35] R. Diestel, Graph theory 4th Edition, Springer (2010)

[36] M. Hein, W. Dur, J. Eisert, R. Raussendorf, M. Van den Nest, and H.J. Briegel, Entanglement in Graph

States and its Applications, arXiv:quant-ph/0602096 (2006)

[37] R.J. Wilson, Introduction to graph theory 5th Edition, Prentice Hall (2010)

[38] M. Hein, J. Eisert, and H.J. Briegel, Multi-party entanglement in graph states, Phys. Rev. A 69, 062311

(2004)

[39] O. Gühne, M. Cuquet, F.E.S. Steinhoff, T. Moroder, M. Rossi, D. Bruß, B. Kraus, and C. Macchiavello,

Entanglement and nonclassical properties of hypergraph states, J. Phys. A: Math. Theor. 47, 335303 (2014)

[40] A. Miyake and H.J. Briegel, Distillation of multipartite entanglement by complementary stabilizer

measurements, Phys. Rev. Lett. 95, 220501 (2005)

[41] W. Dur and H.J. Briegel, Entanglement purification and quantum error correction, Rep. Prog. Phys. 70,

1381 (2007)

50

[42] W. Dur, H. Aschauer, and H.J. Briegel, Muliparticle entanglement purification for graph states, Phys. Rev.

Lett. 91, 107903 (2003)

[43] H. Aschauer, W. Dur, and H.J. Briegel, Muliparticle entanglement purification for two-colorable graph

states, Phys. Rev. A 71, 012319 (2005)

[44] C. Kruszynska, A. Miyake, H.J. Briegel, and W. Dur, Entanglement purification protocols for all graph

states. Phys. Rev. A 74, 052316 (2006)

[45] K. Goyal, A. McCauley, and R. Raussendorf, Purification of large bicolorable graph states, Phys. Rev.

A 74, 032318 (2006)

[46] S. Glancy, E. Knill, and H.M. Vasconcelos, Entanglement purification of any stabilizer state, Phys. Rev.

A 74, 032319 (2006)

[47] C. Kruszynska and B. Kraus, Local entanglability and multipartite entanglement, Phys. Rev. A 79, 052304

(2009)

51

Appendix A

Mathematica code for the bipartite entanglement purification protocol (see section 2.3). The function

randomly selects two Bell states (with probability p selecting Φ+¿¿ and with probability 1−p

3

selecting one of {Ψ +¿ ,Φ−¿ ,Ψ −¿} ¿¿ ¿) and applies the bilateral controlled NOT operation between the states.

After this operation, if the second state is one of {Φ+¿ , Φ−¿}¿ ¿ then x is incremented by 1, and if this is

the case and if the first state is Φ+¿¿, then y is incremented by 1 as well. This is repeated for 100000

runs. The value y / x then represents the fidelity of the state Φ+¿¿ after the purification.

EntDist ¿

Mathematica code for the bilateral controlled NOT operation in the bipartite entanglement

purification protocol.

bxor ¿

Mathematica code for the plot of final fidelity against initial fidelity for the bipartite purification

protocol.

Linear ¿DiscretePlot [ {EntDist [ F] , Linear [F ]}, {F ,0,1,0.01 }, Joined →True ,Filling→ {1 → {2 }}, Frame→ True ]

Appendix B

Mathematica code for the pure hypergraph state ¿ H ⟩ and corresponding density matrix ρH.

52

H= 1√8 (

1111111

−1);

DH=H . H† ;

Mathematica code for the Pauli operators acting on each combination of qubits of the three qubit hypergraph state.

X 1=KroneckerProduct [(0 11 0) ,(1 0

0 1) ,(1 00 1)];

X 2=KroneckerProduct [(1 00 1) ,(0 1

1 0) ,(1 00 1)];

X 3=KroneckerProduct [(1 00 1) ,(1 0

0 1) ,(0 11 0)];

X 12=KroneckerProduct [(0 11 0) ,(0 1

1 0) ,(1 00 1)];

X 13=KroneckerProduct [(0 11 0) ,(1 0

0 1) ,(0 11 0)];

X 23=KroneckerProduct [(1 00 1) ,(0 1

1 0) ,(0 11 0)];

X 123=KroneckerProduct [(0 11 0) ,(0 1

1 0) ,(0 11 0)];

Y 1=KroneckerProduct [(0 −ii 0 ) ,(1 0

0 1) ,(1 00 1)];

Y 2=KroneckerProduct [(1 00 1) ,(0 −i

i 0 ) ,(1 00 1)];

Y 3=KroneckerProduct [(1 00 1) ,(1 0

0 1) ,(0 −ii 0 )];

Y 12=KroneckerProduct [(0 −ii 0 ) ,(0 −i

i 0 ) ,(1 00 1)];

Y 13=KroneckerProduct [(0 −ii 0 ) ,(1 0

0 1),(0 −ii 0 )];

Y 23=KroneckerProduct [(1 00 1) ,(0 −i

i 0 ), (0 −ii 0 )];

Y 123=KroneckerProduct [(0 −ii 0 ) ,(0 −i

i 0 ) ,(0 −ii 0 )];

Z 1=KroneckerProduct [(1 00 1) ,(1 0

0 −1) ,(1 00 1)];

53

Z 2=KroneckerProduct [(1 00 1) ,(1 0

0 −1) ,(1 00 1)];

Z 3=KroneckerProduct [(1 00 1),(1 0

0 1) ,(1 00 −1)];

Z 12=KroneckerProduct [(1 00 −1) ,(1 0

0 −1),(1 00 1)];

Z 13=KroneckerProduct [(1 00 −1) ,(1 0

0 1) ,(1 00 −1)];

Z 23=KroneckerProduct [(1 00 1),(1 0

0 −1) ,(1 00 −1)];

Z 123=KroneckerProduct [(1 00 −1) ,(1 0

0 −1) ,(1 00 −1)];

Mathematica code for each of the noise models presented in section 4.2 (Eqs.(28,29,31)). The noise models are written as functions of the parameter p so that they can be tested for the range 0 ≤ p ≤ 1. Dephase ¿

Depol¿

GlobalNoise ¿

Mathematica code for the plot of the fidelity F against the parameter p for each of the noise models.

Show ¿{p , 0,1 }, PlotStyle→ {ColorData [ SolarColors , 0.25] , ColorData[ DeepSeaColors ,0.25 ] ,ColorData [ AvocadoColors , 0.25] }, Frame→ True¿ , ImageSize→ Small ¿

54

Appendix C

Mathematica code for the CNOT operation between two qubits for the hypergraph purification protocol. The CNOT operation can be expressed as CNOT =|0 ⟩ ⟨0|⨂ I +¿1⟩⟨ 1∨⨂ X, which for the six qubit superstate |H tot ⟩ becomes, when the CNOT operator is applied between qubits 1 and 1' (or qubit 4 as it is called in the code) for example, CNOT 14=|0 ⟩ ⟨0|⨂ I 5 . I 6+|1 ⟩ ⟨1|⨂ I 5. I3⨂ X⨂ I 2.

CNOT 14=KroneckerProduct ¿IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2]¿ .KroneckerProduct ¿IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2]¿+¿KroneckerProduct ¿IdentityMatrix [2] , IdentityMatrix [2]¿ . KroneckerProduct ¿

IdentityMatrix [2] ,(0 11 0), IdentityMatrix [2] , IdentityMatrix [2]¿;

CNOT 25=KroneckerProduct ¿IdentityMatrix [2] , IdentityMatrix [2]¿ . KroneckerProduct ¿IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2]¿+¿KroneckerProduct ¿IdentityMatrix [2]¿ . KroneckerProduct ¿

IdentityMatrix [2] ,(0 11 0), IdentityMatrix [2]¿ ;

CNOT 36=KroneckerProduct ¿IdentityMatrix [2] , IdentityMatrix [2]¿ . KroneckerProduct ¿IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2]¿+¿KroneckerProduct ¿IdentityMatrix [2]¿ . KroneckerProduct ¿

IdentityMatrix [2] , IdentityMatrix [2] ,(0 11 0)¿ ;

Mathematica code for the PE projector (see section 4.3.2) between two qubits.

Entangle 25=KroneckerProduct ¿ IdentityMatrix [2]¿ . KroneckerProduct ¿IdentityMatrix [2] , IdentityMatrix [2]¿+KroneckerProduct ¿

IdentityMatrix [2] , IdentityMatrix [2] ,(0 00 1) , IdentityMatrix [2]¿ .KroneckerProduct ¿

IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2]¿ ;

55

Entangle 36=KroneckerProduct ¿

IdentityMatrix [2] ,(1 00 0)¿ . KroneckerProduct ¿

IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2]¿+KroneckerProduct ¿

IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2] ,(0 00 1)¿ .

KroneckerProduct ¿IdentityMatrix [2]¿ ;

Entangle 14=KroneckerProduct ¿IdentityMatrix [2] , IdentityMatrix [2]¿ . KroneckerProduct ¿IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2]¿+KroneckerProduct ¿

IdentityMatrix [2] , IdentityMatrix [2] ,(0 00 1) , IdentityMatrix [2] , IdentityMatrix [2]¿ .

KroneckerProduct ¿ IdentityMatrix [2]¿ ;

Mathematica code for the X+¿ ¿ operator on one qubit (see section 4.3.2).

Plus2=KroneckerProduct ¿IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2]¿ ;

Plus3=KroneckerProduct ¿

(12

12

12

12), IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2]¿ ;

Plus1=KroneckerProduct ¿IdentityMatrix [2] , IdentityMatrix [2] , IdentityMatrix [2]¿ ;

56

Appendix D

Mathematica code for the operations of the hypergraph purification protocol (see section 4.3.2). As the density matrix formalism is being used, the operators must be applied to the left of the density matrix, and also must be conjugated and applied to the right of the density matrix.

ops1=Plus3 . Entangle36 . Plus2 . Entangle 25 . Plus1 .CNOT 14 ;

conjops 1=CNOT 14† . Plus 1† . Entangle 25† . Plus2† . Entangle 36† . Plus3† ;

ops 2=Plus1 . Entangle 14 . Plus3 . Entangle36 . Plus2 . CNOT 25 ;

conjops 2=CNOT 25† . Plus 2† . Entangle 14† . Plus 1† . Entangle 36† . Plus3† ;

ops3=Plus2 . Entangle25 . Plus1 . Entangle 14 . Plus3 .CNOT 36 ;

conjops 3=CNOT 36† . Plus 3† . Entangle14† . Plus1† . Entangle25† . Plus 2† ;

Mathematica code for the hypergraph purification protocol on one qubit. The division by the trace is required so that the density matrix stays normalised throughout the purification protocol, necessary so that the fidelity of the output state and the pure hypergraph state can be calculated.

Distill 1¿

Distill 2¿

Distill 3¿

Mathematica code for the hypergraph purification protocol on three qubits. Again the division by the trace is required so that the density matrix stays normalised. The code is written for the three different orders of qubit purification 1,2,3, 3,1,2, and 2,3,1.

Distill123 ¿[[1,1;; 8,1;; 8] ] ,{ops1 . KroneckerProduct [h , h] . conjops1 }[[1,1 ;; 8,1;; 8]]¿ . conjops 2}[[1,1;; 8,1 ;;8 ]] ,{ops2 .KroneckerProduct ¿{ops1 . KroneckerProduct [h , h] .conjops 1 }[[1,1 ;; 8,1;; 8]]¿ . conjops 2}[[1,1;; 8,1 ;;8] ]¿ . conjops 3 }[[1,1;; 8,1;; 8] ]/¿Tr ¿[[1,1;; 8,1;; 8] ] ,{ops1 . KroneckerProduct [h , h] . conjops1 }[[1,1 ;; 8,1;; 8]]¿ . conjops 2}[[1,1;; 8,1 ;;8 ]] ,{ops2 . KroneckerProduct ¿{ops1 .KroneckerProduct [h ,h] .conjops 1 }[[1,1 ;; 8,1;; 8]]¿ . conjops 2}[[1,1;; 8,1 ;;8] ]¿ . conjops 3 }[[1,1;; 8,1;; 8] ]¿

Distill 312 ¿[[1,1;; 8,1;; 8] ] ,{ops3 . KroneckerProduct [h , h] . conjops3 }[[1,1 ;;8,1 ;;8 ]]¿ . conjops1 }[[1,1 ;; 8,1;; 8]] ,{ops1 . KroneckerProduct ¿

57

{ops3 . KroneckerProduct [h , h] . conjops3 }[[1,1;; 8,1 ;;8] ]¿ .conjops 1 }[ [1,1 ;;8,1 ;; 8]]¿ . conjops2 }[[1,1;; 8,1;; 8] ]/¿Tr ¿[[1,1;; 8,1;; 8] ] ,{ops3 . KroneckerProduct [h , h] . conjops3 }[[1,1 ;;8,1 ;;8 ]]¿ . conjops1 }[[1,1 ;; 8,1;; 8]] ,{ops1 . KroneckerProduct ¿{ops3 .KroneckerProduct [h ,h] . conjops3 }[[1,1;; 8,1 ;;8] ]¿ .conjops 1 }[ [1,1 ;;8,1 ;; 8]]¿ . conjops2 }[[1,1;; 8,1;; 8] ]¿

Distill 231 ¿[[1,1;; 8,1;; 8] ] ,{ops2 . KroneckerProduct [h , h] . conjops2 }[[1,1;; 8,1;; 8]]¿ . conjops 3 }[[1,1 ;;8,1 ;; 8]] ,{ops3 . KroneckerProduct ¿{ops2 .KroneckerProduct [h ,h] . conjops2 }[[1,1 ;; 8,1;; 8]]¿ . conjops 3}[ [1,1 ;;8,1 ;; 8]]¿ . conjops1 }[[1,1;; 8,1;; 8] ]/¿Tr ¿[[1,1;; 8,1;; 8] ] ,{ops2 . KroneckerProduct [h , h] . conjops2 }[[1,1;; 8,1;; 8]]¿ . conjops 3 }[[1,1 ;;8,1 ;; 8]] ,{ops 3 .KroneckerProduct ¿{ops2 . KroneckerProduct [h , h] . conjops2 }[[1,1 ;; 8,1;; 8]]¿ . conjops 3}[ [1,1 ;;8,1 ;; 8]]¿ . conjops1 }[[1,1;; 8,1;; 8] ]¿

58

Appendix E

Mathematica code for plots of the fidelity F against parameter p for the purification protocol applied to one qubit. The noise model used for these plots is the dephasing channel. To distinguish between each qubit, the plot for the purification of qubit 1 is a solid line, the plot for the purification of qubit 2 is a dashed line, and the plot for the purification of qubit 3 is a dotted line.

Show ¿PlotRange→ {{0,1}, {0,1 }}, PlotStyle →{ColorData [ DeepSeaColors ,0.5] ,ColorData [ DeepSeaColors , 0] }, Frame→ True¿ , ImageSize→ Small¿

Show ¿PlotRange→ {{0,1}, {0,1 }}, PlotStyle →{Directive [ColorData [ DeepSeaColors ,0.5 ] , Dashed ] ,Directive [ColorData [ DeepSeaColors ,0 ] , Dashed ]}, Frame→ True¿ , ImageSize→ Small¿

Show ¿PlotRange→ {{0,1}, {0,1 }}, PlotStyle →{Directive [ColorData [ DeepSeaColors ,0.5 ] , Dotted ] ,Directive [ColorData [ DeepSeaColors ,0 ] , Dotted ]}, Frame → True¿ , ImageSize→ Small ¿

Mathematica code for plots of the fidelity F against parameter p for the purification protocol applied to one qubit, but now the noise model used for these plots is the dephasing channel. Again, to distinguish between each qubit, the plot for the purification of qubit 1 is a solid line, the plot for the purification of qubit 2 is a dashed line, and the plot for the purification of qubit 3 is a dotted line.

Show ¿PlotRange→ {{0,1}, {0,1 }}, PlotStyle →{ColorData [ SolarColors , 0.5] ,ColorData [ SolarColors , 0]}, Frame→ True¿ , ImageSize→ Small¿

Show ¿PlotRange→ {{0,1}, {0,1 }}, PlotStyle →{Directive [ColorData [ SolarColors , 0.5 ] ,Dashed ] ,Directive[ColorData [ SolarColors ,0 ] , Dashed ]}, Frame →True ¿ , ImageSize → Small ¿

Show ¿PlotRange→ {{0,1}, {0,1 }}, PlotStyle →{Directive [ColorData [ SolarColors , 0.5 ] ,Dotted ] ,Directive[ColorData [ SolarColors ,0 ] , Dotted ]}, Frame →True¿ , ImageSize → Small¿

Mathematica code for plots of the fidelity F against parameter p for the purification protocol applied to one qubit. Now the noise model used for these plots is the global white noise model. Again, to distinguish between each qubit, the plot for the purification of qubit 1 is a solid line, the plot for the purification of qubit 2 is a dashed line, and the plot for the purification of qubit 3 is a dotted line.

Show ¿

59

PlotRange→ {{0,1}, {0,1 }}, PlotStyle →{ColorData [ AvocadoColors , 0.5] ,ColorData [ AvocadoColors ,0]}, Frame→ True¿ , ImageSize→ Small ¿

Show ¿PlotRange→ {{0,1}, {0,1 }}, PlotStyle →{Directive [ColorData [ AvocadoColors ,0.5 ] , Dashed] ,Directive [ColorData [ AvocadoColors , 0 ] , Dashed ]}, Frame→ True¿ , ImageSize→ Small¿

Show ¿PlotRange→ {{0,1}, {0,1 }}, PlotStyle →{Directive [ColorData [ AvocadoColors ,0.5 ] , Dotted ] ,Directive [ColorData [ AvocadoColors , 0 ] , Dotted ]},Frame →True ¿ , ImageSize → Small ¿

60

Appendix F

Mathematica code for the plot of fidelity F against the parameter p for purification protocol applied to the whole state ¿ H ⟩, with the qubits purified in the order 1,2,3. The noise model used for this plot is the dephasing channel.

Show ¿Frame→ True ,Filling→ {1 →{{2 }, {Opacity [0.5 ,ColorData [ DeepSeaColors , 0]] ,¿PlotStyle →{ColorData [ DeepSeaColors , 0.5] ,ColorData[ DeepSeaColors ,0]}¿ , ImageSize→ Small ¿

Mathematica code for the numerical calculation of the value of the parameter p for which the purification protocol starts to produce output states with a higher fidelity than an initial state subjected to noise by the dephasing channel, as well as the fidelity of the initial state for this value of p.

NSolve [H † . Dephase [ p] . H=¿ H † . Distill 123[ Dephase[ p]]Tr [Distill 123 [Dephase [ p] ]]

. H , p , Reals]

H † . Dephase[ p] . H / . NSolve [H † .Dephase [ p] . H=¿ H † .Distill 123 [ Dephase [ p ] ]

Tr [ Distill 123 [ Dephase [ p ] ] ]. H , p , Reals ] [[2] ]

Mathematica code for the plot of fidelity F against the parameter p for purification protocol applied to the whole state ¿ H ⟩, with the qubits purified in the order 1,2,3. The noise model used for this plot is the depolarising channel.

Show ¿Frame→ True ,Filling→ {1 →{{2 }, {Opacity [0.5 ,ColorData [ SolarColors , 0]] , ¿PlotStyle →{ColorData [ SolarColors , 0.5] , ColorData[ SolarColors , 0]}¿ , ImageSize→ Small ¿

Mathematica code for the numerical calculation of the value of the parameter p for which the purification protocol starts to produce output states with a higher fidelity than an initial state subjected to noise by the depolarising channel, as well as the fidelity of the initial state for this value of p.

NSolve [H † .Depol [ p] . H=¿ H† . Distill123 [Depol [ p] ]Tr [Distill 123[Depol [ p ]]]

.H , p , Reals ]

H † . Depol[ p] . H / . NSolve [ H † . Depol [ p ] . H=¿ H† .Distill123 [ Depol [ p ] ]

Tr [ Distill 123 [Depol [ p ] ] ]. H , p , Reals ][[2]]

Mathematica code for the plot of fidelity F against the parameter p for purification protocol applied to the whole state ¿ H ⟩, with the qubits purified in the order 1,2,3. The noise model used for this plot is the global white noise model.

Show ¿PlotRange→ {{0,1}, {0,1 }}, Frame →True ,Filling→ {1→ {{2 }, {Opacity [0.5 ,ColorData [ AvocadoColors , 0]] ,¿PlotStyle →{ColorData [ AvocadoColors ,0.5] , ColorData[ AvocadoColors ,0 ]}¿ , ImageSize → Small ¿

61

Mathematica code for the numerical calculation of the value of the parameter p for which the purification protocol starts to produce output states with a higher fidelity than an initial state subjected to noise by the global white noise mode, as well as the fidelity of the initial state for this value of p.

NSolve [H † .GlobalNoise[ p] . H=¿ H† . Distill 123[GlobalNoise [ p ]]Tr [Distill 123 [GlobalNoise[ p]]]

.H , p ,Reals ]

H † .GlobalNoise [ p ] . H /.

NSolve [H † .GlobalNoise [ p] . H=¿ H† .Distill 123 [GlobalNoise [ p ] ]

Tr [ Distill123 [GlobalNoise [ p ]] ]. H , p , Reals ][[2]]

62

Appendix G

Mathematica code for the plot of the fidelity F against the parameter p for the purification protocol applied to a state subjected to noise from the dephasing channel with the order O1 (see section 5.2.1).

Show ¿PlotRange→ {{0,1}, {0,1 }}, Filling→ 2 →{{1}, {¿PlotStyle →{ColorData [ DeepSeaColors , 0.5] , ColorData[ DeepSeaColors , 0]}, Frame→ True¿ ,ImageSize→ Small¿

Mathematica code for the plot of the fidelity F against the parameter p for the purification protocol applied to a state subjected to noise from the depolarising channel with the order O1.

Show ¿PlotRange→ {{0,1}, {0,1 }}, Filling→ 2 →{{1}, {¿PlotStyle →{ColorData [ SolarColors , 0.5] , ColorData[ SolarColors , 0]}, Frame→ True¿ ,ImageSize→ Small¿

Mathematica code for the plot of the fidelity F against the parameter p for the purification protocol applied to a state subjected to noise from the global white noise model with the order O1.

Show ¿PlotRange→ {{0,1}, {0,1 }}, Filling→ 2 →{{1}, {¿PlotStyle →{ColorData [ AvocadoColors , 0.5] , ColorData[ AvocadoColors ,0 ]}, Frame →True ¿ ,ImageSize→ Small¿

63

Appendix H

Mathematica code for the plot comparing the performance of the purification protocol for the order of purifying qubits O1 and the order O2 (see section 5.2.2). This plot is for an initial state subjected to noise from the dephasing channel.

Show ¿Distill 2[Distill 2[Distill 2[Distill 1[ Distill 1[ Distill 1[Dephase [ p] ]]]]]]¿¿¿ /¿Tr [ Distill 3[Distill 3[ Distill 3[Distill 2[Distill 2 [Distill 2[ Distill 1[Distill 1[Distill 1[ Dephase[ p]]] ]]]]]] ]]¿ .H }, {p , 0,1 }, Filling→1 → {{2 }, {¿PlotRange→ {{0,1 } , {0,1 }} , PlotStyle → {ColorData [ DeepSeaColors ,0.5 ] , ColorData [ DeepSeaColors , 0 ] } ,Frame→ True¿ , ImageSize → Small¿

Mathematica code for the plot comparing the performance of the purification protocol for the order of purifying qubits O1 and the order O2. This plot is for an initial state subjected to noise from the depolarising channel.

Show ¿Distill 2[Distill 2[Distill 2[Distill 1[ Distill 1[Distill 1[Depol [ p ]]]]]] ]¿ ¿¿ /¿Tr [ Distill 3[Distill 3[ Distill 3[ Distill 2[Distill 2 [Distill 2[ Distill 1[Distill 1[Distill 1[Depol [ p]] ]]]]]] ]]]¿ .H }, {p , 0,1 }, Filling→1 → {{2 }, {¿PlotRange→ {{0,1 } , {0,1} } , PlotStyle → {ColorData [ SolarColors , 0.5 ] ,ColorData [ SolarColors , 0 ] } ,Frame→ True¿ , ImageSize → Small¿

Mathematica code for the plot comparing the performance of the purification protocol for the order of purifying qubits O1 and the order O2. This plot is for an initial state subjected to noise from the global white noise model.

Show ¿Distill 2[Distill 2[Distill 2[Distill 1[ Distill 1[ Distill 1[GlobalNoise [ p]] ]]]]]¿¿¿ /¿Tr [ Distill 3[Distill 3[ Distill 3[Distill 2[Distill 2 [Distill 2[ Distill 1[Distill 1[Distill 1[GlobalNoise[ p]]]] ]]]]]] ]¿ .H }, {p , 0,1 }, Filling→1 → {{2 }, {¿PlotRange→ {{0,1 } , {0,1 } } , PlotStyle → {ColorData [ AvocadoColors ,0.5 ] ,ColorData [ AvocadoColors ,0 ] } ,Frame→ True¿ , ImageSize → Small¿

64

Appendix I

Mathematica code for the plot comparing the performance of the purification protocol for the order of purifying qubits O1 and the order O3 (see section 5.2.2). This plot is for an initial state subjected to noise from the dephasing channel.

Show ¿

H † . Distill 231[ Distill 312[Distill 123 [Dephase [ p]] ]]Tr [Distill 231[ Distill 312[Distill 123[Dephase [ p ]]]]]

.H },{p , 0,1 }, Filling→ 2→ {{1}, {¿

Opacity [0.5 , ColorData[ DeepSeaColors ,0]]}}, PlotRange →{{0,1 }, {0,1 }},PlotStyle →{ColorData [ DeepSeaColors , 0.5] , ColorData[ DeepSeaColors , 0]}, Frame→ True¿ ,ImageSize→ Small¿

Mathematica code for the plot comparing the performance of the purification protocol for the order of purifying qubits O1 and the order O3. This plot is for an initial state subjected to noise from the depolarising channel.

Show ¿

H † . Distill 231[ Distill 312[Distill 123 [Depol [ p] ]]]Tr [Distill 231[ Distill 312[Distill 123[Depol [ p ]]]]]

. H },{p ,0,1 },Filling→2 →{{1 }, {¿

Opacity [0.5 ,ColorData[ SolarColors , 0]]}}, PlotRange →{{0,1 }, {0,1 }},PlotStyle →{ColorData [ SolarColors , 0.5] , ColorData[ SolarColors , 0]}, Frame→ True¿ ,ImageSize→ Small¿

Mathematica code for the plot comparing the performance of the purification protocol for the order of purifying qubits O1 and the order O3. This plot is for an initial state subjected to noise from the global white noise model.

Show ¿

H † . Distill 231[ Distill 312[Distill 123 [GlobalNoise[ p]]] ]Tr [Distill 231[ Distill 312[Distill 123[GlobalNoise [ p] ]]]]

.H }, {p , 0,1 }, Filling→ 2→ {{1},{¿

Opacity [0.5 ,ColorData[ AvocadoColors ,0 ]]}}, PlotRange→ {{0,1}, {0,1 }},PlotStyle →{ColorData [ AvocadoColors , 0.5] , ColorData[ AvocadoColors ,0 ]}, Frame →True ¿ ,ImageSize→ Small¿

65

Appendix J

Mathematica code for the plot of fidelity F against parameter p for the optimal purification protocol with order O3

4 (see section 5.2.3) applied to an initial state subjected to noise from the dephasing channel.

Show ¿Distill 312[Distill 123[Distill 231[Distill 312[ Distill 123[ Dephase[ p ]]]] ]]¿¿¿¿¿¿¿Tr ¿Distill 231[ Distill 312[Distill 123 [Dephase [ p]] ]]¿ ¿¿¿¿¿¿¿¿¿¿ . H ,{p ,0,1 }, PlotRange→ {{0,1}, {0,1.05 }},PlotStyle →ColorData [ DeepSeaColors , 0] ,Frame → True¿ , ImageSize→ Small ¿

Mathematica code for the calculation of the value of the parameter p for which the output fidelity is F=1 for the order O3

4. Note that this value of p cannot be found with the techniques available to the Mathematica function NSolve and so the protocol must be tested for values of p around p=0.8 (determined from the above plot). The desired value of p is determined to a precision of three decimal places. Once this value is found, a Mathematica code is used to determine the fidelity of the initial state corresponding to this value of p.

H † .¿Distill 231 [ Distill 312[Distill 123 [Dephase [0.803 ]]]]¿¿¿¿¿¿¿¿¿/¿Tr ¿Distill 231[ Distill 312[Distill 123 [Dephase [0.802] ]]]¿ ¿¿¿¿¿¿¿¿¿¿ . H

H † .¿Distill 231[Distill 312[Distill 123 [Dephase [0.802] ]]]¿ ¿¿¿¿¿¿¿¿ /¿Tr ¿Distill 231[Distill 312[Distill 123 [Dephase [0.801] ]]]¿ ¿¿¿¿¿¿¿¿¿¿ . H

H † . Dephase[0.801] .H

Mathematica code for the plot of fidelity F against parameter p for the optimal purification protocol with order O3

4 applied to an initial state subjected to noise from the depolarising channel.

Show ¿Distill 312[Distill 123[Distill 231[Distill 312[ Distill 123[ Depol[ p]]] ]]]¿ ¿¿¿¿¿¿Tr ¿Distill 231[ Distill 312[Distill 123 [Depol [ p] ]]]¿¿¿¿¿¿¿¿¿¿¿ . H , {p ,0,1},PlotRange →{{0,1 }, {0,1.05 }},PlotStyle →ColorData [ SolarColors ,0] , Frame → True¿ , ImageSize→Small ¿

Mathematica code for the calculation of the value of the parameter p for which the output fidelity is F=1 for the order O3

4. Again this value of p cannot be found with the techniques available to the Mathematica function NSolve and so the protocol must be tested for values of p around p=0.8 (determined from the above plot). The desired value of p is determined to a precision of three decimal

66

places. Once this value is found, a Mathematica code is used to determine the fidelity of the initial state corresponding to this value of p.

H † .¿Distill 231[ Distill 312[Distill 123 [Depol [0.799]]]]¿¿¿¿¿¿¿¿¿/¿Tr ¿Distill 231[ Distill 312[Distill 123 [Depol [0.799]]]]¿¿¿¿¿¿¿¿¿¿¿ . H

H † .¿Distill 231[ Distill 312[Distill 123 [Depol [0.798]]]]¿ ¿¿¿¿¿¿¿¿/¿Tr ¿Distill 231[Distill 312[Distill 123 [Depol [0.798]]]]¿ ¿¿¿¿¿¿¿¿¿¿ .H

H † . Depol[0.798] . H

Mathematica code for the plot of fidelity F against parameter p for the optimal purification protocol with order O3

4 applied to an initial state subjected to noise from the global white noise model.

Show ¿Distill 312[Distill 123[Distill 231[Distill 312[ Distill 123[GlobalNoise [ p ]]]]] ]¿¿¿¿¿¿¿Tr ¿Distill 231[ Distill 312[Distill 123 [GlobalNoise [ p]] ]]¿ ¿¿¿¿¿¿¿¿¿¿ . H ,{p ,0,1 }, PlotRange→ {{0,1}, {0,1.05 }},PlotStyle →ColorData [ AvocadoColors ,0] , Frame →True ¿ , ImageSize → Small ¿

Mathematica code for the calculation of the value of the parameter p for which the output fidelity is F=1 for the order O3

4. This time the protocol must be tested for values of p around p=0.6 (determined from the above plot). The desired value of p is determined to a precision of three decimal places. Once this value is found, a Mathematica code is used to determine the fidelity of the initial state corresponding to this value of p.

H † .¿Distill 231[Distill 312[Distill 123 [GlobalNoise [0.602]] ]]¿ ¿¿¿¿¿¿¿¿ /¿Tr ¿Distill 231[Distill 312[Distill 123 [GlobalNoise [0.602]] ]]¿ ¿¿¿¿¿¿¿¿¿¿ . H

H † .¿Distill 231[ Distill 312[Distill 123 [GlobalNoise [0.601]] ]]¿ ¿¿¿¿¿¿¿¿ /¿Tr ¿Distill 231[ Distill 312[Distill 123 [GlobalNoise [0.602]] ]]¿ ¿¿¿¿¿¿¿¿¿¿ . H

H † . GlobalNoise [0.601] . H

67