Non-Abelian Quantum Error Correction Weibo Feng

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2015 Non-Abelian Quantum Error Correction Weibo Feng

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COLLEGE OF ARTS AND SCIENCES

NON-ABELIAN QUANTUM ERROR CORRECTION

WEIBO FENG

A Dissertation submitted to the Department of Physics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

2015

Weibo Feng defended this dissertation on August 31, 2015. The members of the supervisory committee were:

Nicholas E Bonesteel Professor Directing Dissertation

Philip L Bowers University Representative

Jorge Piekarewicz Committee Member

Kun Yang Committee Member

Peng Xiong Committee Member

The Graduate School has veriﬁed and approved the above-named committee members, and certiﬁes that the dissertation has been approved in accordance with university requirements.

ii ACKNOWLEDGMENTS

Here I present my deepest gratitude to my advisor, Nick Bonesteel, without whom the work can never easily be done. Prof. Bonesteel has been constantly supportive of my study here at the Maglab from the first day I joint it. I thank him not only for introducing me to the field of topological quantum computation, but also for his patient guidance and motivation through my research in this field. I shall be forever benefited from what I have learnt from him during this precious period of time. Also to the committee members of my defense, Prof. Philip Bowers, Prof. Takemichi Okui, Prof. Jorge Piekarewicz, Prof. Oskar Vafek, Prof. Pen Xiong and Prof. Kun Yang, I thank them for their challenging questions and helpful comments which have inspired me greatly both in my Prospectus and final Dissertation. I would like to thank my colleagues here in the Maglab, Daniel Zeuch and Julia Wildeboer for their continuous support and feedback throughout the writing and defending of my Dissertation. I thank Julia for the encouragement she gave to me academic-wise and nonacademic-wise. The cakes and cookies she brought to me when I was striving hard to finish my Dissertation were the best! My sincere thanks also go to my good friends in Tally: the list of which can be a really long one and I want to thank them for all the fun we have experienced together here in this lovely town. Our friendship has always been the greatest comfort to me when I was in my hardest time. Finally, I want to thank my parents and my sister. In this past several years their love and support to me are the anchor of my life here in a foreign land. They are the reasons why I never give up on this long life voyage. May my work here and what I shall achieve in the future repay them well.

iii TABLE OF CONTENTS

ListofFigures ...... vi Abstract...... xiii

1 Introduction 1 1.1 Classical and Quantum Computation ...... 1 1.1.1 FromBitstoQubits ...... 2 1.1.2 QuantumGates...... 4 1.1.3 Decoherence and Quantum Error Correction ...... 7 1.2 Topological Quantum Computation ...... 9 1.3 Non-Abelian Quantum Error Correction ...... 11 1.4 BriefOverviewofDissertation...... 12

2 Fibonacci Anyons And Levin-Wen Models 13 2.1 Fibonacci Anyons ...... 13 2.1.1 Basic Algebraic Theory of Fibonacci Anyons ...... 16 2.1.2 Pentagon and Hexagon Equations ...... 21 2.1.3 Quantum Computing by Braiding Fibonacci Anyons ...... 29 2.2 FibonacciLevin-WenModels ...... 33 2.3 Summary ...... 44

3 Non-Abelian Quantum Error Correction 45 3.1 Previous Work on Measuring Qv and Bp ...... 45 3.1.1 VertexErrors...... 47 3.1.2 Plaquette Errors Measured by Plaquette Reduction ...... 49 3.2 New Method for Measuring and Correcting Vertex and Plaquette Errors ...... 60 3.2.1 VertexErrorsonStringEnds ...... 61 3.2.2 Plaquette Swapping ...... 65 3.2.3 Plaquette Swapping: Abelian Case ...... 73 3.3 MovingErrors ...... 75 3.3.1 MovingVertexErrors ...... 76 3.3.2 PlaquetteErrorsRevisited ...... 79 3.3.3 MovingPlaquetteErrors...... 81 3.4 FusingErrors ...... 85 3.5 SummaryofResults ...... 87

4 Summary of Dissertation and Future Work 89

iv Appendices A Quantum Circuits to Verify the Pentagon Equation 92

B Quantum Circuits to Verify the Hexagon Equation 94

Bibliography ...... 96 BiographicalSketch ...... 99

v LIST OF FIGURES

1.1 A classical and quantum bit. (Right) A classical bit can be represented by an arrow which points up if it is in the state 0 or down if it is the state 1. (Left) A qubit can be represented by an arrow which points to a point on the Bloch sphere. Like a classical bit, when the arrow is pointing up or down along the z axis the qubit is in the state 0 or 1 , respectively. However, unlike the classical bit, the qubit can point in an | i | i arbitrary direction, corresponding to a quantum superposition of the states 0 and 1 . 3 | i | i 1.2 One and two qubit quantum gates shown in quantum circuit notation. (a) A single- qubit gate. This gate which applies the unitary operation U to a qubit in the state ψ . The horizontal line represents the qubit with time ﬂowing from left to right. (b) | i A controlled-U two-qubit quantum gate. This gate applies a unitary operation to the bottom qubit in the state ψ if the control qubit is in the state 1 , but otherwise | i | i doesnothing...... 5

1.3 Universal gate set and example of a quantum circuit. (a) The set of all single qubit operations together with the ability to carry out controlled-NOT gates is a universal set of gates, meaning that any unitary operation acting on many qubits can be carried out using them. (b) Example of a quantum circuit acting on 7 qubits...... 6

1.4 Toﬀoli-class quantum gates. (a) A controlled-controlled-U three-qubit gate. This gate applies the operation U to the bottom qubit in the state ψ if and only if the top two | i qubits are both in the state 1 , otherwise it does nothing. (b) A controlled-controlled- | i NOT gate which is also known as a Toﬀoli gate. (c) A controlled-controlled-controlled- U gate. Such multiqubit controlled gates play an important role in the quantum circuits developed in Chapter 3 of this Dissertation...... 8

2.1 A recursive calculation shows that the total quantum dimension of n Fibonacci anyons fusing together follows the Fibonacci series. The first term on the right hand side describes the case when the first two particles fused into a total charge 1, which turns out to be Nn−1. The second term shows when the fused charge of the first two is 0, in which case the number fusion channels is Nn−2 ...... 15

2.2 Two different ways of combining a group of 3 anyons b, a and d are connected by a unitary operation, i.e. the F -tensor. The oval diagram on the left shows how we choose to combine different pairs of anyons( b,a or a,d ). The world line diagram { } { } on the right tells the same story with time flows from top to bottom...... 17

2.3 The R-move switches two anyons before combining into one. The diﬀerence between ab the original and the rotated wave functions is the so called R-matrix Rc ...... 18 2.4 The tube version of the R-move. The space-time line of each anyon is represented by a elastic tube that can track how exactly the line is twisted. The anyon pair a and b then undergoes a counter-clockwise exchange. After that we proceed to ”tighten” the

vi diagram by pulling the tubes away from each other, which results in a clockwise twist for both of the anyon a and b. Note that the anyon c which represents their total charge is still not rotated yet. To ﬁnish the exchange we have to continue to untwist the tube c, giving us a counter-clockwise π rotation...... 19

2.5 The topological spin of a Fibonacci anyon a, which is unique to the 2+1 dimensional phases of matter. Rotating the quasi-particle in 2π is equivalent to a full twist on its tube version, which results in an overall phase θa...... 21

2.6 The pentagon equation which tells the story of two equivalent path of basis changing. Starting from the basis to the very left, one can end at the basis to the very right either through two F -moves(the upper path), of three F -moves(the lower path), putting together to be a pentagon diagram. The diagram is also isotropic: meaning that every basis can be the starting point or the end point. There are always two equivalent path connecting in between — one only needs to change the direction of the F -moves accordingly...... 22

2.7 The world line version of the pentagon equation. Time ﬂows from top to bottom. Each line segment represents an anyon before it merges with another one to create a new anyon that has their total charge...... 23

2.8 A deformed version of the F -move equation, which better demonstrates the symmetries underneath its structure. The dashed red lines mark the symmetric lines of the equation...... 24

2.9 The F -move for Fibonacci anyons in its every component. For each fusion pattern(tree diagram) the black thick line represents a particle of charge 1, while the light thin line marks a charge 0 anyon. (a)&(b) are the cases where the target particle e is solely | i determined by the fusion rule Eq. 2.1. (c) shows the only two non-trivial F -moves here, which needs to be solved in further discussions...... 25

2.10 The hexagon equation which involves both the F - and R-moves. Again it can be seen as composed by two identical paths which can bring one of the six fusion patterns to another(one needs to pay attention to the directions of the moves since R becomes R⋆ when reversed). The diagram contains 3 F -moves and 3 R-moves that interlace with one another, making a hexagonal shape. Notice that in the representation where the equation is FRF = RFR, the R-move always exchange the anyons in a same direction. 26

2.11 The world line version of the hexagon equation. Time ﬂows from top to bottom. Each line segment represents an anyon before it merges with another one to create a new anyon that has their total charge. To avoid confusion the double arrow that marks the R-move is only labeled at the end of the action...... 28

2.12 Constructing the iX single qubit gate using Solovay and Kitaev’s algorithm. The braids produce an operation that approximately realizes the iX rotation with the errors that are in the order of ǫ = 10−5. The algorithm can achieve arbitrary accuracy with the braid length L ln ǫ c where c 4...... 33 ∼| | ∼

vii 2.13 A compilation of the Controlled-NOT gate into braids. Only two of the anyons of the control logical qubit(the top one) are “weaved” around the anyons of the controlled logical qubit. As the total charge of this two determines the state of the control qubit. Given diﬀerent total charge of the two qubits, 0 or 1, the braids can approximate the CNOT gate to a distance ǫ 1.8 10−3 and 1.2 10−3,respectively...... 34 ≈ × × 2.14 An array of Fibonacci anyons that are suitable for the Levin-Wen models. A hexagonal lattice is formed by placing edges on each of the particles. We deﬁne the Qv operator on the three qubits that converge at the same vertex v. And the Bp operator on the plaquette p is so constructed that all the 12 qubits associated to it, including the 6 qubits on the perimeter and the 6 outside, are involved...... 35

2.15 (a) A vertex v in a hexagonal lattice, it has three associated qubits that determines the value of Qv. (b) Lattice conﬁgurations that will have Qv = 1. (c)Lattice conﬁgu- rations that violates the vertex constraints, hence have Qv = 0...... 36

2.16 (a) An example of the Qv ground state, in which the typical conﬁgurations are “loops that allows branches”. (b) Excited states are created by breaking the branching rules. Asindicatedbytheredmark...... 37

2.17 The fattened lattice picture of the Levin-Wen ground state. In this picture, we set the rule that no strings are allowed to pass through the center of the plaquette, as if there is a “puncture”(shaded circles) there for each plaquette. Then, acting the projection operator Bp onto the lattice is equivalent to adding “vacuum strings”(dashed circles) aroundthosepunctures...... 39

2.18 Encoding a logical qubit by cutting a defect(a “hole”) from the lattice. In this case one stops to measure the Bp values of the 10 plaquettes inside the “hole”, as well as the Qv values of the vertices that have been wiped out. In that sense we obtain a degree of freedom from neglecting these operators. Like the defects in Kitaev’s surface code, this degree of freedom is protected globally. And the logical qubit will have the behaviour of a Fibonacci anyon...... 43

3.1 Quantum circuit used to measure the value of the Qv operator defined on the vertex shown left. To perform the measurement one has to add an extra qubit(orange line, initialized in the state 0 ) so that the result can be read out...... 47 | i 3.2 An F -move involves five qubits that are connected in the fusion pattern defined in Chapter 2. One can essentially treat the hexagonal lattice of the Levin-Wen model as a much bigger fusion diagram in which the F -moves can be applied to change its topological structure. The control qubits(black lines) and the target qubit(red line) aremarkedout...... 50

3.3 A rebuild version of Fig. 3.3 which better exhibits the lattice deformation caused by the F -moveintheFibonaccicodes...... 51

viii 3.4 (a) The deﬁnition of the F -move in terms of the F -tensor with each of the qubit labelled. The labels abcdee′ are consistent with those in Fig. 2.8. (b) Quantum circuit which carries out the F -move in (a) for the Fibonacci code. The F rotation acting on the qubit e isdeﬁnedinEq.3.1...... 52

3.5 Four-qubit reduced F -move obtained by identifying the qubits labeled a and d in Fig.3.4...... 53

3.6 Reduction of a hexagonal plaquette to a tadpole through a sequence of six F -moves. The last step is a reduced F -move since two of the control qubits are represented by thesameedge...... 54

3.7 (a) A one loop plaquette with a single leg, often referred as the “tadpole”. (b) A simple quantum circuit which can be used to place a one loop plaquette in the state with Bp = 1. The matrix S isgiveninthetext...... 55

3.8 A quantum circuit which can be used to measure Bp for a one loop plaquette. One extra syndrome qubit initialized to the state 0 is needed to perform the non-demolition | i measurement...... 56

3.9 Quantum circuit which can be used to measure Bp for the Fibonacci code on a hexagonal plaquette based on the plaquette reduction shown in Fig. 3.6. It must be veriﬁed that Qv = +1 on each of the six vertices of the plaquette before carrying out the circuit. 57

3.10 (a) Hexagonal lattice with string end qubits initialized in the state 0 associated with | i each vertex. (b) Possible result of carrying out one round of Qv measurements for each vertex. The remaining string ends represented by thick lines are in the state 1 | i andcorrespondtovertexerrors...... 61

3.11 (a) Two additional qubits, labeled α and β(orange lines), are needed for each vertex v. (b) Unitary transformation which can be used to draw out vertex errors, and (c) a quantum circuit which carries out this transformation and determines whether a vertex error is present. Both the two additional qubits are initialized in the state 0 . | i After the full circuit is carried out qubit α is measured. If it is found in the state 0 then there is no vertex error, and qubits α and β can be safely removed. If it is | i found in the state 1 then there is a vertex error which has now been moved to qubit | i α. From this point on, qubit α will always be in the state 1 and it is therefore not | i necessary to include it explicitly so it can be removed. However, we must keep qubit β which will in general no longer be in the same state as qubit 3...... 62

3.12 (a) A chain of error syndromes in the thickened lattice which consists of only vertex errors. By defaut, the vertex errors are represented by string ends that is placed above the lattice. (b) Combining diﬀerent types of vertex errors and the plaquette errors to form a ribbon graph...... 63

3.13 The idea of doing syndrome measurement and error correction by swapping the target hexagonal plaquette out with a tadpole. The Bp value for the target is unknown and

ix to be measured, however the tadpole has Bp = 1 as already set in the ground state. Three extra qubits, α, β, γ arerequiredhere...... 66

3.14 Swapping a hexagonal plaquette with a good plaquette through a sequence of F /reduced F -moves. At the start a tadpole plaquette is placed at the center of the hexagonal plaquette(the patient). The red line shows the place where the F /reduced F -move is applied at each step of the path. The green ﬁlling indicates it is a “good” plaquette with Bp = 1 while the grey ﬁlling shows that Bp is not known...... 67

3.15 The result of the plaquette swapping process in the thickened picture. The plaquette errors are deﬁned as threads going through each of the plaquettes. By swapping the target plaquette with the “good” tadpole we will end up trapping the error thread insideatadpole(needleeye)...... 68

3.16 Full quantum circuits that realize the plaquette swapping process together with a syndrome measurement for the Fibonacci code on a hexagonal plaquette. The ﬁrst part of it initializes the tadpole that was put in. The second part consists of F /reduced F moves that swap the two plaquette. The third part is for measuring the shrunk targetplaquette...... 70

3.17 The ﬁrst SWAP action which was needed after the ﬁrst F -move in Fig. 3.16. It exchanges the labels of the qubit 7 and γ so that after the F -moves the qubit 7 will stay in its original place while we keep the qubit that was acted by the F -move always being the qubit γ...... 71

3.18 The Abelian version of the F -moves which is all predictable by the Abelian vertex constraint that does not allow branches...... 73

3.19 The Abelian version of the plaquette swapping process on a particular plaquette con- ﬁguration of the Abelian Levin-Wen model. (a) “Swapping” a tadpole for which the head qubit is in the state 0 with the target hexagonal plaquette by applying the | i Abelian F -moves in Fig. 3.18 will bring back the same pattern. (b) “Swapping” a tadpole for which the head qubit is in the state 1 will somehow, ﬂip all the qubits | i that sit on the inner edges of the target plaquette...... 74

3.20 (a) The result of the plaquette swapping process when applied to the Abelian Levin- Wen model. (b) The quantum circuit which realizes the procedure of (a). The tadpole qubit is ﬁrst initialized in the state 0 . It becomes the ground state + after passing | i | i through a Hadamard gate H. At the end, the tadpole qubit is once more transformed by another Hadamard gate and measured to determine the value of Bp...... 75

3.21 An R move used to swing a string end over an edge in order to move it out of a plaquette. When carrying out this move single-qubit rotations must be applied to the two edge qubits. A simple quantum circuit is shown on the right which can be applied to the edge qubits when carrying out an R move...... 76

x 3.22 Use of an F -move to move a string end from one edge to another within a given plaquette...... 77

3.23 (a) Sequence of R and F moves that move a string end around a vertex. The statement that the ﬁnal state must be the same as the initial state (up to an irrelevant phase) is equivalent to the hexagon equation. (b) A quantum circuit, which can be used to verify the hexagon equation. The result of this circuit should be equivalent to the identity when acting on all the possible states which satisfy the vertex constraint. . . 78

3.24 Continuation of the error correction round begun in Fig. 3.10 for Bp measurements. (a) Bp = 1 one loop plaquettes are inserted into all those target plaquettes with no string end pointing inwards. (b) After carrying out the plaquette swap procedure discussed in the text for each of the target plaquettes, the possible result of carrying out one round of Qv measurements for each vertex is shown. The one loop plaquettes have been removed from those plaquettes which were in the state Qv = 1. The remaining plaquettes represented by red one loop plaquettes correspond to plaquette errors ...... 79

3.25 Continuation of one round of error correction. The plaquettes with string ends pointing inwards have not been measured. To measure them one must ﬁrst swing the string ends out of the plaquettes using R moves and then insert freshly initialized one loop plaquettes. The pattern on the right shows one possible result of carrying out plaquette swaps for which one of the plaquettes had Bp = 0 while the other two had Bp = 1. At this point every vertex operator Qv and every plaquette operator Bp has been measured. The vertex and plaquette errors are now entirely contained in the stringendsandoneloopplaquettes...... 80

3.26 A single F -move which moves a bad single-sided plaquette inside of larger plaquette. 81

3.27 Sequence of two reduced F -moves used to pull a bad single-sided plaquette through anedge...... 82

3.28 The Abelian version of moving a good tadpole through a plaquette edge. The Abelian F -move used here is shown in Fig. 3.18. The two diﬀerent case where the head qubit is in the state (a) 0 and (b) 1 are discussed separately here...... 83 | i | i 3.29 (a) Two plaquette errors in the Abelian Levin-Wen model. (b) Moving one of the plaquette error to another is equivalent to applying a series of Z operations along the path...... 84

3.30 Some of the vertex and plaquette errors detected in a round of error correction are moved using F and R moves until they are all entirely within one plaquette on a single edge. Using F moves these errors can then always be combined to form an error tree. 85

3.31 If the root qubit is in the state 1 the errors have not fused into the vacuum. In this | i case one can replace the error tree with a single string end. However, as for (b), it is necessary to check if there is also a plaquette error. This can be done by swinging

xi the string end out of the plaquette using an R move, and then performing a plaquette swap. The possible results of these measurements are shown in the ﬁgure...... 86

3.32 If the root qubit is in the state 0 , then the rest of the tree qubits can be removed. | i However, it is still necessary to check for a plaquette error. This can be done via a plaquette swap and the two possibilities are shown. If one ﬁnds that Bp = 1 then the errors have fused into the vacuum. If one ﬁnds that Bp = 0 the errors have fused into asingleplaquetteerror...... 87

A.1 (a) The pentagon equation. The red qubits mark the target qubits where the F - moves are applied. At the end, the pattern goes back to itself only with qubit 5 and 6 swapped. (b) The quantum circuit which carries out the pentagon equation of (a). We use the box notation introduced in Chapter 3. The equation only holds if the vertex constraints are satisﬁed for all the vertices shown up in (a)...... 93

B.1 (a) The quantum circuit for the R-move deﬁned in Eq. 3.18. Both the qubit a and b undergoes an opposite twist against the qubit c. (b) The circuit for the R−1-move. . . 94

B.2 (a) The hexagon equation on a vertex, which has the same topological structure with that in Eq. 2.11. (b) The quantum circuit identity which reﬂects (a). The box notations for the R and R−1-moves are given in Fig. B.1...... 95

xii ABSTRACT

A quantum computer is a proposed device which would be capable of initializing, coherently manipulating, and measuring quantum states with suﬃcient accuracy to carry out new kinds of computations. In the standard scenario, a quantum computer is built out of quantum bits, or qubits, two-level quantum systems which replace the ordinary classical bits of a classical computer. Quan- tum computation is then carried out by applying quantum gates, the quantum equivalent of Boolean logic gates, to these qubits. The most fundamental barrier to building a quantum computer is the inevitable errors which occur when carrying out quantum gates and the loss of quantum coherence of the qubits due to their coupling to the environment (decoherence). Remarkably, it has been shown that in a quantum computer such errors and decoherence can be actively fought using what is known as quantum error correction. A closely related proposal for ﬁghting errors and decoherence in a quantum computer is to build the computer out of so-called topologically ordered states of matter. These are states of matter which allow for the storage and manipulation of quantum states with a built in protection from error and decoherence. The excitations of these states are non-Abelian anyons, particle-like excitations which satisfy non-Abelian statistics, meaning that when two excitations are interchanged the result is not the usual +1 and -1 associated with identical Bosons or Fermions, but rather a unitary operation which acts on a multidimensional Hilbert space. It is therefore possible to envision computing with these anyons by braiding their world-lines in 2+1-dimensional spacetime. In this Dissertation we present explicit procedures for a scheme which lives at the intersection of these two approaches. In this scheme we envision a functioning “conventional” quantum computer consisting of an array of qubits and the ability to carry out quantum gates on these qubits. We then give explicit quantum circuits (sequences of quantum gates) which can be used to create and maintain a topologically ordered state with non-Abelian anyon excitations using the “conventional” qubits of the computer. Our circuits perform measurements on these qubits which detect “errors” corresponding to deviations from the topologically ordered ground state of interest. We also give circuits which can be used to move these errors and eventually fuse them with other errors to eliminate them.

xiii CHAPTER 1

INTRODUCTION

In this introductory Chapter we review some key concepts in the theory of quantum information and quantum computation, emphasizing those concepts which will be important for understanding the central results of this Dissertation. We then go on to review the properties of so-called non- Abelian anyons and describe how they can be used for quantum computation. Finally, we give a brief overview of the main results of the Dissertation.

1.1 Classical and Quantum Computation

Ever since the ﬁrst electronic programmable Turing-complete computer, ENIAC, was built in the mid 1940s, the industry of manufacturing electronic computers has undergone enormous devel- opment, transitioning from the use of antique vacuum-tube based circuits to transistor based ones. Today, even a common personal computer built with VLSI (Very-Large-Scale Integration) technology can operate with a calculation speed that is billions of times faster than that of the ENIAC. However, despite these tremendous leaps forward, the basic principals on which computations are performed has not changed. Regardless of the particular storage media used, in any “classical” digital computer computational information is typically stored in systems which can exist in one of two possible discrete states, 0 or 1, referred to as “bits.” Logical gates based on Boolean algebra can be carried out on these bits (through suitably designed electrical circuits in an electronic computer) to perform computations. Even though a thorough understanding of today’s transistor-based computers requires some knowledge of quantum mechanics, in any actual computation all the fundamental elements (i.e., the bits and logical gates) that are used can be treated as “classical,” meaning that the laws of physics governing them are those of classical mechanics. The notion of a quantum computer, in which the computations themselves are governed not by classical but by quantum mechanics, was ﬁrst suggested by Feynman [1]. At the time, Feynman was motivated by the observation that simulating quantum systems on classical computers can be

1 extremely difficult, primarily because the dimensionality of the Hilbert space needed to describe these systems grows exponentially with their size. It was therefore natural to suppose that a quantum computer, equipped with a similar exponentially large Hilbert space, would be able to efficiently simulate quantum systems, even when classical computers could not. Subsequently, aside from a very small group of pioneering researchers, the idea of quantum computation was not actively pursued by the broader physics and computer science communities until a key breakthrough due to Peter Shor [2]. What Shor showed was that a quantum computer could factor a large composite number C into primes in a time which scaled polynomially with the number of digits of C. This is a dramatic qualitative improvement over the best known classical algorithm, which scales exponentially with the number of digits of C. This realization that quantum computers could perform fundamentally different kinds of computations led to an explosion of experimental and theoretical work which continues to this day.

1.1.1 From Bits to Qubits

In a quantum computer, ordinary classical bits, which can exist in the states 0 or 1 exclusively, are replaced by so-called quantum bits or “qubits.” A qubit is any two-level quantum systems with basis states 0 and 1 . Because of their quantum nature, qubits possess many peculiar properties | i | i that make them computationally powerful yet diﬃcult to manipulate. Figure 1.1(a) shows a simple depiction of the two states of a classical bit, with an arrow which points up if the bit is in the state 0 or down if it is in the state 1. Like a classical bit, a qubit can also be in the state 0 or 1 . But unlike a classical bit, a qubit can be placed in a quantum | i | i superposition of 0 and 1 . Such a superposition can also be represented by an arrow pointing to | i | i a point on the Bloch sphere, as shown in Fig. 1.1(b). If the arrow is pointing up or down along the z-axis, the qubit is in the state 0 or 1 , respectively. But if it is pointing in any other direction | i | i the qubit is in a quantum superposition which, in general, can be written as

Ψ = α 0 + β 1 . (1.1) | i | i | i For such a state we can no longer say with certainty that a qubit is in the state 0 or 1 , but rather | i | i it has the possibility to be in either of the two states. Moreover, quantum mechanics tells us that a local measurement on the qubit will “collapse” the quantum superposition (1.2) into a measured state which is determined by the basis in which the measurement is performed. For example, in the

2 Classical Bit Quantum Bit Z 0 0 0 θ

Y ϕ

X 1 1 1 θ θ ψ =cos 0 + eiϕ sin 1 2 2

Figure 1.1: A classical and quantum bit. (Right) A classical bit can be represented by an arrow which points up if it is in the state 0 or down if it is the state 1. (Left) A qubit can be represented by an arrow which points to a point on the Bloch sphere. Like a classical bit, when the arrow is pointing up or down along the z axis the qubit is in the state 0 or 1 , respectively. However, | i | i unlike the classical bit, the qubit can point in an arbitrary direction, corresponding to a quantum superposition of the states 0 and 1 . | i | i

“computational” basis 0 , 1 , the result of measuring the state (1.2) will be to ﬁnd the qubit in {| i | i} the state 0 with probability α 2, and in the state 1 with probability β 2 (here we assume that | i | | | i | | the state Ψ is normalized, so that α 2 + β 2 = 1.) | i | | | | An even more bizarre property of qubits is that they can exhibit a property known as quantum entanglement. A simple example of an entangled state would be a two-qubit (a and b) system in the state

1 Ψ ab = ( 0 a 1 b + 1 a 0 b) . (1.2) | i √2 | i | i | i | i

3 This state is an example of a so-called maximally entangled state. In such a state it is not possible to write the quantum state separately for each qubit. More precisely, one can not rewrite the overall state Ψ as a tensor product of two single-qubit states, i.e., | iab Ψ = Ψ Ψ (1.3) | iab 6 | ia ⊗| ib In that sense the two qubits in the state (1.2) are said to be “entangled” with each other. This means that if we measure, for example, qubit a and finds that it is in state 0 , then we immediately | i know that qubit b must be in state 1 . Thus, even though we only touched qubit a, the other qubit | i “magically” collapsed to a known state. Quantum entanglement is considered one of the most interesting phenomenon in the quantum mechanical world. It has been a rich research topic in many fields. In quantum computing, it provides both challenges and opportunities. On the one hand, qubits can be entangled with un- wanted degrees of freedom that could bring additional error into the system (this is the physical mechanism behind “decoherence”). On the other hand, states consisting of many entangled computational qubits are exponentially more complex than states of the same number of classical bits, a fact which, apparently, makes quantum computers significantly more powerful than their classical counterparts when dealing with certain problems.

1.1.2 Quantum Gates

To build a quantum computer, it will not be enough to simply have a collection of qubits. It will, of course, also be necessary to manipulate them according to the laws of quantum mechanics. A standard way to think about how such manipulations would be carried out is in terms of quantum gates — operations which describe the quantum time evolution of small numbers of qubits at a time. The language of quantum gates is both useful theoretically for describing quantum computation, and also experimentally for given a set of minimal requirements for carrying out quantum computation. Quantum gates play an important role in the original work presented in this Dissertation, and so we review some basic facts about them below. Quantum gates can be viewed as quantum versions of the classical Boolean logic gates that lie at the heart of classical computers. Like classical gates, which transform the state of a set of ordinary bits, quantum gates transform the quantum state of a set of qubits. Such transformations are described mathematically in terms of operators acting on the Hilbert space of the qubits involved.

4 a) ψ U U ψ

b) 0 0 1 1

ψ U ψ ψ U U ψ

Figure 1.2: One and two qubit quantum gates shown in quantum circuit notation. (a) A single- qubit gate. This gate which applies the unitary operation U to a qubit in the state ψ . The | i horizontal line represents the qubit with time ﬂowing from left to right. (b) A controlled-U two- qubit quantum gate. This gate applies a unitary operation to the bottom qubit in the state ψ if | i the control qubit is in the state 1 , but otherwise does nothing. | i

These operators must be unitary, a fact which follows from the normalization condition. Consider a quantum gate which takes the initial state Ψ to a final state U Ψ where U is the operator | i | i which describes the gate. Since Ψ Ψ = 1 and Ψ U †U Ψ = 1, it must be the case that UU † = 1 h | i h | | i and thus U is unitary. Figure 1.2 shows examples of single-qubit and two-qubit quantum gates. These gates are depicted using quantum circuit notation. In this notation, each horizontal line represents a qubit, with time flowing from left to right. In Fig. 1.2(a) a single-qubit operation, U, represented by the box labeled U is enacted on a single qubit represented by the single line. The ability to carry out such single-qubit gates is crucial for quantum computation, but clearly not sufficient. It is also necessary to interact qubits with each other in a controlled way in order to produce entangled states. Figure 1.2(b) shows an important class of two-qubit quantum gates which can be used for this purpose. These gates are so-called “controlled” gates. For such a controlled-U

5 a) A universal gate set:

0 1 X =   U 1 0 X  

Single qubit operations Controlled-NOT b) X

U X U X U U X X U

Figure 1.3: Universal gate set and example of a quantum circuit. (a) The set of all single qubit operations together with the ability to carry out controlled-NOT gates is a universal set of gates, meaning that any unitary operation acting on many qubits can be carried out using them. (b) Example of a quantum circuit acting on 7 qubits.

gate a unitary operation is carried out on the bottom qubit (the target qubit) if the top qubit (the control qubit) is in the state 1 and otherwise does nothing. | i A fundamentally important concept in the theory of quantum computation is the notion of a “universal set” of quantum gates (see, e.g., [3]). Universal gate sets have the property that they can be used to carry out arbitrary unitary operations on an arbitrary number of qubits. This means that any quantum algorithm which acts on any number of qubits can be decomposed into a series of quantum gates, drawn from the universal set, each of which acts on only a small number of qubits. A standard choice for a universal gate set is shown in Fig. 1.3(a). This gate set consists of all possible single qubit operations, and a single nontrivial two-qubit controlled gate. For the two-qubit gate, a standard choice is to take the controlled operation acting on the target qubit to be a NOT

6 gate,

0 1 X = . (1.4) 1 0 (Here we follow standard notation in quantum information theory and use the notation X for this gate, rather than the Pauli matrix σx more familiar to physicists.) The resulting two-qubit gate is known as a controlled-NOT (CNOT) gate. In the two qubit basis

ab = 00 , 01 , 10 , 11 (1.5) | i {| i | i | i | i} the matrix which denotes a CNOT gate can be written as 1 0 0 0 0 1 0 0 UCNOT =   (1.6) 0 0 0 1  0 0 1 0      With a universal set of quantum gates in hand, one can construct quantum circuits which carry out any unitary operation. Figure 1.3(b) shows a typical example of a quantum circuit. It is often useful to expand gate sets to including quantum gates which act on more than two qubits. Examples of an important class of such multiqubit gates are shown in Fig. 1.4. Figure 1.4(a) shows a three qubit controlled-controlled-U gate. This gate carries out a unitary operation U on the bottom (target) qubit only if the top two (control) qubits are in the state 1 , and otherwise | i acts as the identity. Such gates are referred to as Toffoli-class gates because they generalize the fundamental Toffoli gate, a gate which is equivalent to a controlled-controlled-NOT gate, shown in Fig. 1.4(b). The notion of a Toffoli-class gate can be generalized to many qubits as shown, for example, in Fig. 1.4(c), which depicts a four-qubit controlled-controlled-controlled-U gate. Of course any Toffoli-class gate can be expressed in terms of simpler one and two-qubit gates drawn from a universal gate set. Nevertheless, such multi-qubit gates will play an important role in the original work of this Dissertation presented in Chapter 3.

1.1.3 Decoherence and Quantum Error Correction

In addition to providing a useful theoretical framework for thinking about quantum computation, the quantum gate model provides a “recipe” for how to build one. Any proposed experimental quantum computing scheme must provide a physical realization of qubits, and procedures for carrying out a universal set of quantum gates on these qubits. In the 15 years since the discovery ∼

7 a) a a 1 1

b b 1 1

 U   U U  ab 00,01,10

b) c)

X U

Figure 1.4: Toﬀoli-class quantum gates. (a) A controlled-controlled-U three-qubit gate. This gate applies the operation U to the bottom qubit in the state ψ if and only if the top two qubits are | i both in the state 1 , otherwise it does nothing. (b) A controlled-controlled-NOT gate which is also | i known as a Toﬀoli gate. (c) A controlled-controlled-controlled-U gate. Such multiqubit controlled gates play an important role in the quantum circuits developed in Chapter 3 of this Dissertation.

of Shor’s algorithm, there has been enormous experimental progress on both of these fronts. This progress has occurred for a variety of physical realizations of qubits, including trapped ions, electron spins trapped in semiconductor quantum dots, and superconducting qubits (see, e.g., [4], for a review of progress realizing such qubits, and carrying out quantum gates using them). There is, however, one enormous hurdle which will have to be overcome in order to build a functioning quantum computer. The computer will have to protected against the unavoidable eﬀects of decoherence. Decoherence is the loss of quantum coherence of a quantum system which occurs when its degrees of freedom become entangled with its environment [5]. Because it is impossible to completely isolate any system from the rest of the universe, such deocoherence is an inevitable process. In a quantum computer, decoherence, if left unchecked, would eventually spoil any quantum com-

8 putation. Remarkably, it was shown soon after the discovery of Shor’s algorithm that there exist procedures for protecting quantum information from decoherence. These procedures are collectively known as quantum error correction [6]. The essential idea of quantum error correction is to store the logical qubits of the computer (i.e., the qubits one is actually computing with) by encoding them into highly entangled states of many physical qubits (i.e., the inevitably imperfect two-level systems used to build the computer). If these highly entangled states, referred to as quantum error correcting codes, are chosen properly, it is possible to perform measurements on them in order to determine whether or not an error has occurred, without disturbing the quantum state of the encoded logical qubits. These errors can then be corrected before they become severe enough to degrade the quantum information stored by the code. In this way the quantum coherence of the encoded information is protected through the constant vigilance of repeated measurements checking for errors. While these procedures result in signiﬁcant overhead in both qubits and processing time, this overhead still can be shown to scale only polylogarithmically with the size of the problem be solved and so does not change the complexity class in the computer science sense of any given problem [6]. One of the most promising quantum error correcting codes are the so-called surface codes [7]. These codes can be viewed as ground states of certain lattice Hamiltonians. The prototypical surface code is based on Kitaev’s Abelian toric code [8, 9]. Procedures for using the Kitaev surface code to encode and process quantum information, while simultaneously detecting and correcting errors, have been worked out in great detail [10, 11, 12]. There are good reasons for believing this approach, which appears to have a high error threshold, is one of the most promising routes to building a functioning fault-tolerant quantum computer.

1.2 Topological Quantum Computation

An alternate approach to building a fault-tolerant quantum computer is based on using so-called non-Abelian anyons — exotic quasiparticle excitations of certain topologically ordered states of matter in two space dimensions — to store and manipulate quantum information in an intrinsically fault-tolerant manner [13]. According to the spin-statistic theorem all real world particles (quarks, photons, electrons, etc.) are classiﬁed into two fundamental categories: bosons and fermions. The reason to do so is based

9 on the intrinsic fact that they are identical particles. Imagine we have a system consisting of N such identical particles. Exchanging any two of them would apparently yield the same physical state, since there will only be a change on the labellings of each particle. Then the wave functions which describe the states before and after the exchanging should only differ by a phase eiφ. In 3 + 1-dimensional space-time (our real world), one would expect two sequential exchange of the same pair of identical particles is trivial since their 4-dimensional world lines after the exchange can be topologically deformed into the initial one, which means they are essentially equivalent. This gives (eiφ)2 = 1. Therefore the phase difference can only be φ = 0 or φ = π, leading to a possible sign change of the wavefunction. Each of them corresponds to bosons and fermions, respectively. Now if we consider the case when we confine particles to a 2-dimensional plane (as can occur, e.g., in semiconductor heterostructures), i.e. (quasi)particles living in the 2 + 1-dimensional space- time, exchanging two particles twice will not necessarily return to the original state. In fact, the result is always nontrivial. This can be easily understood when we consider the 3-dimensional world line of the two particles, moving these particles around will cause their world lines to be “twisted” together into “braids”, which is topologically non-trivial (the worldlines can not be topologically deformed into the initial state) and leads to exotic particle statistics. For these particles, the phase differences caused by moving them around one another can be expanded from the discrete values of 0 and π to a continuous spectrum on the interval [0, π], namely the Abelian anyons, or even characterized by a unitary matrix Uij, for the case of non-Abelian anyons. It is due to the observation that when world-lines of non-Abelian anyons are braided they carry out unitary operations on a Hilbert space (i.e., they perform a kind of quantum gate) that Alexei Kitaev proposed they might be used to build a novel kind of quantum computer, a “topological” quantum computer [8]. The motivation for Kitaev’s proposal was that, unlike other conventional quantum computing schemes, e.g., those based on superconducting qubits, electron spins, trapped ions, etc., which are relatively fragile and susceptible to error, the quantum information associated with non-Abelian anyons is well protected by its topological robustness. In a topological quantum computer unitary operations are carried out by adiabatically exchanging anyons on a 2-dimensional plane. The world lines of the computational anyons under such operations will form a braid pattern and the resulting unitary operation only depends on the

10 its topology. Therefore local perturbations on the space-time trajectories of the anyons will not aﬀect the overall braiding topology as long as no additional errors are involved. There are a number of physical systems in which it is conjectured that non-Abelian anyons may arise naturally [13]. The simplest type of non-Abelian anyon for which braiding alone is suﬃcient to carry out arbitrary quantum computation (i.e., for which the braids form a universal set in the sense described above) are the Fibonacci anyons [14] (so-called because the Hilbert space degeneracy associated with N such particles grows as the Fibonacci sequence 1,1,2,3,5,8,13, ). In ··· Chapter 2 we will review the properties of these anyons and also describe the procedures which have been suggested for how they might be used to carry out quantum gates in a topological quantum computer.

1.3 Non-Abelian Quantum Error Correction

One route to computing with non-Abelian anyons is to follow the same approach as in the Kitaev surface code. In this approach, rather than physically realizing a Hamiltonian whose ground state has topological order and non-Abelian anyon excitations with the desired properties, one imagines realizing these states using a “conventional” quantum computer in essentially the same way that the ground state of the Kitaev toric code model is realized in the Kitaev surface code. This is done by repeatedly measuring the local terms in the Hamiltonian (which, for the Hamiltonians of interest are all mutually commuting) to detect deviations from the ground state. The results of these measurements are referred to as syndromes, and after extracting these syndromes, any detected errors are then decoded and corrected in order to restore the ground state. An outline for realizing such non-Abelian surface codes based on the Levin-Wen models, a class of model Hamiltonians which generalize the Kitaev toric code and can have non-Abelian excitations, was presented by K¨onig, Kuperberg, and Reichardt [15]. The speciﬁc case they focused on was the Levin-Wen model for so-called doubled Fibonacci anyons. One advantage of using this model is that Fibonacci anyons are universal for quantum computation, meaning it is possible to carry out all quantum operations purely by braiding them, without the need for non-topological operations and magic state distillation which are both required when using the Abelian Kitaev surface code, as well as topological quantum computation using Ising anyons [16].

11 The main results of this Dissertation are a new set of quantum circuits which can be used for syndrome extraction and error correction for the ground state of the Fibonacci Levin-Wen model.

1.4 Brief Overview of Dissertation

The rest of the Dissertation is organized as follows. In Chapter 2 we review the mathematical properties of non-Abelian anyons, focusing on the case of Fibonacci anyons, and describe in some detail how they can be used for quantum computation. We then go on to review the Levin-Wen models, focusing again on the Fibonacci case. In Chapter 3, we turn to the main new results of the Dissertation. First, we give explicit quantum circuits for detecting and “healing” vertex and plaquette errors, at the cost of introducing additional defects in the lattice. We then give quantum circuits which can be used to carry out a novel procedure for detecting plaquette errors using what we refer to as plaquette swapping. This procedure swaps a freshly initialized one-loop plaquette for the full plaquette under investigation, trapping any error that was in the original plaquette in the loop. We then go on to addresses the question of how the additional vertex defects and trapped plaquette errors can be moved through the lattice, and eventually fused in order return the system to the Levin-Wen ground state.

12 CHAPTER 2

FIBONACCI ANYONS AND LEVIN-WEN MODELS

In this Chapter we ﬁrst review the theory of Fibonacci anyons. Due to the simplicity of their nature, as well as the fact that they can be used to carry out a universal set of quantum gates [17, 18, 19], Fibonacci anyons are considered one of the most promising anyon types for performing topological quantum computation [13]. In addition to describing the mathematical properties of these anyons, we review proposed methods for using them to encode qubits and carry out quantum gates. After this review of Fibonacci anyons, we turn to the so-called “Levin-Wen models.” These are lattice spin models with non-Abelian anyon quasiparticle excitations. We focus on the particular Levin-Wen model which realizes Fibonacci anyons which we refer to as the “Fibonacci Levin-Wen model.” This review sets the stage for Chapter 3, where we present the main new results of this Dissertation on how the ground states of the Fibonacci Levin-Wen model can be realized and protected against error using a “conventional” quantum computer.

2.1 Fibonacci Anyons

As pointed out in the previous chapter, anyons are of great interest among frontier physicists not only because of their bizarre properties shown in fractional quantum hall system, but also the possibility to utilize them in topological quantum computations. Fibonacci anyons, one of the simplest yet non-trivial non-Abelian anyons, immediately draw our attention ﬁrst because of their potential for performing universal quantum computation. Unlike Abelian anyons that are used in Kitaev surface code, where non-topological procedures (e.g. so-called “magic state distillation [20]) must be used to realize arbitrary quantum gates, Fibonacci anyons can be used to carry out quantum gates purely by braiding them around each other [18, 17, 19]. In what follows we will elaborate on the nature of Fibonacci anyons and also introduce the mathematical framework for describing and computing their braiding properties.

13 In any anyon model, particles, and collections of particles, are characterized by a quantum number known as “topological charge” or, simply, charge. For Fibonacci anyons there are only two such charges, which we label 0 or 1. The charge 0 is the same charge as the “vacuum” (or many-body ground state in the condensed-matter physics context) and, unless we state otherwise, when we refer to a Fibonacci anyon we mean a particle with charge 1. General anyon theories describe both particles and their antiparticles. For Fibonacci anyons the charge 1 particle acts as its own antiparticle which is one reason why Fibonacci anyons are simple in their nature. A key ingredient for deﬁning any anyon model is specifying how they behave when fused together. This behavior is encoded mathematically in what are known as “fusion rules.”’ For Fi- bonacci anyons, the set of fusion rules are given below,

0 0 = 0 × 0 1 = 1 × 1 0 = 1 × 1 1 = 0 + 1. (2.1) ×

The last fusion rule is the core of what makes Fibonacci anyons non-Abelian. Unlike Abelian anyons for which the fusion result of two particles is always unique, non-Abelian anyons can be fused in diﬀerent “channels”. For instance the fusion rule above states that two charge 1 Fibonacci anyons can be fused to have either a total charge 1 or a total charge 0. What this means is that the Hilbert space of two charge 1 Fibonacci anyons is two-dimensional, with basis states where the total charge of the two anyons is 0 and 1. Thus we see that fusing into multiple channels leads to a multidimensional Hilbert space which is a clear indication of non-Abelian anyons. The reason why anyons that obey the above fusion rules are called “Fibonacci anyons” is because, as we add particles, the dimensionality of their Hilbert space grows as the Fibonacci series. The proof is as follows. For a group of n (n> 0) anyons which obey the above fusion rules, consider the case where all of the anyons are Fibonacci anyons (i.e. they all have charge 1) and they are fused one by one in such a way that their total charge is 0. How many fusion channels are there? The answer can be easily calculated using a recursive method. Suppose that the total number of the fusion channels for n anyons is Nn, Let us ﬁrst put our attention on the two anyons which are fused at the beginning. As predicted by the fusion rules there are two possible results

14 n 1 1 1 1 1 1 1 1 n-2 1 n-2 1 1 1 1 0 1 1 1 1

0 0 0

Figure 2.1: A recursive calculation shows that the total quantum dimension of n Fibonacci anyons fusing together follows the Fibonacci series. The first term on the right hand side describes the case when the first two particles fused into a total charge 1, which turns out to be Nn−1. The second term shows when the fused charge of the first two is 0, in which case the number fusion channels is Nn−2 of them fusing together: a total charge 1 where it continues to fuse with the third anyon and etc., or a total charge 0 which adds no complexity to the rest of the fusions, since it has to fuse with the third anyon into a total charge 1 thus there is only one fusion channel next. The first possible result will give us nothing else but the case when n 1 charged 1 anyons(considering the first-fused − two anyons as one particle plus the rest n 2 ones) fusing all together, hence a total number of − Nn−1 fusion channels. While in the second case we can safely “remove” the first two anyons which leaves us the same problem that has Nn−2 anyons. Therefore the following recursive equation can be derived,

Nn = Nn−1 + Nn−2, (2.2) given the initial conditions that a) N1 = 0 because clearly there is no way a charged 1 anyon can be fused into charge 0 and b) N2 = 1 by the fusion rules. Solving the equation gives the famous Fibonacci sequence

N = 1, 1, 2, 3, 5, 8, 13, 21, , (2.3) n ··· n = 2, 3, 4, 5, . ···

A detailed illustration of the above deduction is shown in Fig. 2.1. In this ﬁgure we use so-called “fusion tree” diagrams, in which the tree branches (labeled 1 in this ﬁgure) represent Fibonacci

15 anyons, and fusion occurs when two branches meet. Also note that for large numbers of n the dimensions of the Hilbert space grow asymptotically in the exponential function as

N (φ)n , (2.4) n → where φ =(√5 + 1)/2 is the golden mean, a characteristic number for the Fibonacci anyons. Based on the above result, we can determine the so-called “quantum dimension” of Fibonacci anyons. This quantity, denoted ds, where s is the charge of the particle being considered, determines the rate at which the Hilbert space of n of these particles grows in the limit of large n, with this n dimensionality being proportional to ds . For the Fibonacci anyon model, the quantum dimensions for each type of particle now can be deﬁned conveniently as:

d0 = 1

d1 = φ. (2.5)

While the behavior of the charge 1 Fibonacci anyons is evident in the above illustration, the fact that the quantum dimension of charge 0 particles is 1 is immediately follows from the fact that the fusion rules for these particles shows no branching. Given the quantum dimensions of all particle types in an anyon theory it is possible to deﬁne the so-called “total quantum dimension,”

2 D = di , (2.6) sXi where, for Fibonacci anyons, D = φ2 + 1. p 2.1.1 Basic Algebraic Theory of Fibonacci Anyons

To define an anyon theory, it is not enough just to know how anyons are fused. For example, it is important for us to know the properties of the Hilbert space that the Fibonacci anyons have spanned. Here a natural question would be: how do we perform a basis change in that space? Given a number of n anyons, and again suppose that the total charge of all of them is fixed (either 0 or 1), one choice of fusing these anyons one by one in pairs represents a group of basis of the degenerate Hilbert space. However, we are always allowed to do a basis change, i.e to choose different ways to combine these particles in another particular order granted that the fusion pattern is still inside the same Hilbert space. From the experiences where we were dealing with the other Hilbert spaces

16 b a d b a d

F abe b a d b a d e Fcde′ e’ e e’ c c c c

Figure 2.2: Two different ways of combining a group of 3 anyons b, a and d are connected by a unitary operation, i.e. the F -tensor. The oval diagram on the left shows how we choose to combine different pairs of anyons( b,a or a,d ). The world line diagram on the right tells the same story { } { } with time flows from top to bottom.

we know that, the only price we have to pay for choosing another set of basis is the multiplication of a unitary transformation that is represented by a tensor featuring the inner product of old and new basis vectors. Here in the anyon theory we call it the F -tensor, and the action of basis change is called the F -move. A simple example would be to consider the case when 3 anyons b, a and d fuse together, shown in Fig. 2.2. In principle there are two ways to combine these anyons: one could either choose to fuse anyon b and a first, getting a total charge e which then fuses with anyon d to get the total charge c or, alternatively, choose to fuse a and d first to have a total charge e′, then combine with anyon b to again get c. In Fig. 2.2 we show two different diagrams representing this basis change: the “oval diagram” on the left which circles the pair of anyons in the fusing order, or the “tree diagram” which is essentially the world lines of this fusing process. Since we have known that different fusion patterns actually represents different choice of basis for this 3-anyon Hilbert space, we may conveniently connect these two patterns with the F -tensor named above. Intuitively, by looking at the oval diagram of the F -move one can easily tell that after the fusion pattern is switched, the total charge c of the three anyons remains the same. Therefore it will be convenient for us to think of the anyon b, a, c which is the combination of the three, and d as four “control particles”(as more clearly illustrated in the tree diagram) since they will be invariant under the F -move. The anyon e which is the combination of b and a and which then becomes total charge of a and d as e′, can be named the “target particle”. In other words, we can put the wavefunction multiplier, the F -tensor solely onto the target particle. We will see this conclusion useful in the later contents.

17 a b b a a b

R c a b b a c c Rab c c c

Figure 2.3: The R-move switches two anyons before combining into one. The diﬀerence between ab the original and the rotated wave functions is the so called R-matrix Rc

Another important property of fusing anyons comes from the fact that fusion is invariant under rotation (clockwise or counter-clockwise). For a pair of two anyons, exchanging their position shall intuitively not aﬀect the total charge. In other words, the fusion Hilbert space only undergoes an isomorphism with the exchange. Hence similarly there should be a unitary operator connecting in between, which is called the R-matrix. The operation of exchanging two anyons is then named an R-move. Figure 2.3 shows the exchange of two anyons a and b with a total charge of c, the sense of the exchange is taken counter-clockwise as a default. To understand how the R-matrix aﬀects the two particle states with total charge 0 and 1, it will be necessary to peek inside the details of the exchange process. One has to consider very carefully what actually occurs when switching the positions of two anyons by rotating them as a whole particle, since this will inevitably rotate each of the anyons themselves as well. Here it is more convenient if to consider “tubed” versions of the anyon world lines as this allows us to better visualize the details of the rotations. Later on we will show how this new picture can be used to simplify calculations involving R-moves. Figure 2.4 shows a counter-clockwise exchange of the two anyons a and b . The rotations of { } { } all the concerned anyons a,b,c are tracked by adding an arrow on top of each one of them. As { } clearly seen in the picture, after the exchange all of the three tube branches (representing a, b and c separately) will undergo a π twist after stretching them out. Note that both of the fused anyons a and b in the end will result in a state which is equivalent to the original state getting rotated in the opposite direction (clockwise in this case) of the exchange, while their fused version c is already rotated in the same direction of the exchange. Therefore it is reasonable for us to believe that the

18 b a b a a b b a

c c c

Figure 2.4: The tube version of the R-move. The space-time line of each anyon is represented by a elastic tube that can track how exactly the line is twisted. The anyon pair a and b then undergoes a counter-clockwise exchange. After that we proceed to ”tighten” the diagram by pulling the tubes away from each other, which results in a clockwise twist for both of the anyon a and b. Note that the anyon c which represents their total charge is still not rotated yet. To ﬁnish the exchange we have to continue to untwist the tube c, giving us a counter-clockwise π rotation.

R-move can be actually decomposed into three single anyon rotations (or “twists”) which share the same absolute value of the angle of rotation, only with exactly opposite signs. Now if we deﬁne the topological spin [21] of an anyon as “the topological phase acquired by rotating the particle by 2π”, similar to a regular spin, then it is possible to characterize the R-matrix by the topological spins of each anyon involved. Fig. 2.5 shows the deﬁnition of the topological

i2πsa spin for the Fibonacci anyons where particles are their own anti-particles. Note that θa = e is given when the anyon a undergoes a counter-clockwise self-rotation. The clockwise case will { } ⋆ −i2πsa actually generate its complex conjugate version, θa = e .

Generally for non-Abelian anyons, the topological spin θa has a matrix form, depending on the charge a of a given quasiparticle. For Fibonacci anyons that have only two types of quasiparticle (charge a = 0 and 1), ei2πs0 0 θ = . (2.7) a 0 ei2πs1 And it is easy to conjecture that

θ0 = 1, s0 = 1 (2.8)

i2πs1 while θ1 = e is the component that contains the non-trivial information we need. For Abelian anyons, the topological spin may reduce to a single phase eiθ where θ is not a matrix, which well reﬂects the Abelian nature. Moreover, we will have the particles being bosons if θ = 0 and the fermions are obtained if θ = π.

19 c Going back to the R-move representation now it is clear how the R-matrix Rab is decomposed in terms of topological spins. Since exchanging two anyons is equivalent to rotating each of them by π in the opposite direction, which is depicted in Fig. 2.4, we conclude that two consecutive position exchanging(which is essentially acting the R-move twice) on the same anyon pair yields the 2π self rotations of the anyons themselves, including the one that represents the total charge c. Therefore,

i2πsc 2 θ e (Rc ) = c = , (2.9) ab i2πsa i2πsb θaθb e e or equivalently,

eiπsc Rc = , (2.10) ab eiπsa eiπsb

As we did for a single particle, we can also reverse the exchanging direction of the R-move, leading us to the opposite chirality, the R⋆-move.

e−iπsc (R⋆)c = . (2.11) ab e−iπsa e−iπsb

In practice we would like to discuss speciﬁcally about the diﬀerent components of the R-matrix. For example, it is clear that R-move acting on a single world line is trivial(the fusion channel of 1 0 = 1): × iπs1 1 1 e R10 = R10 = = 1. (2.12) eiπs1 eiπs0

The non-trivial part comes when both the fused anyons are in the state 1 : | i iπs 0 e 0 −i2πs1 1 R = iπs iπs = e = 11 e 1 e 1 θ1 (2.13)  iπs  1 e 1 −iπs1  R11 = eiπs1 eiπs1 = e . For convenience in the later chapters we deﬁne the 2 2 matrix R to represent the core of the × R-move as R0 0 e−i2πs1 0 R 11 = . (2.14) ≡ 0 R1 0 e−iπs1 11 So far we have not speciﬁed the R and F matrices for Fibonacci anyons. It turns out that the fusion rules place strong self consistency conditions on the possible values of these matrices. In the next section we focus on these conditions and use them to “solve” for R and F for Fibonacci anyons.

20 i2π ⋅sa a e

Figure 2.5: The topological spin of a Fibonacci anyon a, which is unique to the 2+1 dimensional phases of matter. Rotating the quasi-particle in 2π is equivalent to a full twist on its tube version, which results in an overall phase θa.

2.1.2 Pentagon and Hexagon Equations

The braiding property of Fibonacci anyons are determined by the F -move and R-move introduced above. One can essentially braid Fibonacci anyons by using these two basic moves to construct quantum gates. However, the unitary matrix associated with F and R-moves are not chosen arbitrarily. In fact, they (as well as the case with any other anyon categories) will have to satisfy some certain self consistency conditions. These conditions, which are build up by examining some equivalent ways of fusing and braiding anyons, will finally boil down to two equations: the pentagon equation and the hexagon equation. Consider a group of four Fibonacci anyons a,b,c,d placed from left to right in order, without { } switching any two of them there shall be five different ways in total to fuse them, which means five different fusing patterns there (see Fig.2.6). Here the statement is made as follows, that these five patterns have the topology that they can be connected with each other solely using the F -moves: we can start with choosing any one of the five fusing patterns as the initial point, and choose another one of the five as the final pattern. Then there are always two distinct “path”(each consisted of a sequence of F -moves), that both connect between this initial and final pattern. For instance we choose the starting point to be a,b,c,d fused together from left to right (the { } pattern furthest to the left in Fig.2.6). And the ending point is where the four are fused in the opposite direction(the pattern furthest to the right in which c,d are fused first, then fuse with { } b , then a ). Now one can first do an F -move so that the pairs a,b and c,d are each fused { } { } { } { } simultaneously(to do so we treat a,b as a single anyon and then do a fusion base change together

21 a b c d e f’ F g F

a b c d a b c d e f e’ f’ g g F F F

a b c d a b c d e’’ f e’ e’’ g g

Figure 2.6: The pentagon equation which tells the story of two equivalent path of basis changing. Starting from the basis to the very left, one can end at the basis to the very right either through two F -moves(the upper path), of three F -moves(the lower path), putting together to be a pentagon diagram. The diagram is also isotropic: meaning that every basis can be the starting point or the end point. There are always two equivalent path connecting in between — one only needs to change the direction of the F -moves accordingly.

with the rest of the particles), as shown in the upper side of Fig.2.6. Then to reach the pattern on the right again another F -move is performed on anyons a , b and f ′ (anyons c,d as a { } { } { } { } whole). On the other hand, again starting with the same initial fusion pattern one can follow the lower side of the Fig.2.6 and choose to do a base change(F -move) on anyons a,b,c so b,c can { } { } be fused first. A similar trick on a , e′′ (fused b and c ) and d then follows. Last but not { } { } { } { } { } least, acting an F -move onto anyons b,c,d will transform the pattern to the final one. { } The identity implied in the above process is called the pentagon equation, literally because of the overall five F -moves contained in the diagram of Fig.2.6. It is one of the two fundamental equations that defines any anyon category, called the Moore-Seiberg polynomial equations [22]. The other one is the hexagon equation which will be introduced later. Once these conditions are satisfied the fact that resulting anyon theory is consisent for any number of anyons follows from the MacLane coherence theorem [23]. In the case of the pentagon equation, it is the simplest identity of any of those that consists of only the F -moves. On the other side, it is also the only irreducible equation made of only the F -moves. That mean there will be no more similar but independent equations for the F -moves. A fusion tree version of the pentagon equation is also drawn in Fig.2.7

22 a b c d

e f’ g F a b c d F a b c d

e f’ f e’ g g F F

a b c d a b c d

e’’ e’’ F f e’ g g

Figure 2.7: The world line version of the pentagon equation. Time ﬂows from top to bottom. Each line segment represents an anyon before it merges with another one to create a new anyon that has their total charge.

for the purpose of better spotting out which F -tensor is used in each step. The pentagon equation can in some cases be used to uniquely determine the mathematical form of the F -tensor [14]. It is shown below that this is in fact the case for Fibonacci anyons. Again, looking at Fig.2.7, let us first mark the state of the basis choice furthest to the left of the figure as α , which is the starting point of our basis evolution. The state of the basis furthest to the right | i is then called β where the two paths converges. If we choose to represent the F -moves using the | i F -tensor defined previously, then the underlying equations that is implied by the upper part of the above diagrams now can be written down as

cef bae ′ ′ α = Fgdf ′ Fgf e β . (2.15) | i ′ ′ | i Xe f The lower part of the diagrams is a three F -move chain which gives

′′ bae e af cbe′′ ′′ ′ ′ α = Ffce Fgde′ Fe df β . (2.16) | i ′′ ′ ′ | i eXe f

23 a d b a d b a d a d abe e F abe e’ e Fcde′ e’ cde′ b c c c b c

Figure 2.8: A deformed version of the F -move equation, which better demonstrates the symmetries underneath its structure. The dashed red lines mark the symmetric lines of the equation.

Putting both of this two equations together we will obtain

′′ cef bae bae e af cbe′′ ′ ′ ′′ ′ ′ Fgdf ′ Fgf e = Ffce Fgde′ Fe df . (2.17) ′′ Xe Daunting as it seems to be at ﬁrst, actually solving this pentagon equation does not require much analytical work. First of all, the F -tensor is proven to be symmetric for certain exchanges of the indices among the control particles. Such a conclusion is partially apparent by just looking at the symmetry of the fusion pattern itself, see Fig. 2.8. For instance, exchanging the control particle pair a,b and c,d will obviously yield the same equation due to the mirror symmetry { } { } demonstrated by the topology of the fusion diagram, so will the pair a,d and b,c . Thus we { } { } have the following identities: abe cde bae Fcde′ = Fabe′ = Fdce′ (2.18)

In what follows we will utilize these symmetries to simplify the process of solving the F -tensor. As we move on to find the exact form of the F -tensor in fact, most of the components of the pentagon equation are already fixed by the fusion rules. Fig. 2.9 shows the F -move transformations acting on all various of fusion diagrams that are enumerated by the different choices of the four control particles a,b,c,d . Notice that only those diagrams that satisfy the fusion rules are listed { } here, since by definition the F -tensor is assumed to vanish when acting on states that don’t satisfy the fusion rules (of course, such states are not part of the physical Hilbert space of the anyon theory). In Fig. 2.9(a)&(b) where at least one of the control particles is in the state 0, the result of the F -move is trivial: the fusion rules have already predicted the state of the target particle e′ { } according to the four control particles. Mathematically, the F -tensor components in such cases can

24 a) b) c)

b a d b a d b a d b a d b a d b a d b a d φ −1 φ − 21 e e’ e e’ e e’ + e’ c c c c c c c

b a d b a d b a d b a d b a d b a d b a d b a d b a d φ − 21 φ −1 e e’ e e’ e e’ e e’ - e’ c c c c c c c c c

b a d b a d b a d b a d b a d b a d

e e’ e e’ e e’ c c c c c c

b a d b a d b a d b a d b a d b a d = 0 = 1

e e’ e e’ e e’ c c c c c c

Figure 2.9: The F -move for Fibonacci anyons in its every component. For each fusion pattern(tree diagram) the black thick line represents a particle of charge 1, while the light thin line marks a charge 0 anyon. (a)&(b) are the cases where the target particle e is solely determined by the | i fusion rule Eq. 2.1. (c) shows the only two non-trivial F -moves here, which needs to be solved in further discussions.

be written down as

00e e 0 F00e′ = δ0δe′ 11e 00e e 1 F00e′ = F11e′ = δ0δe′ 10e 01e e 0 F01e′ = F10e′ = δ1δe′ 01e 10e e 1 F01e′ = F10e′ = δ1δe′ ; (2.19)

11e 10e 01e 11e e 1 e F ′ = F ′ = F ′ = F ′ = δ1δ ′ (F0) ′ . (2.20) 10e 11e 11e 01e e ≡ e The only non-trivial part of the F -tensor comes when all of the four control particles have charge 1. For this particular case we would denote the F -tensor component as an F -matrix:

11e e F ′ (F1) ′ , 11e ≡ e F F1. (2.21) ≡ To solve for this F -matrix we need to look back into the pentagon equation for any clues that may be helpful. By setting all the particles that only play as “control particles” role (i.e. particles a,b,c,d,g ) in all the F -moves to the charge 1, we will have the equation component that actually { }

25 R

a b c b c a e’ d e’ d F F a e’’

a b c b c a e d e’’ d

a b a c R R F

b a c b a c e d e’’ d

Figure 2.10: The hexagon equation which involves both the F - and R-moves. Again it can be seen as composed by two identical paths which can bring one of the six fusion patterns to another(one needs to pay attention to the directions of the moves since R becomes R⋆ when reversed). The diagram contains 3 F -moves and 3 R-moves that interlace with one another, making a hexagonal shape. Notice that in the representation where the equation is FRF = RFR, the R-move always exchange the anyons in a same direction.

deﬁnes the F -matrix. While all the other combinations turn out to be trivial. We will not list the detailed proof here but only give the result of the search. From Eqn. 2.17 now we have

′′ 1ef 11e 11e e 1f 11e′′ ′ ′ ′′ ′ ′ F11f ′ F1f e = Ff1e F11e′ Fe 1f . (2.22) ′′ Xe According to the deﬁnitions of Eqn. 2.20 and Eqn. 2.21 we can rewrite the above one as

f e e f e′′ ′ ′ ′′ ′ ′ ′ (Fe)f Ff e′ = (Ff )e′′ (Fe )e (Fe )f . (2.23) e′′ X Solving the above equation we will obtain the exact form of the F -matrix:

1 1 φ− eiθφ− 2 1 F = − − −1 (2.24) e iθφ 2 φ ! − where θ is an arbitrary phase that can be normally set to 0 for convenience, and φ is again the golden mean. Now if we allow neighbouring anyons to swap positions when manipulating the fusion diagram, there will be another self-consistency equation coming out, called the hexagon equation. This is the second of the Moore-Seiberg polynomial equations [22]. In this new equation both F -moves and R-moves are involved. It is therefore independent of the pentagon equation. Again, the hexagon

26 equation is the fundamental relationship for the two basic moves: that any two identical path which are both non-trivially made of the F &R-moves and connects two fusion diagrams, shall eventually reduce to the hexagon equation. Consider a group of only three Fibonacci anyons a,b,c fusing in to an anyon with total charge { } d , initially the anyons a,b are fused first, as shown in Fig. 2.10. Starting from the pattern { } { } furthest to the left we are able to find out two paths which lead it to the same destination, where anyons a,b,c are fused from right to left, the exact opposite direction, see the pattern furthest to { } the right. The first path consists of a sequence of F R F moves: an F -move at the beginning sets − − up a new base where anyons b,c can be fused first. Then we do a rotation, which is essentially { } the R-move, on the positions of anyon a and e′′ (the fused b,c ). In the end another F -move { } { } { } will take the anyon c to be fused with a . { } { } The second path also takes three steps but instead of two F -moves and one R-move here we have one F -move sandwiched between two R-moves. In this path we first do an R-move which swaps the positions of anyons a,b , followed by an F -move that changes the pattern to where { } anyons a,c instead of b,a are now fused. Next we only need to exchange between anyons c { } { } { } and a (an R-move again) before we end up with the same fusion pattern as we have in the first { } path. To be strict on what we have derived here we want to emphasize a small yet crucial detail in the above hexagon diagram. Looking at the directions of how two anyons (including those fused ones) are swapped, It is important for us to always stick to the same choice, clockwise or counter- clockwise. As emphasized during the introduction of the R-move, we need to carefully navigate ourselves among all the intricate braiding patterns by keeping track on the chiralities of every R-moves. For the hexagon diagram, inside the equation we either only use the R-moves alone or replace all of them with R⋆-moves. Both cases reflect the same law. For convenience the default is to exchange anyon pairs counter-clockwise, hence the R-move. Going back to the graphic representation of the hexagon equation(Fig. 2.11) we now proceed to solve for the R-matrix: the hexagon equation is exactly the ruler equation that caliberates the R-matrix provided that the F -tensor is already calculated using the pentagon equation. In Fig. 2.11 the fusion diagram furthest to the left is now called α . In this diagram anyons a,b are fused | i { } first. While the fusion diagram furthest to the right is marked as β , in which anyons b,c are | i { }

27 a b c R b c a F e’ e’ F d d a b c b c a

e e’’ d d

R b a c F b a c R

e e’’ d d

Figure 2.11: The world line version of the hexagon equation. Time ﬂows from top to bottom. Each line segment represents an anyon before it merges with another one to create a new anyon that has their total charge. To avoid confusion the double arrow that marks the R-move is only labeled at the end of the action.

now fused ﬁrst. Translating directly from the two paths that are described previously we will have the following mathematical equations:

abe d cbe′ α = Fdce′ Rae′ Fdae′′ β (2.25) | i ′ ′′ | i Xe e which describes the top path. The bottom path gives

e abe e′′ α = RabFdce′′ Rac β . (2.26) | i ′′ | i Xe Combining these two we get

e abe e′′ abe d cbe′ RabFdce′′ Rac = Fdce′ Rae′ Fdae′′ . (2.27) ′ Xe Again similar to the pentagon equation, the hexagon equation 2.27 here is in fact a collection of its components: some of them are trivial, while the non-trivial ones can be used to solve the R-matrix. Actually the only non-trivial case is when all the control particles of the involved F - moves( a,b,c,d ) are set to have charge 1: { }

e 11e e′′ 11e 1 11e′ R11F11e′′ R11 = F11e′ R1e′ F11e′′ . (2.28) ′ Xe

28 Using the notations introduced previously for the F and R we have

e e 1 e′ (R)e (F )e′′ (R)e′′ = (F )e′ R1e′ (F )e′′ e′ X e 0 e 1 =(F )0 (F )e′′ +(F )1 (R)1 (F )e′′ . (2.29)

At this point we would exploit the fact that the F -matrix is already nailed down by the pentagon equation, which is Eqn. 2.24. Simply substitute it for that in Eqn. 2.29 we obtain the R-matrix as

e4πi/5 0 R = (2.30) 0 e−3πi/5 Note that the reversed version of R: the R⋆-matrix is also one of the two solutions to the hexagon equation. Which means it actually describes the same model. Both of them together represent the two chiralities of the Fibonacci anyon model.

2.1.3 Quantum Computing by Braiding Fibonacci Anyons

One of the reason why we choose Fibonacci anyon as a candidate for topological quantum computation is because an arbitrary quantum gate can be realized purely by braiding these anyons in the 2+1 dimension space-time. Given a set of n labelled Fibonacci anyons that fuse in order with a fusion wavefunction α1 2 ··· . The exchange of the particles inside actually gives a representation | , , ,ni of the braid group Bn. More speciﬁcally, if we denote the action of exchanging the jth and j + 1th neighbouring particles as σj, then arbitrary braid B is a composition of these nearest-neighbour braids, hence we would call the σj’s as the “elementary braids” and have

B = σ1, ,σ , ,σ . (2.31) n { ··· j ··· n} One can name a few properties of the braid group, the ﬁrst one is the far commutativity. Apparently two elementary braids commute with each other if they are far enough apart and do not share strands:

i j

σi σj = σj σi = (2.32) · · ··· ··· ··· for i j 2. | − |≥

29 Another important relation between the elementary braids is call the YangBaxter equation in the braid group theory, as shown below

σ σ +1 σ = = σ +1 σ σ +1 (2.33) i · i · i ≡ i · i · i

Now with the two basic actions in the anyon theory presented — the basis change(the F -move) and the fusing position exchange(the R-move), we are actually able to construct the braid group using the braid generators that combines this two moves. The reason is stated as follows: First of all, exchanging two neighbouring anyons that have a deﬁnite total charge is already well known: it is exactly the R-move. For example if the ﬁrst two particle are fused, the exchange between them

σ1. However, to change the positions of two anyons that do not get fused directly in the vector basis α1 2 ··· , we will have to apply the F -move(s) ﬁrst to change the original basis into another | , , ,ni one in which the R-matrix for these two anyons is block diagonal. Therefore, we would envision that all of the braids can be represented by a sequence of F and R’s. For a clear picture we consider the example where there are three anyons( , , ) fusing from left to right( ). Remember that theoretically anyons are indistinguishable particles, we mark them with colors only to track the strands of the braids they form. All possible braids among these three particles are actually generated by two elemental braids: The ﬁrst one is where the world lines of the anyon and are braided, which is σ1. While the other one is done by exchanging and instead, labelled σ2.

One can easily tell that σ1 is actually the R-move upon those two anyons:

σ1 = = = R. (2.34)

30 Again note that the direction of the exchange(clockwise or counter-clockwise) needs to be carefully monitored as they corresponds to diﬀerent topology in braids and diﬀerent R/R⋆.

For σ2, The structure of the braid of anyon and is not so straightforward. We are not allowed to apply the R-move directly upon them since they do not have a definite total charge here. Yet we can first do an F -move to change the fusion basis into where and are fused. A well defined R-move then carries the braiding. In the end we need to act a reversed F -move(which is essentially the F -move since for the F -matrix, (F )−1 = (F )) to bring back the original basis. Therefore,

σ2 =

F R F (2.35) −→ −→ −→

which implies

σ2 = F R F. (2.36) · · An interesting application of the above results is that we can verify the hexagon equation by rewriting it as braid patterns and examining if both sides have the same topology. In fact, the upper part of Fig. 2.10 describes the braid:

R F R

R F R = , (2.37) −→ −→ −→

while the lower part turns out to be

31 F R F

F R F = , (2.38) −→ −→ −→ the same braid as the upper part. In reality when we want to construct arbitrary quantum gates by braiding anyons around, the first thing needs to be done is to encode logical qubits using Fibonacci anyons. In order to form braids there should be at least two anyons presented. However, this is not enough for our needs. Taking the single qubit rotations for instance, braiding two anyons(which is essentially applying R and R−1 over and over again) will clearly not change their total topological charge, hence resulting in only the rotations around the z axis of the Bloch sphere. To perform any single qubit rotations we would need at least three fusing Fibonacci anyons with topological charge 1( ), for which the Hilbert space is three dimensional. Among the three orthogonal basis states we encode the logical qubits 0 and 1 using only two of them, while the third one is treated as the “non-computational” | i | i basis NC . During the real computational process we have to carefully manage the braids so that | i there is no leakage errors caused by the transitions into this state. 0 = , | i 0 1  (2.39)   1 = ; | i 1 1  NC = . (2.40) | i 1 0 It has been proven [24] that the braid group generators for the Fibonacci anyons are “sufficient” enough to generate all the rotations that covers the SU(2). That is, any single qubit rotations can be realized by braiding the quasiparticles inside the logical qubits given enough times of braid operations. In fact, there are numbers of ways to compile arbitrary single qubit gates. For example, one can do a brute force search of the sequence for some short braids. And a clever algorithm which stems from the Solovay-Kitaev theorem would allow for systematic improvement in compiling those gates. More details about such work can be seen in [19, 17]. Fig. 2.12 here gives one of the nice result of finding the braid sequence that realizes a specific single qubit gates, the iX rotation where 0 i iX = . (2.41) i 0

32 ψ iX

0 i    −5    + O )10(  ψ  i 0 

Figure 2.12: Constructing the iX single qubit gate using Solovay and Kitaev’s algorithm. The braids produce an operation that approximately realizes the iX rotation with the errors that are in the order of ǫ = 10−5. The algorithm can achieve arbitrary accuracy with the braid length L ln ǫ c where c 4. ∼| | ∼

Moreover, two-qubit gates can be constructed by braiding the Fibonacci anyons in the logical qubits as well. A much better term here would be “weaving”, which is essentially a special class of braiding where only some of the anyons are moved around, while the space-time world lines of the others remain still in a graphic representation. For example, the controlled-NOT(CNOT) operation can be realized by weaving part of the control qubit (see Fig. 2.13) around the anyons of the target qubit. Again for Fibonacci anyons, the braiding operations are dense enough on the Hilbert space of the two-qubit unitary transformations. That grants us the theoretical basis of performing arbitrary quantum gates purely by braiding, since single qubit rotations plus the CNOT gate will give us a universal set of quantum gates. We can do similar things with even larger numbers of qubits. Then we are safe to say that the Fibonacci anyons are a perfect candidate for realizing topological quantum computation, provided that we have the way to practically get these anyons. The question is: where are they?

2.2 Fibonacci Levin-Wen Models

To answer the question of the last section, one thought is that we may seek a physical system in which non-Abelian anyons may arise naturally. There are proposals for “engineering” them and

33 X

Figure 2.13: A compilation of the Controlled-NOT gate into braids. Only two of the anyons of the control logical qubit(the top one) are “weaved” around the anyons of the controlled logical qubit. As the total charge of this two determines the state of the control qubit. Given different total charge of the two qubits, 0 or 1, the braids can approximate the CNOT gate to a distance ǫ 1.8 10−3 and 1.2 10−3, respectively. ≈ × × manipulating them to carry out braiding. In fact, with the bizarre topological phases of matter discovered, the work carried out in the fractional quantum Hall system has already shed a silver lining on the topological quantum computation field. However, there are still an enormous amount of issues needs to be dealt with. An important one is that of errors due to stray non-Abelian excitations, including those inevitably produced by thermal fluctuations. Recent work has begun to addresses the important question of how such non-Abelian errors would be decoded and corrected, pointing to a number of complexities which arise for non-Abelian errors which are not present for Abelian errors [25, 26]. Another route to computing with non-Abelian anyons is to follow the same approach as in the Kitaev surface code. In this approach, rather than physically realizing a Hamiltonian whose ground state has topological order and non-Abelian excitations with the desired properties, one imagines realizing these states using a “conventional” quantum computer in essentially the same way that the ground state of the Kitaev toric code model is realized in the Kitaev surface code. This is done by repeatedly measuring the local terms in the Hamiltonian (which, for the Hamiltonians of interest, are all mutually commuting) to detect deviations from the ground state which are viewed as errors, and then decoding and correcting these errors. An outline for realizing such non-Abelian surface codes based on the Levin-Wen models, a class of model Hamiltonians which generalize the Kitaev toric code and can have non-Abelian excitations, was presented by König, Kuperberg, and Reichardt.[15] The specific case they focused on was the

34 Qv

Figure 2.14: An array of Fibonacci anyons that are suitable for the Levin-Wen models. A hexagonal lattice is formed by placing edges on each of the particles. We deﬁne the Qv operator on the three qubits that converge at the same vertex v. And the Bp operator on the plaquette p is so constructed that all the 12 qubits associated to it, including the 6 qubits on the perimeter and the 6 outside, are involved.

Levin-Wen model for so-called doubled Fibonacci anyons. One advantage to using this model is that Fibonacci anyons are universal for quantum computation, meaning it is possible to carry out all quantum operations purely by braiding them, without the need for non-topological operations and magic state distillation which are both required when using the Abelian Kitaev surface code, as well as topological quantum computation using Ising anyons [16]. So far we have been spending many pages on describing the basic concepts and the braiding rules of the Fibonacci anyons, as they are crucial for motivating our study of the Fibonacci Levin-Wen models. Now we would briefly introduce the concept of the Levin-Wen models. The Levin-Wen models are a class of lattice Hamiltonians defined on two-dimensional trivalent lattices in which s-level quantum systems are assigned to each lattice edge (in some situations they are also referred to as “strings”). These models can be constructed given any consistent anyon theory. As described above, such an anyon theory is defined by a set of s distinct particle labels, or topological “charges,” fusion rules for these charges, and an F matrix, which is related to basis changes for these particles and which must satisfy the pentagon equation. The Levin-Wen

35 a) b) c)

= 0 = 1

Figure 2.15: (a) A vertex v in a hexagonal lattice, it has three associated qubits that determines the value of Qv. (b) Lattice conﬁgurations that will have Qv = 1. (c)Lattice conﬁgurations that violates the vertex constraints, hence have Qv = 0.

models are time reversal invariant and have anyon excitations corresponding to both possible anyon chiralities [27]. Here we focus on the Fibonacci Levin-Wen model, i.e. the Levin-Wen model constructed using the theory of Fibonacci anyons reviewed above. For this model the particle labels can be either 0 (trivial) or 1. The model is realized on a trivalent lattices with qubits (i.e. two-level systems because there are only two particle types) assigned to each edge. Because particles with charge 1 are their own antiparticles the corresponding Levin-Wen model is unoriented. As for all Levin-Wen models, the Hamiltonians are written as a sum of mutually commuting vertex (Qv) and plaquette

(Bp) operators,

H = Qv Bp (2.42) − v − p X X where the sum over v / p is over all vertices / plaquettes of the lattice. The vertex operator is simply the operator associated to each vertex. Suppose that for a vertex v in which the states of the three edges connected to that vertex are i,j, and k, we denote the state of this vertex as ijk . The vertex operator Qv is so deﬁned that it becomes diagonal in the | i standard qubit basis. When Qv acts on a vertex state ijk the result is simply | i

Qv ijk = δ ijk (2.43) | i ijk| i

36 a) b)

= 0 = 1

Figure 2.16: (a) An example of the Qv ground state, in which the typical conﬁgurations are “loops that allows branches”. (b) Excited states are created by breaking the branching rules. As indicated by the red mark.

where the tensor δijk encodes the fusion rules for the relevant anyon model. The only non-Abelian fusion rules for which a consistent anyon theory for particles with only one nontrivial charge are the Fibonacci fusion rules(Eqn. 2.1) The corresponding δ tensor is

1 if ijk = 000, 011, 101, 110, 111 δ = (2.44) ijk 0 otherwise.

In that sense, the Qv operator is in fact a projection operator that has only two eigenvalues: 0 and

1. The ground state of the Qv operator is then deﬁned if δijk = 1 (or more often referred to Qv = 1 in the later Chapters), while the Qv = 0 state is considered as the excited state. Since we will be using the ground state of the Fibonacci Levin-Wen model to perform quantum computations, the

Qv = 0 excited states are treated as “vertex errors”. Because of the two possible fusion results when combining two particles with charge 1, the states which satisfy the vertex constraint Qv = 1 on each vertex consist of configurations in which edges in the state 1 form branching loops. A typical such configuration is also shown in Fig. 2.16. | i The plaquette operator is in general significantly more complex and has the form,

1 0 1 Bp = B + dB , (2.45) 1+ d2 p p 37 where

j1 jn m2 mn p s Bp j2 + m3 m5 j3 j4

′ ′ j1 j m2 n mn ′ ′ ′ ′ p s,j1j2···jn−1jn j′ = Bp,j1j2···jn−1jn (m1m2 mn−1mn) 2 , (2.46) ··· + j1,··· ,jn X m3 ′ ′ m5 j3 j4 m4 and

′ ′ ′ ′ s,j1j2···jn−1jn B (m1m2 m −1m ) (2.47) p,j1j2···jn−1jn ··· n n m1jnj1 m2j1j2 mn−1jn−2jn−1 mnjn−1jn = F ′ ′ F ′ ′ F ′ ′ F ′ ′ . sj1jn sj2j1 ··· sjn−1jn−2 sjnjn−1

ijk We have been quite familiar with the F -tensor Flmn from the description in the last chapter. It satisﬁes certain self-consistency conditions (essentially the pentagon equation), and d is the anyon quantum dimension. As for Fibonacci anyons, the quantum dimension is simply the golden mean: φ, which is proven in section 2.1. Like the vertex operators, the plaquette operators are also projection operators that have eigenvalues either 0 or 1. In the quantum computation scheme which we will be discussing later on, the ground state is deﬁned when the plaquettes have Bp = 1. And the excited states, or called

“plaquette errors”, are those states where Bp = 0. Again, given the branching of the fusion rules it is not possible to write the Levin-Wen Hamilto- nian down in a form in which the plaquette operator acts on not just the six edges of each hexagonal plaquette, but rather it must also act on the six edges that connect to the vertices of the plaquette. This is the generic case for non-Abelian Levin-Wen models, and on the hexagonal lattice this means the plaquette operator acts on 12 qubits and is thus highly nontrivial. A useful way to think about the ground state of the Levin-Wen model is to envision a ”fattened” lattice (see [27]) as shown in Fig. 2.17. The shaded circles at the center of each plaquette can be

38 Figure 2.17: The fattened lattice picture of the Levin-Wen ground state. In this picture, we set the rule that no strings are allowed to pass through the center of the plaquette, as if there is a “puncture”(shaded circles) there for each plaquette. Then, acting the projection operator Bp onto the lattice is equivalent to adding “vacuum strings”(dashed circles) around those punctures.

thought of as holes through which strings are not allowed to pass. Each of these holes is encircled by a “vacuum” string 1 = + φ (2.48) 2 1+ φ where = 0 which is often omitted asp blank( = 0 ) and = 1 . | i | i | i By applying the diagrammatic rules:

= (2.49)

= φ (2.50)

1 1 = φ 2 φ− 2 (2.51) −

One can always reduce the graph to the point that there is a single string, either in the state 0 or state 1, for each edge, and the states of this string can be represented by the qubits on each edge. Note that in Eq. 2.48 the loops are not allowed to contract to a scalar using the second rule(Eq. 2.50) because of the puncture the vacuum loops has inside. The third rule(Eq. 2.51) is

39 actually obtained by combining the two non-trivial F -moves in Fig. 2.9. A few useful derivations of the above rules are also listed here. First of all,

1 1 = φ 2 φ− 2 = 0 (2.52) − which implies = 0. (2.53)

Another case when “bubbles” can be removed is

1 = φ 2 . (2.54)

More generally, we can rewrite Eq. 2.53 and Eq. 2.54 as

i i dmdn m n = δij (2.55) di j r where the quantum dimension di is deﬁned in Eq. 2.5. Now we are able to interpret the action of the plaquette operators as adding loops into the lattice, and simplify it using the above diagrammatic rules. Taking the hexagon plaquette for instance, to project one plaquette onto its Bp ground state is to add a vacuum loop around its shaded center.

m1 m1

j1 j6 j1 j6 m2 m6 m2 m6 p Bp j2 j5 = j2 j5 (2.56) + + m3 m5 m3 m5 j3 j4 j3 j4

m4 m4

s More speciﬁcally, acting the Bp component of the plaquette operator is equivalent to encircle a type s string around the center:

m1 m1

j1 j6 j1 j6 m2 m6 m2 m6 p s Bp j2 j5 = j2 j5 (2.57) + + m3 m5 m3 s m5 j3 j4 j3 j4

m4 m4

40 To further reduce the diagram we need to apply multiple F -moves to the edges (remember that the strings are not allowed to pass through the shaded area, one would imagine there is a “p uncture” attached to every plaquette, otherwise the s string loop will easily contract into a scalar φ according to rule 2.50). We are able to do this by seeing there are type 0 edges connecting the two plaquettes.

m1 m1

j1 j6 j1 j6 m2 m6 m2 m6 0 0 0 j2 j5 j2 j5 = 0 + s 0 + 0 m3 s m5 m3 m5 j3 j4 j3 j4

m4 m4

′ j1 j6 ′ m2 j1 j6 m6 j1 j6 j2 s j5 j1j10 j2j20 j3j30 j4j40 j5j50 j6j60 ′ s s ′ = F ′ F ′ F ′ F ′ F ′ F ′ j2 j5 (2.58) ssj1 ssj2 ssj3 ssj4 ssj5 ssj6 j2 s s j5 j1,··· ,j6 s + j3 j4 X m3 ′ ′ m5 j3 j4 j3 j4 m4

We continue applying F -moves onto the “corners” of the plaquette so the “s arc” can be moved to each edge:

j1 j6 m2 m6 p s Bp j2 j5 + m3 m5 j3 j4

j1j10 j2j20 j3j30 j4j40 j5j50 j6j60 m1 = F ′ F ′ F ′ F ′ F ′ F ′ ssj1 ssj2 ssj3 ssj4 ssj5 ssj6 · ′ ′ j1,··· ,j6 j1 j1 j6 j6 m2 ′ ′ m6 X j1 j6 j′ s s j′ m1j6j1 m2j1j2 m3j2j3 m4j3j4 m5j4j5 m6j5j6 2 5 F ′ ′ F ′ ′ F ′ ′ F ′ ′ F ′ ′ F ′ ′ j2 s s j5 (2.59) sj1j6 sj2j1 sj3j2 sj4j3 sj5j4 sj6j5 ′ ′ j2 j5 ′ s s ′ + m3 j3 j4 m5 j3 ′ ′ j4 j3 j4 m4

Note that we can actually construct an F -move diagrammatic equation which solves the F -

j1j10 tensor of the form F ′ , that is ssj1

41 j1 s j1 j1j10 ′ F ′ = j1 s (2.60) ssj1 j1 using the basic diagrammatic rules(e.g. Eq. 2.55) both sides can be reduced to

j1 ′ j1 j1j10 dj1 ds F ′ ds = . (2.61) ssj1 · s dj1

Therefore we have ′ j1j10 dj1 F ′ = . (2.62) ssj1 sdsdj1 Now note that we can also eliminate the “bubbles” inside the plaquette using Eq. 2.55 as well. For example, ′ ′ j1 j1 dj1 ds j1 s = (2.63) d ′ ′ s j1 j1 which cancels exactly with Eq. 2.62. Thus eventually:

j1 j6 m2 m6 p s Bp j2 j5 + m3 m5 j3 j4

j′ j′ m2 1 6 m6

m1j6j1 m2j1j2 m3j2j3 m4j3j4 m5j4j5 m6j5j6 ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ j j = Fsj j Fsj j Fsj j Fsj j Fsj j Fsj j 2 5 (2.64) 1 6 2 1 3 2 4 3 5 4 6 5 + j1,··· ,j6 X m3 ′ ′ m5 j3 j4 m4

42 Figure 2.18: Encoding a logical qubit by cutting a defect(a “hole”) from the lattice. In this case one stops to measure the Bp values of the 10 plaquettes inside the “hole”, as well as the Qv values of the vertices that have been wiped out. In that sense we obtain a degree of freedom from neglecting these operators. Like the defects in Kitaev’s surface code, this degree of freedom is protected globally. And the logical qubit will have the behaviour of a Fibonacci anyon.

In short, when applying the Bp operator onto a plaquette p we have

m1 m1

j1 j6 j1 j6 m2 m6 m2 m6 p Bp j2 j5 = j2 j5 + + m3 m5 m3 m5 j3 j4 j3 j4

m4 m4

j′ j′ m2 1 6 m6

m1j6j1 m2j1j2 m3j2j3 m4j3j4 m5j4j5 m6j5j6 ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ j j = Fsj j Fsj j Fsj j Fsj j Fsj j Fsj j 2 5 (2.65) 1 6 2 1 3 2 4 3 5 4 6 5 + s,j1,··· ,j6 X m3 ′ ′ m5 j3 j4 m4

Now we shall be able to encode logical qubits using the excited states of the Fibonacci Levin- Wen model, given that the ground state is deﬁned by the Levin-Wen Hamiltonian (Eq. 2.42). To obtain the excited states we would like to borrow the idea from Kitaev’s toric code: to introduce extra degrees of freedom by creating defects on the lattice where system is located or more visually,

43 “cutting holes” on it so that the local operators(the vertex and plaquette operators) are omitted for measurement. See Fig. 2.18, in which the 10 hexagonal plaquette is cut oﬀ from the Levin-Wem ground state, meaning that these plaquettes can stay in any allowed states while the rest of the plaquettes around are still required to be in their ground states, i.e. they will have the vacuum loop added in the fattened lattice picture. Same is true vertex-operator-wise. Note that the physical qubits are not removed — they are just ignored for computational purposes. These defects with the extra degrees of freedom are suggested to behave like Fibonacci anyons: they can be fused, moved throughout the two dimensional lattice space, or even braided to perform topological quantum computations. The quantum codes that use the above constructions and have features like this are called the Fibonacci codes. A much more detailed illustration can be found in K¨onig, Kuperberg, and Reichardt’s paper “Quantum computation with Turaev–Viro codes”[15].

2.3 Summary

To summarize, we have reviewed the theory of Fibonacci anyons by ﬁrst introducing their fusion rules and the properties of the Hilbert space formed by fusing them. Two anyon actions, F and R, which play important roles in the later discussions, are also included. We then illustrated two fundamental self consistency equations of Fibonacci anyons, the pentagon and hexagon equations. By solving these equations we obtains the exact form of F and R actions. With this background knowledge understood, we then give a few examples of how to carry out quantum computations using the quantum gates constructed by braiding these Fibonacci anyons. In the second part of this Chapter, we discussed in detail about a type of mathematical lattice model called Fibonacci Levin-Wen models, the ground states of which can be used to perform topological quantum computation. These models can be written as the sum of mutually commuting projection operators, Qv and Bp, associated with the vertices and plaquettes of the lattice. Finally, following the original Levin-Wen paper [27] we reviewed how these operators can be most naturally represented in the so-called “fattened” lattice picture.

44 CHAPTER 3

NON-ABELIAN QUANTUM ERROR CORRECTION

In this Chapter we describe how the ground state of the Fibonacci Levin-Wen model can be realized using a conventional quantum computer. After reviewing some previous work on this problem, we present the main results of this Dissertation: a set of new quantum circuits which can be used to measure the vertex and plaquette operators of the model (Qv and Bp, see Chapter 2) [28].

When Qv and Bp are measured using our new circuits, one either ﬁnds that these operators have the values required in the ground state of the Fibonacci Levin-Wen model (Qv = Bp = 1) or they do not (Qv = 0 or Bp = 0). In the latter case, the vertices or plaquettes which are not in the Levin- Wen ground state can be viewed as errors. Because these errors themselves satisfy non-Abelian statistics they can be considered “non-Abelian” errors. In addition to our new quantum circuits for measuring Qv and Bp we also give procedures for moving any detected vertex or plaquette errors through the lattice, as well as procedures for fusing these errors in order to attempt to restore the state of the computer to the error-free Levin-Wen ground state. We illustrate our procedures by going step by step through an explicit sequence in which i) measurements, ii) error transportation, and iii) error fusion, are carried out for a set of errors. The title of this Dissertation, ”non-Abelian quantum error correction,” refers precisely to this set of procedures.

3.1 Previous Work on Measuring Qv and Bp

Building a topological quantum computer would require a state of matter that has a non-trivial topological statistics, as discussed in the previous chapters. Fortunately we do not have to turn to Nature itself to ﬁnd a natural state that has the desired properties. An alternative path is to “simulate” a quantum system that has topological excitations. This led us to the recently found lattice model called the “Fibonacci Levin-Wen model” [15] (see Chapter 2). The Hamiltonian for this model is constructed so that it will have quasiparticle excitations that behaves like Fibonacci anyons, a type of anyon category that has the simplest non-Abelian statistics. In addition, these

45 anyons are famously known for having sufficiently rich braiding properties that they can generate a universal set of quantum gates. Therefore, it is possible to use the ground states of the Fibonacci Levin-Wen model as quantum codes to develop universal quantum computations. Logical qubits, encoded by Fibonacci anyons are then created from cutting defects on the code. But first of all, it is necessary to constantly measure the lattice Hamiltonian to make sure that the code stays in its ground state, or “is stabilized”. The Fibonacci Levin-Wen Hamiltonian is defined by two types of projection operators (our “non-Abelian version” of stabilizers in Kitaev surface code): the vertex operators and the plaquette operators. Each of these stabilizers acts on a collection of qubits associated with the corresponding vertex or plaquette on the two-dimensional trivalent lattice (Sec. 2.2). By measuring these operators we will be able to possibly locate the errors that emerge inside the code, and correct them before the computational process can proceed. In the quantum information literature, these measurements are known as syndrome measurements, since they can be used to diagnose whether or not an error has occurred. In this chapter, we will present techniques that perform syndrome measurement for both vertex and plaquette errors in the Fibonacci Levin-Wen model. We will refer to the resulting quantum error correcting code as the Fibonacci code, which we consider a non-Abelian generalization of Kitaev’s Abelian surface code. In the end, the ultimate purpose of constructing these quantum circuits for the non-Abelian surface code is to compare them with the Kitaev surface code and see how much more complex they will be and what kind of cost we will have to pay to realize them. Therefore, at the end of each section we will also discuss the complexity of every circuit we obtained for the Fibonacci code in terms of a set of elementary gates (Toffoli gates, CNOT gates and single-qubit rotation gates). This will in turn give us a relatively clearer sense of how advantageous it is to use the Fibonacci code, given that the Kitaev surface code is already proven to be a mature scheme of building a fault tolerant quantum computer.

To provide context for our new circuits for measuring Qv and Bp we ﬁrst review the preliminary related work of [29]. Unlike our new circuits, these preliminary circuits can only be used to measure errors — they cannot be used to correct them. This preliminary work is presented here to provide needed context to understand the new results of this Dissertation, which begin in Sec. 3.2.

46 1 1 2 3 v

2 3 0 X X X X

1−Qv

Figure 3.1: Quantum circuit used to measure the value of the Qv operator deﬁned on the vertex shown left. To perform the measurement one has to add an extra qubit(orange line, initialized in the state 0 ) so that the result can be read out. | i

3.1.1 Vertex Errors

We begin with the simplest syndrome measurement, the stabilizer Qv associated with vertex v. As we will see below, we must consider the vertex error problem before the plaquette error problem because the plaquette operator Bp is well-deﬁned only if Qv = 1 on all vertices (a condition we will refer to as the vertex constraint).

In practice, to detect vertex errors we will have to know the value of Qv, which requires us to develop the technique of measuring the vertex projecting operator for any given vertex v. The result of the measurement determines the error syndrome: if Qv = 1 then we know that the vertex sits in the ground state, however if we find out that Qv = 0 then according to the definition, there is an error on that vertex. To have a clear picture of the vertex errors one may look back at the Figure. 2.15 where all the possible edge configurations for one vertex is listed. The vertex states where the vertex constraints are violated, i.e. Qv = 1, are those states that have only one of the three edges stays in the state 1 , while the other two are in the states 0 . In most of the pictures | i | i of this Dissertation we often marked out the edges in the state 1 using heavy lines, then those | i vertices that violates the vertex constraints, i.e. the vertices that have vertex errors, will appear to have one single heavy edge (or string) ending at there. In that sense, one may simply refer to these errors as “string end”. Before we start, it is worth mentioning that measuring the non-Abelian vertex error is expected to be not significantly more complex than the Abelian Kitaev code where according to its definition of the vertex operator there are only even numbers of state- 1 strings coming in and out a | i

47 vertex. In fact, with a little modification to the Abelian case, the quantum circuit which does a Qv measurement on our non-Abelian Fibonacci code can be constructed as what is shown in Fig. 3.1 [29]. Note that the only difference in this quantum circuit construction compared to the Kitaev surface code is the four-qubit Toffoli gate (see more details about this type of gates in Chapter 1). This difference is closely related to the non-Abelian feature that the Fibonacci code possesses, which will be more commonly seen in what follows. More specifically, this additional Toffoli gate reflects the existence of the possible fusion channel represented by 1 1 = 1. × Quantifying the complexity of a quantum circuit is of great interest when building the error correction protocols. It will to some extent tell us how hard it will be to be practically realized in experiments, despite the fact that it also inevitably depends on the particular environment that the experiments are conducted. Recent discoveries have found that three-qubit Toffoli gate can be carried out experimentally by using superconducting qubits [30, 31, 32] and trapped ions [33]. Therefore it is reasonable for us to use the three-qubit Toffoli gate as one of the primary gates, together with the single-qubit rotation gate, to quantify the complexity of the quantum circuit that is obtained in Fig. 3.1. To do so we first need to be able to reduce the four-qubit Toffoli to these primary gates. In fact it has been proven that a n-qubit Toffoli gate can be decomposed into 4n 12 three-qubit Toffoli gates if n 3 extra qubits are provided [34]. Fortunately these extra − − qubits do not get involved in the calculation and should not be changed after the measurement is done. Therefore in practice we can use those nearby qubits that are not currently taken for operations as the extra qubits. Now to proceed we can count the total number of three-qubit

Toffoli gates, CNOT gates and single-qubit rotations for the Qv quantum circuit in Fig. 3.1: the four qubit Toffoli gate can now be substituted by 4 three-qubit Toffoli gates. Therefore the circuit contains totally 4 three-qubit Toffoli gates(we will simply call them Toffoli gates from now on) and

3 CNOT gates. While in the Kitaev surface code, to measure the Qv value of a trivalent vertex there are only 3 CNOT gates been used. It is understandable due to the fact that the Abelian version of the circuit is obtained by remove the four-qubit Toffoli gate. As an alternate choice, we can also see the n-qubit Toffoli gate as primitive gates, which is motivated by a series of proposals for carrying out single-step n-qubit Toffoli-class gates using superconducting qubits [35], trapped irons [36] and neutral atoms interacting with cavity photons

48 [37] (Moreover, it has been observed that these gates can be efficiently achieved when one of the qubits has n available quantum levels [38]). In that sense, the gate count for the Qv quantum circuit is then 1 four-qubit Toffoli gate and 3 CNOT gates. Indeed, it is possible to take one more step and set the primitive gates as only CNOT gates and single-qubit rotations since a n-qubit Toffoli gate can be furthermore decomposed into 2n 2 CNOT gates and 2n single-qubit rotations, − as proven in [34]. Yet what we will show in the next chapters demonstrates that the ability to carry out n-qubit Toffoli gates will benefit the quantum computation significantly when we are using the non-Abelian code. Thus for now, we will stick ourselves using the Toffoli gate basis unless there are future demands.

Generally, the measurement of the Qv is still quite simple even for the Fibonacci code, with only one Toﬀoli gate that needs to be implemented. It is mainly because the Qv operator is simple in its nature for both of our Abelian and non-Abelian codes. In the next section we will see that the same thing does not apply to the plaquette operators. For the Bp operator in the Fibonacci code, a brute force approach will not be realistic for constructing the quantum circuit which measures this operator. We will have to ﬁnd a much clever way to deal with it.

3.1.2 Plaquette Errors Measured by Plaquette Reduction

We now review the earlier procedure for measuring the plaquette operators in the Fibonacci

Levin-Wen model [29]. It is easy to see that a direct measurement of the Bp operator of a 6-side plaquette can be very hard. As pointed out in Sec. 2.2, the action of the Bp operator on a hexagonal plaquette involves 12 qubits, 6 of which are control qubits and the others are controlled ones. It would be quite problematic if we will have to diagnose the plaquette for each possible state of the involved edges. However, it has been noticed that to measure a one-sided plaquette(often referred to as “tadpole” or “loop”) is much less complicated. Since it only has two qubits where the overall Hilbert space is 4-dimensional. For such a simple device one can essentially ﬁnd out the orthogonal basis of Bp projection operator and then measure the one-sided plaquette accordingly. Now the remaining work is to build up a connection between arbitrary n-side plaquette and a one-sided plaquette, that is so simple that measuring the value of Bp will be easy as discussed before. Fortunately, we have learned one important conclusion from [39] that one of the basic moves in the anyon theories, the F -move actually commutes with the plaquette operator, which

49 F

Figure 3.2: An F -move involves ﬁve qubits that are connected in the fusion pattern deﬁned in Chapter 2. One can essentially treat the hexagonal lattice of the Levin-Wen model as a much bigger fusion diagram in which the F -moves can be applied to change its topological structure. The control qubits(black lines) and the target qubit(red line) are marked out.

may enlighten us to find a path that leads us from the very complicated n-side plaquette to the simplest tadpole plaquette. Here we would like to look back onto the F -move that is introduced in the previous chapter. In Chapter 2 we have illustrated the F -move as an outcome of changing the basis of many anyons fusing together. On a second thought, by looking at the right side of Fig. 2.2 we may also consider the world line diagram of the fusion pattern as a lattice structure which has anyons placed on top of each (world) lines. Indeed, the fusion diagram can be seen as a trivalent lattice that shows up in the Levin-Wen model. With that in mind, it is hence possible to use the F -move to change the configuration of the lattice model used in some specific quantum codes, for instance the Fibonacci code we have been talking about. First of all, we need to emphasize that in the Fibonacci Levin-Wen models where we encode qubits and simulate topological quantum computers, the physical qubits of the grid are generally fixed in space (in practice one would expect a physically rigid lattice on which the simulation is performed). However as pointed out in KKR [15], the topology of the abstract lattice should be the only thing that we are concerned with. And it does not have to be the same with the physical lattice. In fact, the abstract lattice shall be viewed as fluid that constantly changes its shape throughout the computation. These changes are accomplished by applying F -moves with F -matrices applied

50 a d a d a d a d a d a d a d −1 − 21 e e e e’ e’ φ e’ + φ e’ b c b c b c b c b c b c b c

a a d a d a d a d a d d a d a d a d e e e e − 21 −1 e’ e’ e’ φ e’ - φ e’ b c b b c b c b c b c b c c b c b c

a a a d a d d a d d a d e e e e’ e’ e’ b c b c b c b c b c b c = 0 = 1 a d a d a a d a d a d d e e e e’ e’ e’ b c b c b c b c b c b c

Figure 3.3: A rebuild version of Fig. 3.3 which better exhibits the lattice deformation caused by the F -move in the Fibonacci codes.

on the lattice wave function. In other words, we can use the F -move to redraw the lattice locally while reassigning the physical qubits to new lattice edges, if needed. Since the F -move commutes with the plaquette operator, we would envision that, if started from a trivalent lattice that is in its Levin-Wen ground state, after applying an F -move onto it the new trivalent lattice will still be sitting in a Levin-Wen ground state. Fig. 3.2 illustrates what happens when the F -move is applied to a small part of the lattice: it locally redraws the connection between five qubits that are associated with two neighbouring vertex, while the physical positions of them are not affected. In this figure the original hexagonal lattice alters its shape locally, no longer has four hexagonal plaquettes in the area. Instead, the F -move produces two heptagonal plaquettes and two pentagonal plaquettes. The final goal here is to obtain the quantum circuit that realizes plaquette error syndrome measurement. Hence for now before we start it is crucial for us to construct the quantum circuits that represent the F -move. To do so we will have to know the functions that the F -move does. For Fibonacci anyons, we have already solved the pentagon equation and find out the exact form abe of the F tensor(Fcde′ as defined in Fig. 2.2) in Sec. 2.1.1. For a better shape we also redraw the enumerated F -moves in the “lattice version” as seen in Fig. 3.3. Again it needs to be emphasized that we only discuss the states that satisfy the vertex constraint. And this property always preserves after applying the F -moves.

51 a) b) a a d a d b X X e F eba c X X ∑ edc ′ e’ = d b c e′ b c e F X

Figure 3.4: (a) The deﬁnition of the F -move in terms of the F -tensor with each of the qubit labelled. The labels abcdee′ are consistent with those in Fig. 2.8. (b) Quantum circuit which carries out the F -move in (a) for the Fibonacci code. The F rotation acting on the qubit e is deﬁned in Eq. 3.1.

Shown in Fig. 3.4(b) is the five-qubit circuit which carries out the F -move. We mark the five qubits on which an F -moves is acted as a,b,c,d,e , the same notation from Chapter 2. The qubit { } e is colored red here to emphasize that it is the target qubit which undergoes actual rotations { } when applied with F -moves. Even though it is not immediately apparent from the gate structure of the circuit, one can still check by hand that it has the symmetries talked in Eq. 2.18. Note that the defining part of this circuit is the five-qubit controlled F gate showing on the left side of the circuit. It applies an F rotation on the qubit e if all the other control qubits abcd are in the state 1 . The F rotation is | i characterized by the 2 2 unitary matrix in Eq. 2.24 without the arbitrary phase θ: × 1 1 φ− φ− 2 1 F = − −1 (3.1) φ 2 φ ! − The rest of the circuit consists four CNOT gates plus a Toffoli gate. They realize the F -move on those configurations that are quite straightforward by considering the vertex constraint, i.e. when at, least one of the control qubits is in the state 0 . In what follows we would always assume that | i the vertex constraint is satisfied in default. And every vertex already has Qv = 1 before an F -move is applied. In general the F -move applies a controlled unitary operation upon one single qubit, which is marked red in the figures. In that sense we would view the whole quantum circuit as a generalized Toffoli-class gate with inequivalent control qubits a,b,c,d,e that determine what kind of single- { } qubit rotation shall be chose to act upon the target qubit e. The green box on the left of Fig. 3.4(b)

52 a) b) a a a b X X F eba e ∑ eac ′ e’ c = X X b c e′ b c e F X

Figure 3.5: Four-qubit reduced F -move obtained by identifying the qubits labeled a and d in Fig. 3.4. here provides a simpliﬁed notation which can be further embedded in more complicated circuits acting on more numbers of qubits. The inputs a,b,c,d,e correspond to the qubits and indices of { } the F -tensor in Fig. 3.4. Note that the F -quantum circuit can be used as an inverse to itself. Indeed, acting the F -move twice on the same set of qubits will deﬁnitely return it to its original state. Hence

F −1 = F (3.2)

This can also be check by examining the structure of the quantum circuit, which is quite obvious by noticing that the only varying qubit e can only either have F rotation or the X gate acted, which is not aﬀected if we reverse the circuit. During the next couple sections we will also encounter another kind of F -move: a slightly varied one but essentially the same. It is obtained by identifying two of the control qubits labeled a and d from the original F -move, hence it acts on four qubits instead of ﬁve. A quantum circuit which carries out the reduced F -move is shown in Fig. 3.5.

We now turn to constructing the quantum circuit that measures the plaquette operator Bp. Since we have already seen the effect of applying an F -move to the Levin-Wen lattice, the reduction of the size of the affected plaquette may shed some light onto our original question of how to redraw a hexagonal plaquette into a one-sided plaquette: we can definitely try to continue applying the F -moves on the same plaquette until we will get the tadpole plaquette as expected, as long as we always choose the right part of the plaquette to perform the F -move for each step. Fig. 3.6 then shows the whole process of the deformation of a hexagonal plaquette all the way to a one-sided plaquette, i.e. the “tadpole”. Each step here involves an F -move acting upon selected

53 Figure 3.6: Reduction of a hexagonal plaquette to a tadpole through a sequence of six F -moves. The last step is a reduced F -move since two of the control qubits are represented by the same edge.

five qubits of the plaquette, the position of which is marked red, the same style as we used in the F -move definition in Fig. 3.4. Note that the final step is a reduced F -move with the two identified qubits being the head of the tadpole. The plaquette reduction procedure shown in Fig. 3.6 actually resembles the entanglement renormalization scheme brought out in [39]. Without going into too much details of the original paper here we want to again emphasized that such a renormalization process preserves the value of all the projection operators(stabilizers) defined in the Hamiltonian, including Bp, which means that if the target plaquette plaquette originally stays in the Levin-Wen ground state(that has Qv = 1 on each vertex and Bp = 1 for the plaquette), then after each step of the plaquette reduction process, the new lattice will continue to be in the ground state of the Levin-Wen model. And it will be safe to say that Qv = 1 on each vertex and Bp = 1 for the reduced plaquette. Therefore, to measure the Bp of a given plaquette one only has to measure the Bp value of the corresponding tadpole after a reduction process. A tadpole, or loop, is a one-sided plaquette that consists of two qubits(see Fig. 3.7(a)): one is the plaquette itself(head of the tadpole), marked as “h” and the other one is the external line(tail of the tadpole), marked as “t”. The Levin-Wen ground state for a plaquette as simple as a tadpole actually takes the following form

1 ψBp=1 = 0 t ( 0 h + φ 1 h) . (3.3) | i 1+ φ2 | i | i | i assuming that Qv = 1 is always granted.p The space which corresponds to the Bp = 0 projection will then be orthogonal to the ψ =1 state. It is spanned by two of the following state: | Bp i 1 ψBp=0,x = 0 t (φ 0 h 1 h) , (3.4) | i 1+ φ2 | i | i −| i ψ =0,y = 1 1 . (3.5) | Bp i |pit| ih

54 a) b) h h t 0 Bp =1 0 t h t

Figure 3.7: (a) A one loop plaquette with a single leg, often referred as the “tadpole”. (b) A simple quantum circuit which can be used to place a one loop plaquette in the state with Bp = 1. The matrix S is given in the text.

We can initialize a tadpole to be placed in the Bp = 1 ground state for further uses. Shown in Fig. 3.7(b) is the quantum circuit that performs such preparation. Both of the qubits t and h in the tadpole from Fig. 3.7(a) are ﬁrst initialized in the state 0 . We know for sure that the “tail” | i qubit has to be in the state 0 in order to have the Bp = 1 ground state. While for the “head” | i qubit, all we need is a special single qubit operation that rotates it from the state 0 to the state | i ( 0 + φ 1 ) / 1+ φ2, which is called the S-rotation. | i | i p 1 1 φ S = . (3.6) 2 φ 1 1+ φ − p t More generally, for a two-qubit operation we can deﬁne a tensor Shh′ acting on the tadpole as.

h h′ t S ′ . (3.7) −→ hh h′ t X t

Remember that since the vertex constraint is satisﬁed by default, we don’t need to worry about the situation where the tail is in the state 1 but the head is in the state 0 . Then the only valid | i | i 0 1 components of S tensor are the Shh′ , which takes the form of Eq. 3.6, and the S11 = 1, obviously.

Now to actually measure the value of Bp of a tadpole we will need a few more steps compared to its Bp ground state initializing. Firstly we can examine the tail qubit t: if t = 1 we imme- | i diately know that the tadpole is in the state ψ =0,y (Eq. 3.5). However if t = 0 the answer | Bp i | i will still be unclear. The tadpole can be in the superposition of the states ψ =1 (Eq. 3.3) and | Bp i ψ =0,x (Eq. 3.4). We need to further prepare the two-qubit system so that this two basis can be | Bp i

55 h t x x t h

Bp =1 0 x

1− Bp Bp = 0

Figure 3.8: A quantum circuit which can be used to measure Bp for a one loop plaquette. One extra syndrome qubit initialized to the state 0 is needed to perform the non-demolition measurement. | i well separated. To do so we can combine the S-rotation with the other elementary gates in such a way that the head of the tadpole will be transformed into two distinct states according to the two different values of Bp(for example, state 0 if Bp = 1 and state 1 if Bp = 0). | i | i Fig. 3.8 shows the quantum circuit of how we exactly execute the separation. In what follows we will refer to it as the S-circuit. The core of the circuit is the controlled single-qubit S rotation. By definition, it acts an S rotation on the target qubit(here it is the head qubit h) if the control qubit is in the state 1 , while it does nothing if the control qubit is in the state 0 . Note that here | i | i the control qubit is actually the tail qubit t being flipped by a NOT gate. After the controlled S rotation on the head qubit we only need to apply another NOT gate on it to bring it back. Notice that the S-circuit can also be further decomposed into the elementary CNOT gate and single-qubit rotations according to [34].

Overall, If the tadpole is initially in the state ψ =1 , after two NOT gate acting on its tail | Bp i qubit and a controlled S rotation acting on the head qubit, it will end up with the state 0 0 . | it| ih While if the tadpole are in the two-dimensional Hilbert space spanned by the state ψ =0,x | Bp i and ψ =0,y at the beginning, after the process the space will be transformed to the one that | Bp i has 0 1 and 1 1 as its basis(veriﬁcations can be done by explicitly calculating the result | it| ih | it| ih of multiplying Eq. 3.3, 3.4 and 3.5 by the S-matrix). In both cases the state of the head qubit determines the value of Bp. We can pass down this information by implement a CNOT gate that has the head qubit as the control qubit and an extra syndrome qubit initialized to the state 0 as | i

56 1 b b 2 b b 2 3 b b 4 b b 1 8 9 3 5 b b 6 c c 7 p 10 7 e c c e 8 a e c c e a 6 12 11 4 9 a e c c e a 10 a e c c e a 5 11 a e X X e a 12 d d d d a a d d d d

0 X

1− Bp

Figure 3.9: Quantum circuit which can be used to measure Bp for the Fibonacci code on a hexagonal plaquette based on the plaquette reduction shown in Fig. 3.6. It must be veriﬁed that Qv = +1 on each of the six vertices of the plaquette before carrying out the circuit.

the target qubit. If the head qubit is in the state 0 then through the CNOT gate the syndrome | i qubit will be ﬂipped which tells that 1 Bp = 1, thus we know Bp = 0. On the other side if the − head qubit is in the state 1 then the syndrome qubit will remain in the state 0 , and Bp = 1, the | i | i tadpole is in the ground state.

With the Bp measurement done we want to restore the tadpole to its original state. It can be done by simply applying the NOT gates and controlled S rotation once again in a reversed way since S = S−1, (3.8) as seen in Fig. 3.8. Moreover, to reconstruct the whole lattice it is necessary to undo the F -moves as well. Fig. 3.9 shows the full quantum circuit that measures the Bp operator of a hexagonal plaquette p including both the plaquette reduce and restore steps we have discussed here. We use the box notations which are introduced in Fig. 3.4 and 3.5 here for the F and reduced F -circuits with letter labeling the various “inputs”. This quantum circuit construction process can be easily generalized to the case of an arbitrary n-sided plaquette.

57 It is important for us to make sure that every vertex constraint Qv = +1 is satisfied at the start of the Bp measurement. Otherwise the circuit may not be functioning as expected. For instance, if the vertex constraint on one of the vertices on a plaquette is violated, i.e.Qv = 1, applying the Bp circuit will, with a certain possibility, gives a wrong result of Bp = 1. Yet in definition, for that plaquette the Bp value should be 0 as it is for any plaquette that has vertex constraints violated. The cause to such a problem is due to the fact that the dimensionality of the subspace which satisfies the vertex constraint is much less than it is for the entire Hilbert space. Take an n-sided plaquette with 2n qubits as the example, The full Hilbert space is 22n-dimensional, if the vertex constraint is not applied. After the Bp measurement by using the quantum circuit in 2n Fig. 3.9 this 2 -dimensional Hilbert space is divided into two subspaces. Each has Bp = 1 and 2n−1 Bp = 0, separately. And the dimensionality for both of them would be 2 . Now if we take the vertex constraint into account then the whole Hilbert space will be drastically reduced. As for the Levin-Wen ground states which have Qv = 1 and Bp = 1, the projected Hilbert space has a dimensionality of Dn[Bp = 1] = F2n−1(Fn is the nth Fibonacci number which which is defined with F0 = 0 and F1 = 1), while for states which satisfy the vertex constraint but have Bp = 0, the dimensionality will be Dn[Bp = 0] = F2n+1. Hence in total the Hilbert space with Qv = 1 is

Dn = F2n−1 + F2n+1. A simple calculation will show that for the hexagonal plaquette we used in Fig. 3.9, if the vertex constraint is satisfied beforehand, the overall 212 = 4096 dimensional qubit space will be projected down to a D6 = F11 + F13 = 322 dimensional space with the subspace that has the Levin-Wen ground states to be D6[Bp = 1] = F11 = 89 dimensional. Similar to what has been done for the vertex operator measurement, here we want to quantify the complexity of the Bp operator measurement as well. To do so we need to figure out the “gate spectrum” of each inputs in the quantum circuit constructed in Fig. 3.9. For the F -moves, each box contains 1 five-qubit controlled-F gate, 1 Toffoli gate and 4 CNOT gates. It has been noticed that since the F -matrix satisfies F 2 = 1 and det F = 1, it is possible to construct the five-qubit − controlled-F gate using one five-qubit Toffoli gate and two single-qubit rotations as

= (3.9)

F Ry ( θ) X Ry ( θ) − −

58 θ ±i X where Ry ( θ) e 2 are the rotation operators about they ˆ axis(The angle θ is given by tan θ = 1 ± ≡ φ− 2 . It is easy to verify that F = R (θ) XR ( θ) hence the two circuits in Eq. 3.9 are indeed y y − equivalent). Therefore for the F -move alone we count 1 five-qubit Toffoli gate, 1 Toffoli gate, 4 CNOT gates and 2 single-qubit rotations. Or 9 Toffoli gates, 4 CNOT gates and 2 single-qubit rotations if we take only (three-qubit) Toffoli gate as primitives since any n-qubit Toffoli gate can be further carried out by 4n 12 Toffoli gates [34]. For the reduced F -move the only difference is − to have 1 four-qubit Toffoli gate instead of a five-qubit gate, or 4 less Toffoli gates. The same analysis can be done to the “controlled-S gate” in Fig. 3.8 as also we have S2 = 1 and det S = 1. The following identity can be observed −

X X = (3.10)

S X Ry ( ρ) X Ry (ρ) · − where the angle ρ is given by tan ρ = φ−1. And the equation is verified by the fact that S = R (ρ) XR ( ρ). It involves 1 CNOT gate plus 2 single-qubit rotations. y y − ′ To sum up, the Bp circuit in Fig. 3.9 contains 8 F -moves, 2 reduced F -moves(F ), 2 S-circuits which are basically the left-hand side of Eq. 3.10(one can see it by noticing that the NOT gate on its right can actually be cancelled out in the Bp circuit), and one CNOT gate to read out the final measurement. It can be generalized to the n-sided plaquette p for which the Bp measurement will take 2n 4 F -moves, 2 reduced F -moves(F ′), 2 S-circuits plus one CNOT gate. Substituting what − has been discussed above we then know that overall the quantumc circuit for measuring Bp will take 2n 4 five-qubit Toffoli gates, 2 four-qubit Toffoli gates, 2n 2 Toffoli gates, 8n 5 CNOT − − − gates and 4n single-qubit rotations. Or if we take only the three-qubit Toffoli gates as primitives, the circuit will have 18n 26 Toffoli gates, 8n 5 CNOT gates and 4n single-qubit rotations. − − We can now compare the Bp measurement in the Fibonacci code to that in the Kitaev surface code. In the later case to measure a n-sided plaquette(the plaquette stabilizer) it only takes n CNOT gates which can be applied simultaneously. It is reasonable to expect a more demanding requirement for the non-Abelian code due to the fact that the plaquette operator is truly more complicated. Like pointed out before, the advantage for using the Fibonacci code, despite the number of quantum gates it needs, is that one can realize the quantum computation purely by

59 braiding. One shall weigh both of the two types of quantum codes thoroughly before it comes down to experiment.

3.2 New Method for Measuring and Correcting Vertex and Plaquette Errors

The vertex and plaquette error measuring methods reviewed in the last section have at least served as a start on the road to the full non-Abelian quantum error correction scheme. But that is not enough! The methods in the last section only addressed the issue of how to measure those errors. In order to do error correction after knowing where the errors are we also have to eliminate them! The challenging question is how to do it. This requires us to find a new method to finish what is left after the error measurement, which essentially leads to the core result of this Dissertation. To correct the errors from the quantum computer, theoretically if we knew when and where the exact event that created a group of errors occurred, we would simply reverse the event to bring the errors back together so they can be annihilated with one another. For example, in the Kitaev surface code the errors are always created in pairs due to their Abelian nature, so the core work there is to identify and pair up those error so they can be cancelled out from the code. Here in our non-Abelian code the errors are not necessarily showing up in pairs (a good example is when two Fibonacci anyons are in the state 1 and we can not say that they will be fused into 0 as the | i | i fusion rule 1 1 = 0 + 1 states). To correct the errors we would anticipate that the procedure is × much more complicated, since the non-Abelian errors now can be connected in a long complicated chain and well entangled (imagine that errors now can form branched “trees” instead of threads that only have two ends). One of the consequences here is that the errors are no longer able to be corrected in one shot as is the case for the Kitaev surface code. In that case, once the error pair is identified, the only thing we have to do is to act in parallel with a series of quantum gates along any path that connects the two errors and the pair will annihilate instantly. While for the Fibonacci code, the non-Abelian feature would prevent us from using similar tricks. Instead, we will have to move those errors from plaquette to plaquette to bring them together before attempting to remove them, a much more demanding requirement compared to the Abelian code (one would make a rough analogy that those non-Abelian errors can be seen as carrying non-trivial informations therefore they can only travel under a certain speed limit, rather than a instant transportation). To achieve

60 a) b)

Figure 3.10: (a) Hexagonal lattice with string end qubits initialized in the state 0 associated with | i each vertex. (b) Possible result of carrying out one round of Qv measurements for each vertex. The remaining string ends represented by thick lines are in the state 1 and correspond to vertex | i errors. this, in this section we will introduce the new syndrome measurement methods for both the vertex and plaquette errors. As a major diﬀerence, these new methods provide the preparation for the further moving and fusing schemes discussed in subsequent sections.

3.2.1 Vertex Errors on String Ends

The vertex syndrome measurement in Sec. 3.1.1 is indeed a straightforward and eﬃcient way to rapidly determine whether a vertex error has occurred. The only requirement for realizing its quantum circuit is one extra syndrome qubit (per site) that is used to read out the measurement.

However, this is not the end of the job since so far we only know the value of Qv. The measurement itself does not give a process to deal with the errors if Qv = 0. Based on what we have learned so far we know that the vertex errors can be nailed down on their own vertices using the syndrome measurement provided. Now if we want to move them to some other places of course we can again use the F -moves as before. By doing this, the lattice will have to be distorted to some degree, a situation that we want to avoid at this stage. So the idea comes up here that we can try to relocate the errors to an “ancillary anyon lattice”, an extra lattice that is required in addition to the original lattice (see the gray edges in Fig. 3.10(a)). The new lattice serves as a place where those errors can be stored for future disposal and it does require extra physical qubits to be formed. This tricky method actually comes out from the idea that the vertex errors are in fact “string ends” if we view the lattice model as a “string net” [27].

61 a) 1 b) c) v  1  2 2 3 3 2 1  v 0 X X 1Q    0 X X X v 3

Figure 3.11: (a) Two additional qubits, labeled α and β(orange lines), are needed for each vertex v. (b) Unitary transformation which can be used to draw out vertex errors, and (c) a quantum circuit which carries out this transformation and determines whether a vertex error is present. Both the two additional qubits are initialized in the state 0 . After the full circuit is carried out qubit α | i is measured. If it is found in the state 0 then there is no vertex error, and qubits α and β can | i be safely removed. If it is found in the state 1 then there is a vertex error which has now been | i moved to qubit α. From this point on, qubit α will always be in the state 1 and it is therefore | i not necessary to include it explicitly so it can be removed. However, we must keep qubit β which will in general no longer be in the same state as qubit 3.

In Sec. 2.2 we have already seen the example of how vertex errors look like in the “string net” picture(Fig. 2.16(b)). Now imagine that we can pull out those string ends to this ancillary lattice. Then we will be free to move them wherever they are needed. In what follows we will also show the tools that are needed for moving vertex errors. They are exactly the two basic moves introduced before: the F -moves and the R-moves (we will have more details in Sec. 3.3). A detailed picture which illustrate this string end extraction procedure is presented in Fig- ure. 3.11. To make sure that we have the syndrome measurement for each vertex we will have to plant the same set of extra qubits near each vertex. In general two extra qubits are needed per site: one is of course the string end qubit α that will have any on-site vertex error pulled onto it; the other one is actually a result from planting the string end qubit β. Putting extra qubits on the lattice also means adding an additional vertex to the system which will also have to be put in the ground state. Naturally, the string end qubit is ﬁrst initialized to 0 and the additional | i qubit on edge β is placed in the same state as the qubit on the edge labelled 3 in Fig. 3.11(a) by a controlled-NOT operation. We then carry out the unitary operation deﬁned in Fig. 3.11(b). The result of this operation is that the vertex v is now guaranteed to satisfy the vertex constraint, as well as the new vertex formed by edges α, β, and 3.

62 a) b)

Figure 3.12: (a) A chain of error syndromes in the thickened lattice which consists of only vertex errors. By defaut, the vertex errors are represented by string ends that is placed above the lattice. (b) Combining diﬀerent types of vertex errors and the plaquette errors to form a ribbon graph.

The state of the string end qubit is then measured. If the result of this measurement is 0 | i there was no vertex error and the two additional qubits can be safely removed. If the result of this measurement is 1 there was a vertex error, but this error has now been “pulled” into the string | i end. At this point the string end qubit can again be removed, since we know with certainty that it is in the state 1 , though we must keep track of its location in the lattice. The second ancillary | i qubit, β, cannot be removed since it is no longer guaranteed to be in the same state as qubit 3. The full quantum circuit for carrying out the above procedure is shown in Fig. 3.11(c). This circuit is a modiﬁed version of that introduced in Fig. 3.1. As noted in Sec. 3.1.1, the appearance of a four-qubit Toﬀoli gate in this circuit is characteristic of non-Abelian error correction. Figure 3.10 shows an example of one round of vertex error correction on a hexagonal lattice. In Fig. 3.10(a) two ancillary qubits have been added to each vertex, forming the ancillary lattice as proposed. The quantum circuit shown in Fig. 3.11(c) is then carried out for each vertex and a possible result is shown in Fig. 3.10(b). The remaining string ends, shown as heavy lines to indicate they are in the state 1 , all correspond to vertex errors. Those vertices for which there is no longer | i an associated string end were found to satisfy the vertex constraint. After one round of vertex error correction all the vertices v of the original trivalent lattice now satisfy the vertex constraint Qv = 1. However, this has come at the cost of introducing string ends wherever a vertex error was detected. These string ends modify the lattice and so before turning to the plaquette errors, it is necessary for us to specify the state for a given set of string ends for which the plaquette errors are zero. We do so again using the thickened lattice. However, at this

63 stage we also thicken the lattice in the third dimension. A string end then corresponds to adding a charge 1 string to the lattice, and pulling it over the vacuum string of the plaquette where the string end appears, see Fig. 3.12. We denote such string ends that are pulled over the lattice as type (1, 0) errors. Accordingly there are also type (0, 1) errors which corresponds to the string ends that are pulled under the lattice. The resulting ribbon graph can again be reduced to a string net with one string per edge by using the rules given above along with the Kaufmann bracket,

3 3 = e−i 5 π + ei 5 π (3.11)

The thickened lattice picture will be a better illustration on the structure of the error syndromes. For instance the error chain in Fig. 3.12(a) describes how the three vertex errors (string ends) are related in the absence of plaquette errors. The error chain can be even more sophisticated if combined with plaquette errors (Fig. 3.12(b)). Yet keep in mind that these chains will be hidden in the “black box” in a practical error correction process, i.e. there is no way for us to know their virtual structure before we can annihilate them. The only thing we can do is trying to bring these errors together and see if they can fuse to vacuum. A detailed explanation on how to fuse errors can be found in Sec. 3.4. At the end, the string ends do affect the plaquette error measurement in such a way that it changes the shape of the plaquettes. If there are no plaquette errors the value of Bp will be 1 for each plaquette that does not enclose a string end. This includes those plaquettes for which one or more of the legs are string ends. For those plaquettes which do enclose one or more string ends it will be necessary to move these ends out of the plaquette before measuring Bp. We will describe precisely how to do this in Sec. 3.3. However, before doing so we need to first present our plaquette measurement scheme, which will be shown next. The total number of quantum gates used in this vertex syndrome extraction process now can be counted in Fig. 3.11(c). Compared to the method used in Fig. 3.1 of Sec. 3.1.1, it has one more CNOT gate in addition to a four-qubit Toffoli gate (or 4 Toffoli gates) and 3 CNOT gates. Also it needs two extra per site to realize the process. But the advantage of using this new method is that not only the vertex operator Qv is known by the measurement, the vertex will also be cured of any error if there is any, since the errors will be moved to the ancillary lattice with all the original vertices having Qv = 1. In general the vertex syndrome extraction process can be served as a

64 replacement for the former idea of measuring the Qv operator [29], in coordination with the new plaquette error treatment which uses a similar trick, as will be introduced in the next section.

3.2.2 Plaquette Swapping

We now turn to our new procedure for carrying out the most demanding syndrome measurement, that of the plaquette operator Bp. The preliminary measurement circuits of [29] reviewed in Sec. 3.1.2 are useful as proof of principle, but have the limitations when applied to the problem of error correction. In fact after the syndrome measurement in the plaquette reduction method, we again only acquires the value of Bp, the plaquette will then have to be restored to its original shape. During this procedure the old method does not address the issue if there is an error detected

(Bp = 0). However, in our new method which we will be presenting below, the reader will see, after the plaquette operator is measured, the detected errors will also be isolated for further purposes, i.e. ready to be transported and fused.

The new suggestion of measuring Bp is actually inspired by the previous method as discussed in Sec. 3.1.2. Again, the fact that a single sided plaquette, the “tadpole” is easy to measure and manipulate has triggered us with this new idea of the Bp measurement — though instead of reducing the original n-sided plaquette (our target) to a simple tadpole, here we actually would propose replacing it with a good simple plaquette which already have a Bp = 1 (from now on we will refer to those plaquettes that have Bp = 1 as “good” plaquettes). For a 2-qubit tadpole, the state with Bp = 1 (i.e. the Levin-Wen ground state on this simple two-qubit lattice) is already given in Eq. 3.3. The initialization of such a two-qubit ground state can also be accomplished by the S-rotation as pointed out in Fig. 3.7, Sec. 3.1.2. The trickiest part of this new method would be how exactly to swap the patient plaquette with the “good” tadpole we want to plant in. The F -move again plays a crucial role here, as we will elaborate next. Since the F -moves commute with both the Qv and Bp operators, such a nice property could allowed us to transform the lattice freely as we wish(to some extent). For example, we may ﬁnd a way to gradually exchange the position of the two plaquettes through a sequence of F -moves(see below). Nevertheless, the advantage of swapping out the plaquette we want to measure in an error correction attempt, if possible, is that not only will we be able to measure the patient plaquette since it will become separated, but we will also ﬁnish the error correction

65 3

2 4 9 10

8 β 11 α 7 12 1 γ 5

Figure 3.13: The idea of doing syndrome measurement and error correction by swapping the target hexagonal plaquette out with a tadpole. The Bp value for the target is unknown and to be measured, however the tadpole has Bp = 1 as already set in the ground state. Three extra qubits, α, β, γ are required here. procedure as the new plaquette(which was originally the “good” tadpole) is now guaranteed to be the ground state no matter what the syndrome is measured on the old plaquette. Before we start, it must be assumed that one full round of vertex error correction has already taken place, i.e. all the vertex errors are relocated to the ancillary lattice which become strings ends attached to edges of the plaquettes, as described in the previous section. Now if the string ends are attached to the inner edges of the to-be-measured plaquette, we can simply consider them as additional plaquette legs which are known to be in the state 1 . However, if the string ends are | i inside the plaquette, we will have to figure out a way to move them out so the plaquette can be a “normal” one(Sec. 3.3). Either way we must be assured that the vertex error will not bar our way to the plaquette error treatment. Here we outline the exact procedure of the plaquette swapping process. First of all, to initialize the swapping process, we can simply insert the “good” tadpole into the patient plaquette, as shown in Figure. 3.13. It is always safe for us to throw in a part of extra lattice configuration as long as it is in the Levin-Wen ground state, e.g. it has Qv = 1 and Bp = 1 for every vertex v and plaquette p. Note that three ancillary qubits are needed in total in order to introduce this extra one-sided plaquette. Two of them compose the tadpole itself. The other one comes from the intersection of the tadpole to one side of the n-sided plaquette — the side will be broken into two qubits. More specifically, the two qubits α and β are encoded with the Bp = 1 loop which has the state given in

66 Bp = ? Bp = 1

Bp = ?

Bp = 1

Figure 3.14: Swapping a hexagonal plaquette with a good plaquette through a sequence of F /reduced F -moves. At the start a tadpole plaquette is placed at the center of the hexagonal plaquette(the patient). The red line shows the place where the F /reduced F -move is applied at each step of the path. The green ﬁlling indicates it is a “good” plaquette with Bp = 1 while the grey ﬁlling shows that Bp is not known.

Eq. 3.3. Therefore we have α = 0 | i | i , (3.12) β =( 0 + φ 1 ) / 1+ φ2 | i | i | i or in the diagrammatic way: p

β 1 = + φ . (3.13) 2 α 1+ φ   p   As for the qubit γ, since α is in the state 0 the vertex constraint on the vertex associated with | i qubit α, γ and 12 demands that it must be in the same state as qubit 12. We then use the F -move to continue our plaquette swapping scheme. The whole procedure consists of n + 2 steps and each step is an F -move acting on a group of selected qubits. See Fig. 3.14, the red qubit indicates where the F -move should be applied, as deﬁned in the F -move circuit in Fig. 3.4. The key point here is that each time when an F -moves is applied to change the shape of the lattice, the size of the patient plaquette (here the hexagonal one) will be reduced by 1, while the size of the “good” plaquette (colored light green) is increased by 1. In such a way we can expand the “good” plaquette step by step to ﬁll the “cell”(area inside the patient plaquette)

67 Figure 3.15: The result of the plaquette swapping process in the thickened picture. The plaquette errors are deﬁned as threads going through each of the plaquettes. By swapping the target plaquette with the “good” tadpole we will end up trapping the error thread inside a tadpole(needle eye).

and the patient plaquette will be squeezed and eventually turned into another tadpole! Therefore in the end one will have the two plaquettes successfully exchanged while the Bp value for both of them is kept the same along with them. It needs to be pointed out that all of these swapping steps are taken place inside one plaquette(the patient) thus the shape of the lattice does not have to be distorted, which happens in the plaquette reducing method in Sec. 3.1.2. Note that in this “cycle” the first and the last F -moves acts only on four qubits, just like the final step in Fig. 3.6 of the plaquette reducing method. This is due to the fact that at this two steps it is the tail of the tadpole upon which the F -moves are applied, where the head of the tadpole should be considered as two qubits being identified, see the definition of reduced F -move in Fig.3.5.

Again, since the F operator commute with Qv and Bp [39], for each step an F -move is applied the system will stay in the Levin-Wen ground state or the excited state depending on whatever the original state it was in. In the color coding shown in Fig. 3.14 it is clearly marked out that the coloring of the two diﬀerent “cells” is never changed during the swapping process. Therefore as we ﬁnish all the steps, the “good” tadpole is expected to have expanded and become the new n-sided plaquette with Bp = 1, while the patient plaquette will be trapped inside a tadpole which is easy to be dealt with as the simplest plaquette. For example, we can borrow the S-circuit(Fig. 3.8) from the plaquette reducing method in the previous section to measure its Bp value as it is expected to be the same with that of the original patient plaquette. Furthermore, considering the error

68 correction of the Levin-Wen codes we may even want to move these tadpoles(if they are detected the existence of errors) through the lattice, the details of which can be found in Sec. 3.3. The plaquette swapping process actually reﬂects the nature of capturing errors in the thickened picture, in which the plaquette errors can be seen as threads that go through the “punctures” of the plaquettes. In what follows we will see that the swapping process is in fact a way to trap these threads inside a bunch of tadpoles, or a more vivid name, “needles”. Fig. 3.15 shows how a plaquette error is collected under such interpretation. The thick black line that penetrates the puncture as well as the plaquette itself is a presumed plaquette error, denoted as a type (1, 1) error. It can be jointed with the other thick black lines(plaquette errors) and even the string ends(vertex errors) introduced in Sec. 3.2.1 by simply using the Fibonacci fusion rules(Eq. 2.1). For example, a plaquette error (1, 1) combined with a type (1, 0) vertex error can be either turned into a (0, 1) vertex error or another plaquette error:

   (0, 1) :        (1, 0) (1, 1) =  (3.14) ×  

   (1, 1) :         In the swapping process we implant a good tadpole at ﬁrst as described, which then gradually replace the target plaquette under the F -moves. While the target plaquette shrinks and becomes a tadpole(needle) and the error thread is now wrapped inside the needle eye! Note that the needle eye has a puncture inside through which the error thread is penetrating. It is essentially the same puncture as we are only changing the topological structure of the lattice. Now that all the details of plaquette swapping have been described, we can construct the full quantum circuit for carrying out the entire swapping process. Again the reader can be assured that the circuits for all the F -moves including the reduced F -moves are guaranteed with the satis- faction of vertex constraint for every vertex applicable. This is because, as stated before this Bp

69 = SWAP

1 b 2 b 3 b 4 b 5 b 6 b 7 a c 8 a c 9 a c 10 a c 11 a c 12 b a b γ 0 X a e e e e e e a α 0 e c e X X β 0 S c d d d d d d c S S

0 X

1− Bp

Figure 3.16: Full quantum circuits that realize the plaquette swapping process together with a syndrome measurement for the Fibonacci code on a hexagonal plaquette. The ﬁrst part of it initializes the tadpole that was put in. The second part consists of F /reduced F moves that swap the two plaquette. The third part is for measuring the shrunk target plaquette.

measurement is preceded by a round of Qv measurement which presumably have “healed” all the vertex errors. We give the example of a hexagonal target plaquette which has 12 qubits plus the three extra qubits cost by the introduction of the “good” tadpole. The full quantum circuit for the swapping procedure and the syndrome measurement is given in Fig. 3.16. At the ﬁrst step, qubit γ should be in the same state with qubit 12 (here the qubit labeling is that shown in Fig. 3.13), which can be done by carrying out a CNOT gate between them if the qubit γ is initialized to the state 0 . | i Meanwhile, a single-qubit S rotation acting on the head of the tadpole(qubit β which is also in the state 0 at the beginning, and so is the tail qubit α) puts the tadpole into its ground state with | i Bp = 1. After this, we then follow the procedure in Fig. 3.14 and install the F -moves and the reduced F -moves one by one using the box notations.

70 SWAP β γ β β γ : γ 7 7 γ α 7 α 7 α

Figure 3.17: The ﬁrst SWAP action which was needed after the ﬁrst F -move in Fig. 3.16. It exchanges the labels of the qubit 7 and γ so that after the F -moves the qubit 7 will stay in its original place while we keep the qubit that was acted by the F -move always being the qubit γ.

One tricky part in this circuit that one has to be very careful about is the need of a “SWAP” action which appears after some of the F -moves during the swapping procedure. In fact rather than an actual quantum gate it is more an exchange on the labelings of the qubits, which makes the graph a better shape if implanted. Shown in Fig. 3.17 is an example of why the SWAP action is needed: in general we would like to see all the qubits that composes the original target plaquette(qubit 7, 8, , 12 ) to stay where they were after the F moves, which make the qubit γ always the place { ··· } where the F moves are applied. It is important to understand that we are only swapping the labels of those qubits — it does not mean that the two qubits undergoes a swap gate, which is a unitary operation upon their quantum state. We are still able to construct the full quantum circuit in Fig. 3.16 without those SWAP boxes. The cost is that the qubit labeling must be rearranged in some way which may look quite confusing and messy. The syndrome measurement at the end is done again with the S-circuit, the exactly the same procedure we used in Sec. 3.1.2 since after the procedure we once more have a tadpole to be measured. Note that we no longer need to restore the lattice after the swapping procedure, because the shape of plaquettes will remain the same and because it is also a part of the curing process(see elab- orations below). Though we do have to restore the tadpole after it is rotated to the “measurable” position. Thus the circuit is only symmetric for the final measurement part. To summarize all the discussions so far in this section and to make an effective gate count on the full quantum circuit we have obtained, the plaquette swapping method for measuring the Bp operator of a n-sided plaquette can be divided into three parts. The first one is the initialization part which requires only 1 CNOT gate and 1 single qubit rotation(the S-rotation). The second part contains n F -moves plus 2 reduced F -moves. The gate count for the SWAP actions are omitted

71 here since, as pointed out before, they are not real quantum gates. Therefore a total number of n five-qubit Toffoli gates, 2 four-qubit Toffoli gates, n + 2 Toffoli gates, 4n + 8 CNOT gates and 2n + 4 single qubit rotations is expected for this part. The third one is the final syndrome measurement part which has two S-circuit and one read-out CNOT gate, counting 3 CNOT gates and 4 single-qubit rotations. In total, the full quantum circuit in Fig. 3.16 counts n five-qubit Toffoli gates, 2 four-qubit Toffoli gates, n + 2 Toffoli gates, 4n + 12 CNOT gates and 2n + 9 single qubit rotations. Or if we take the (three-qubit)Toffoli gates as primitives, The plaquette swapping procedure would then requires 9n + 10 Toffoli gates, 4n + 12 CNOT gates and 2n + 9 single qubit rotations, a less demanding requirement compared to the plaquette reducing method introduced in Sec. 3.1.2. This is mostly because the old method has to have the “plaquette restoration step” after the measurement while it is not needed for the new method, which reduces the number of F -moves and saves quantum gates. For an n-sided plaquette, the plaquette swapping method will have n 4 less five-qubit − Toffoli gates, n 4 less Toffoli gates, 4n 17 less CNOT gates and 2n 9 less single-qubit rotations, − − − or simply 9n 36 less Toffoli gates, 4n 17 less CNOT gates and 2n 9 less single-qubit rotations. − − − Other than the consumption of less quantum gates, one needs to know that the new method has an even greater advantage over the old plaquette reducing method: It not only measures the plaquette operator Bp, but also cures the target plaquette by replacing it with a good one. One key feature of the plaquette swapping procedure is that if an error is detected, the error is contained entirely within the swapped tadpole, which is represented by the conserved value of Bp associated with it. While the target plaquette becomes the one with Bp = 1, namely “cured”. Furthermore we believe this will be quite useful for the future error correction scheme, because these “error tadpoles” if there is any, can be moved through the lattice using F -moves(Sec. 3.3). If a collection of such errors are actually generated locally out of the “vacuum” (i.e. the Levin-Wen Ground state), then it should be possible to figure out a way to bring them together and “annihilate” them back into the vacuum, which fully corrects the errors. Given the complexity of the plaquette swapping quantum circuit (and other circuits which arise in our work), an important part of the work has been the numerical verification that this circuit does indeed perform the desired operation. Such numerical verifications are already carried out by simulating the circuit using a Mathematica tool pack called “QuCalc” which is specifically designed

72 a d a d a d a d a d a d a d a d e e e e e’ e’ e’ e’

b c b c b c b c b c b c b c b c

a d a d a d a d a d a d a d a d e e e e e’ e’ e’ e’

b c b c b c b c b c b c b c b c

= 0 = 1

Figure 3.18: The Abelian version of the F -moves which is all predictable by the Abelian vertex constraint that does not allow branches.

to numerically model quantum circuits [40]. The result of this simulation veriﬁed that the circuit in Fig. 3.16 does indeed perform a plaquette swap for arbitrary n-sided plaquettes.

3.2.3 Plaquette Swapping: Abelian Case

Before proceeding, we note that the procedure outline above for the non-Abelian case reduces to the usual procedure for measuring the error syndrome for the Abelian toric code. In this case, the tadpole is initially in the state

+ =( 0 + 1 )/√2 . (3.15) | i | i | i or in the diagrammatic way: 1 = + . (3.16) √2 which is the Abelian version of the ground state. In addition, when inserting the tadpole into a plaquette it is no longer necessary to include the two extra ancillary qubits — there is only one tadpole qubit(the head) which is in the state described above. This is because the states of these qubits are completely fixed by the states of the other qubits, a consequence of the lack of branching of the Abelian fusion rules, as reflected in the tensor δijk. Therefore we only draw the head of the tadpole while omitting the tail since it will always be in the state 0 . | i If the tadpole qubit is in the state 0 carrying out the swap procedure outlined above is trivial | i and results in the identity operation, i.e. the final state of the tadpole is again 0 and the state | i of the qubits on the plaquette edges are unchanged, see the example in Fig. 3.19(a). This is quite

73 a) b)

Figure 3.19: The Abelian version of the plaquette swapping process on a particular plaquette configuration of the Abelian Levin-Wen model. (a) “Swapping” a tadpole for which the head qubit is in the state 0 with the target hexagonal plaquette by applying the Abelian F -moves in Fig. 3.18 | i will bring back the same pattern. (b) “Swapping” a tadpole for which the head qubit is in the state 1 will somehow, flip all the qubits that sit on the inner edges of the target plaquette. | i an obvious result if one notices that the F -move for the Abelian anyons(Fig. 3.18) is trivial in the sense that each of its components is totally fixed by the Abelian vertex constraint, which is quite unlike the Fibonacci anyon case where the F -move does have non-trivial parts. However, if the tadpole qubit is in the state 1 , Fig. 3.19(b) shows that carrying out the swap | i using the Abelian F -move (Fig. 3.18) results in a final state in which the tadpole is still in the state 1 , but a NOT operation has been performed on each of the qubits on the edges of the hexagonal i plaquette. Combining the two cases it is now reasonable to make the conclusion that the Abelian swapping process is overall equivalent to applying a controlled-NOT gate to each of the n sides of the target plaquette, with the tadpole qubit being the control qubit. The result is diagrammatically shown in the Fig. 3.20(a). It follows that the plaquette swapping procedure for the Abelian model is carried out by the quantum circuit shown in Fig. 3.20(b). This is, of course, the familiar circuit for measuring plaquette

74 a) b) 6 X

5 X 3 X 2

β X 4 X X

4 X 1 3 X X X X 2 5 6 1 X β 0 H H | i 1 Bp −

Figure 3.20: (a) The result of the plaquette swapping process when applied to the Abelian Levin- Wen model. (b) The quantum circuit which realizes the procedure of (a). The tadpole qubit is ﬁrst initialized in the state 0 . It becomes the ground state + after passing through a Hadamard | i | i gate H. At the end, the tadpole qubit is once more transformed by another Hadamard gate and measured to determine the value of Bp. errors for the Kitaev surface code [41]. Here we see that we can interpret the initial Hadamard gate as a gate which initializes the good plaquette. The Hadamard gate performs a single-qubit rotation which is characterized by the Hadamard matrix: 1 1 1 H = . (3.17) √2 1 1 − It can be seen as the equivalent S-circuit in the Abelian Levin-Wen model. Then the series of CNOT gates carry out the plaquette swap, and the ﬁnal Hadamard gate results in a measurement of the tadpole in order to determine if an error has occurred. If the answer is “no” then the swapped plaquette can simply be removed. However, if the answer is “yes” then it is necessary to move the bad plaquette through the lattice to annihilate with other errors. We will see in the next section that for the Abelian case this again results in the familiar procedure for annihilating pairs of errors in the Kitaev surface code.

3.3 Moving Errors

In this section we discuss what happens after trapping the errors using the new method we introduced in Sec. 3.2. For the vertex errors, they now become string ends, while for the plaquette

75 a a a a T-1 = e π si a e− π si b = b T b b b

Figure 3.21: An R move used to swing a string end over an edge in order to move it out of a plaquette. When carrying out this move single-qubit rotations must be applied to the two edge qubits. A simple quantum circuit is shown on the right which can be applied to the edge qubits when carrying out an R move.

errors, they are now swapped out and conﬁned in the tadpole loops. Having isolated these errors has, in some sense, corrected the vertex and plaquette partially, However, as pointed out at the beginning of this Chapter, the errors still need to be moved before we can proceed to annihilate them and restore the system to the Levin-Wen ground state.

3.3.1 Moving Vertex Errors

To start with, there are completely two diﬀerent types of errors we have to take into account. The vertex errors and the plaquette errors. In the previous section we have showed how to trap the plaquette errors by swapping a freshly initialized Bp = 1 loop with a target plaquette, provided the target plaquette had all of its legs pointing outward, which includes the string ends(vertex errors) the vertex measurement described in Sec. 3.2.1 results in. However, This procedure cannot be directly applied to plaquettes where there is a string end pointing into the plaquette. When this happens it is necessary to move the string end out of the target plaquette before carrying out the loop swap. The string end can be moved from one plaquette to another sharing the same inner edge by the R-move process which is introduced in Sec. 2.1.1: the world line representation of the R-move in Fig. 2.3 is now seen as a lattice conﬁguration change in the Levin-Wen model. Then the move itself in this particular case can be visualized as “braiding” the string end over the edge. In doing so there are two choices, one can either braid the string end above the edge or below the edge. Like

76 1 a 2 b 1 1 3 c 4 4 5 4 d

5 5 2 3 2 3 e

Figure 3.22: Use of an F -move to move a string end from one edge to another within a given plaquette.

the F -move, this operation results in a unitary transformation which must be performed on the relevant qubits to guarantee the system stays in the Levin-Wen ground state for the new lattice. The required unitary operation is shown in Fig. 3.21. As a retrospection, one will understand the origin of this operation by considering the braiding of two Fibonacci anyons of a particular chirality represented by the following fusion diagram,

a b b a b a = Rc . (3.18) c → c ab c

c Here Rab is the R tensor which gives the phases acquired when two anyons with topological charges a and b and total topological charge c are braided around one another. In general, the R tensor can be expressed as a product of the phases

c −iπsa −iπsb iπsc Rab = e e e , (3.19) which is composed by the “topological spins”(Eq. 2.7) of the anyons of topological charges a,b,c { } and for Fibonacci anyons s0 = 1 and s1 = 3/5. Because the string end is always understood to be in the state 1 , and in fact need not be | i represented by a physical qubit, the unitary operation associated with the R move only acts on the two qubits on the edges the string end is being braided over. And because of the factorized form of the R matrix (Eq. 3.19) this operation consists of two single qubit operations, one on each of these two qubits. These operations are “half” of the topological spins which is

1 0 T = (3.20) 0 ei3π/5

77 a) R b) F F 1 a b c b 1 2 b a b c α 3 c b a b β 2 3 1 α d d d R R β e a e a e a F

Figure 3.23: (a) Sequence of R and F moves that move a string end around a vertex. The statement that the ﬁnal state must be the same as the initial state (up to an irrelevant phase) is equivalent to the hexagon equation. (b) A quantum circuit, which can be used to verify the hexagon equation. The result of this circuit should be equivalent to the identity when acting on all the possible states which satisfy the vertex constraint.

and its inverse. The fact that the string end can be either braided over or under an edge is reﬂected in the fact that there are two consistent R matrices (the R matrix given above and its complex conjugate). Here we choose the convention that if the edge and string end are lined up so that the edge is vertical and the string end is on the left before carrying out the R move, then the bottom qubit is acted on by T and the top qubit is acted on by T −1, as shown in Fig. 3.21. Going back to the thickened lattice picture the convention that we choose here would help to have the convenience where we needs only to deal with (1, 0) vertex errors. On the right of Fig. 3.21 a simple quantum circuit which carries out an R move is shown. Even though this circuit, which only involves two single qubit operations, is much simpler than the F -move circuit, we use a similar labeled “box” notation to emphasize that, like the F -move, the R-move is a fundamental operation in any anyon theory. The R-move can be used to move a string end from one plaquette to another, provided these two plaquettes share the edge to which the string end is attached. To move the string end to any arbitrary edge one can also use F -moves to move it within a plaquette, as shown in Fig. 3.22.

The fact that the R move can be used to move a string end without altering the value of Bp for any of the plaquettes follows from the self consistency condition which is equivalent to the hexagon

78 a) b)

Figure 3.24: Continuation of the error correction round begun in Fig. 3.10 for Bp measurements. (a) Bp = 1 one loop plaquettes are inserted into all those target plaquettes with no string end pointing inwards. (b) After carrying out the plaquette swap procedure discussed in the text for each of the target plaquettes, the possible result of carrying out one round of Qv measurements for each vertex is shown. The one loop plaquettes have been removed from those plaquettes which were in the state Qv = 1. The remaining plaquettes represented by red one loop plaquettes correspond to plaquette errors

equation in Sec. 2.1.2. This condition is depicted in Fig. 3.23. In this diagram, a string end is braided over the three edges connected to a single vertex using R-moves and also moved within a plaquette via F -moves. In this diagram it should be understood that after carrying out the full sequence of operations which go all the way around the hexagon, the resulting state need only agree up to an overall phase. The R matrix given above, and its complex conjugate, are the unique solutions to the hexagon equation given the F tensor for Fibonacci anyons. A quantum circuit identity which corresponds to the hexagon equation is shown in Fig. 3.23(b). This circuit plays the same role for the hexagon equation as the quantum circuit representation of the pentagon equation in [29] does for the pentagon equation. It should be understood that the circuit identity only holds for those states which are consistent with the vertex constraint Qv = 1.

3.3.2 Plaquette Errors Revisited

Given a Fibonacci code where quantum error correction needs to be processed, the non-Abelian nature it possesses prevents us from correcting both the vertex error and the plaquette error at the same time. Indeed, all the plaquette treatments we have established so far are all based on the precondition that the vertex error has already been eliminated, or at least identified. In other words, these two different types of errors have to addressed in order: first the vertex errors, then

79 Figure 3.25: Continuation of one round of error correction. The plaquettes with string ends pointing inwards have not been measured. To measure them one must ﬁrst swing the string ends out of the plaquettes using R moves and then insert freshly initialized one loop plaquettes. The pattern on the right shows one possible result of carrying out plaquette swaps for which one of the plaquettes had Bp = 0 while the other two had Bp = 1. At this point every vertex operator Qv and every plaquette operator Bp has been measured. The vertex and plaquette errors are now entirely contained in the string ends and one loop plaquettes.

the plaquette errors. Moreover, if we choose to use the new method of error treatments proposed in Sec. 3.2, when processing the plaquette error one has to face the situation where the plaquettes will have edges pointing inside — the string ends which was planted during the vertex error process.

The Bp for these plaquettes are ill-defined hence the plaquette swapping method can not be applied to them directly. Fortunately there are ways to move these string ends out to make way for the measurement of Bp, thanks to the two basic anyon moves: F and R. Nevertheless, concerning both of the reasons listed above it is required that the first step of our non-Abelian quantum error correction scheme must be a round of vertex error correction, in which the goal is to make sure that every vertex on the original lattice are cured to satisfy the vertex constraint, while the errors are moved to string ends as shown in Fig. 3.10. After the round of vertex error correction is done, some of the ancillary qubits will remain in their initial state 0 since the vertices they are attached to have no errors. So these qubits can be | i safely removed, leaving a “clean” plaquette behind. At this point, we shall be able to do part of the plaquette swapping process by first planting those good Bp = 1 tadpoles inside only the “clean” target plaquettes, then carry on the swap, and remove any good tadpoles left at the end. Fig. 3.24 shows an example of the first part of the plaquette error correction by continuing with the case of the vertex error correction in Fig. 3.10, Sec. 3.2.1. As a result, we have measured and cured all the vertex errors and plaquette errors in this particular example, only except for those plaquette errors

80 1 a 2 b 1 1 4 3 c 4 5 4 d 5 2 3 2 3 5 e

Figure 3.26: A single F -move which moves a bad single-sided plaquette inside of larger plaquette.

that may reside in the plaquettes that have string ends pointing inward, which means they can not be dealt immediately. Having established the tools for braiding string ends out of plaquettes we revisit the problem of measuring the plaquette operators for those plaquettes with a string end pointing into them. Figure 3.25 shows the result of our example round of vertex and plaquette measurement after measuring all vertices followed by those plaquettes with no inward pointing string ends. To measure Bp for the remaining plaquettes we simply braid the string ends out of them and then introduce one loop

Bp = 1 plaquettes, as shown in Fig. 3.25. We can then carry out the plaquette swap of Sec. 3.1.2 and measure the resulting swapped loop. As before, if the swapped loop has Bp = 1, we can simply remove it and conclude that the target plaquette had Bp = 1. However, if the result of this measurement is that the swapped loop qubits is in a state with Bp = 0 then there was an error, but this error is now “packaged” in the loop.

3.3.3 Moving Plaquette Errors

In Sec. 3.3.1 we described how to move string ends corresponding to vertex errors through the lattice by braiding them over lattice edges using R moves in order to move them from one plaquette to a neighboring plaquette, as well as by carrying out F moves to move them within a given plaquette. We now show how to do the same for plaquette errors which now consist of one loop plaquettes with Bp = 0 which we refer to as error loops. A general procedure for moving this tadpole through the lattice, while preserving the fact that the swapped plaquette now has Bp = 1 follows partly from a circuit presented in [29]. There are

81 1 b b 2 2 2 2 c c 3 4 3 e a 4 3 4 3 3 4 a e 1 1 1

Figure 3.27: Sequence of two reduced F -moves used to pull a bad single-sided plaquette through an edge.

two operations necessary, like the case of moving a string end. The first is to move a tadpole within a plaquette by moving the tail of the tadpole from one plaquette edge to another. This can be done by simply applying an F -move on the edge where the tadpole is placed, see Fig. 3.26. The second procedure pulls the tadpole through an edge using reduced F -moves (see Fig. 3.27). The first reduced F -move is applied to the tail of the tadpole and places the head in the middle of the edge, as shown in Fig. 3.27. Then another reduced F -move will pull it to the other side. Note that after the pulling through the head and the tail qubit have been swapped. Doing so preserves the value of Bp for each plaquette, so the corrected full plaquette will continue to have Bp = 1, and the bad tadpole will continue to have Bp = 0. The quantum circuit which carries out this procedure is also shown in Fig. 3.27. As in the previous section, we now show that this procedure reduces to the usual error correction protocol for the Kitaev surface code when we consider the Abelian case. In this case, the bad tadpole is always in the state =( 0 1 )/√2 . (3.21) |−i | i−| i or in the diagrammatic way: 1 = . (3.22) √2 − As in the previous section it is not necessary to explicitly include the two extra ancillary qubits that are needed in the non-Abelian case. Figure 3.28(a) shows the effect of moving a tadpole through an edge using the Abelian F -moves of Fig. 3.18 when the tadpole is initially in the state 0 , showing | i that the result is that the after being pulled through the edge the tadpole is in the same state the edge was originally. Figure 3.28(b) shows that if the tadpole is initially in the state 1 , after | i

82 a) b) a a

0 a 1 a

Figure 3.28: The Abelian version of moving a good tadpole through a plaquette edge. The Abelian F -move used here is shown in Fig. 3.18. The two diﬀerent case where the head qubit is in the state (a) 0 and (b) 1 are discussed separately here. | i | i pulling it through an edge in the state a , the tadpole is in the state a¯ (wherea ¯ denotes the | i | i NOT of a). It follows that the pulling a bad tadpole in the state thorough an edge is equivalent |−i to performing a Z operation on that edge, since we can write down the total wavefunction of the tadpole and the edge as

0 a edge 0 ( a a¯ ) a edge |−i| i·| i −→ | i | i−| i ·| i

=(Z) 0 a edge (3.23) a | i|−i · | i 1 0 where Z = . Note also that since the state of the tail of the tadpole is always 0 in 0 1 | i − this case the process of moving the tail within a plaquette is trivial in the Abelian case. If we then imagine moving one bad tadpole through the lattice to find another bad tadpole, they can then be combined and removed from the lattice. Figure 3.29(a) depicts a hypothetical situation in which two errors are detected in nearby plaquettes. To complete the error correction one needs to bring them back and annihilate them using the technique introduced above, i.e. using a sequence of F moves that pull one tadpole from its original plaquette through the lattice all the way to the plaquette containing the second error. By contrast, for the Abelian case, depicted in Fig. 3.29(b), the procedure is the familiar one whereby one need only Z operations applied to the edges crossed by the string connecting two errors. All of these operations can be performed in parallel. A crucial difference between the Abelian and non-Abelian case is apparent at this point. For two Abelian plaquette errors, we immediately know from the fusion rules that they will annihilate (this is true even if the errors were produced by different events). By contrast, in the non-Abelian

83 a) b)

Figure 3.29: (a) Two plaquette errors in the Abelian Levin-Wen model. (b) Moving one of the plaquette error to another is equivalent to applying a series of Z operations along the path.

case, when we bring two errors( , ) together we do not know a priori whether they will combine( ) to make an object with trivial topological charge. If the errors were indeed δ produced by the same event then this should be the case. If so, the qubit labeled δ at the end of the sequence of F -moves will be in the state 0 and the remaining tadpole qubits can simply be | i removed. However, if we do not know this is the case then we need to somehow measure the result of fusing the two errors. This can be done by measuring the qubit labeled δ. If the result is 1 | i we immediately know the two errors did not fuse into a state with trivial topological charge. To proceed, we can simply replace the four ancillary qubits connected to δ with a single qubit in the state 1, and then continue the error correction by searching for other nearby errors with which to fuse. However, even if we ﬁnd that qubit δ is in the state 0 , it is still not safe to simply remove | i the ancillary qubits. Only if these qubits are encoding a state with total topological charge 0 will such a removal result in no more errors. In this case, after removing the ancillary qubits, one can simply carry out another plaquette swap. If there was no error then everything is ﬁne. However, if there is an error one again has a bad plaquette which must be combined with some other hopefully nearby errors in order to fuse to something with trivial topological charge. In what follows we will present a more detailed discussion on how to fuse these errors in general cases.

84 Figure 3.30: Some of the vertex and plaquette errors detected in a round of error correction are moved using F and R moves until they are all entirely within one plaquette on a single edge. Using F moves these errors can then always be combined to form an error tree.

3.4 Fusing Errors

After following the procedures given above for measuring Qv and Bp for each vertex and each plaquette, the state of the system will in general have a form similar to that shown furthest to the right in Fig. 3.25(c). In this state, all the original vertices and plaquettes of the system are error free and the original vertex and plaquette errors are all now trapped in string ends and loops. The next step will require a decoding procedure to determine which errors should be fused in order to attempt to return the system to its ground state. Some of the diﬃculties associated with performing such decoding for non-Abelian models has been addressed in [25, 26]. Here we assume such a procedure exists and simply describe how the string ends and loops can be fused together and the resulting state measured in order to determine if the errors have fused back into the vacuum or whether they have left an error behind. The ﬁrst step is to use the procedures given in Sec. 3.3.1 and 3.3.3 to move the string ends and loops one would like to fuse through the lattice, bringing them inside a single plaquette, as shown in Fig. 3.30. Then, using a series of F moves, it is always possible to combine these errors into a single fusion tree, which we refer to as the error tree. Once the error tree is formed, the qubit connecting the tree to the plaquette edge, which we refer to as the root qubit, should be measured, since it represents the overall topological charge of the collected errors. If the measured result is the state 1 there is no way to remove the error | i tree, indicating that the result of fusing the errors has nontrivial topological charge, which means

85 Figure 3.31: If the root qubit is in the state 1 the errors have not fused into the vacuum. In this | i case one can replace the error tree with a single string end. However, as for (b), it is necessary to check if there is also a plaquette error. This can be done by swinging the string end out of the plaquette using an R move, and then performing a plaquette swap. The possible results of these measurements are shown in the ﬁgure.

we may have only collected part of the errors that should be fused into vacuum. However, if the result is 0 there is at least a chance that the errors have fused to a topologically trivial state. | i In either case, the natural next step is to ignore all the qubits in the error tree except the root qubit. For the case that the root qubit is in the state 1 this amounts to fusing all of the errors to | i obtain a single string end. However, in general it is also possible that the result of this fusion will have resulted in a plaquette error. In the thickened lattice picture, a parallel explanation would be that the collection of the error tree is equivalent to only one type (1, 0) vertex error, or one type (1, 0) vertex error together with a (1, 1) plaquette error, the former of which can be graphically

represented as tree while the later one is tree .

To determine this, one must braid the fused string end out of the plaquette using an R-move, insert a vacuum loop, and carry out a plaqutte swap to determine the state of the plaquette. If the result of the measurement is Bp = 1 which means there is no plaquette error, then the loop can be removed and the result of the error fusion is a single string end. If the result is Bp = 0 then the fusion has resulted in a string end and a loop(plaquette error). In either case, these errors will then have to be moved through the lattice and fused with other errors in order to return the system to its ground state.

86 Figure 3.32: If the root qubit is in the state 0 , then the rest of the tree qubits can be removed. | i However, it is still necessary to check for a plaquette error. This can be done via a plaquette swap and the two possibilities are shown. If one ﬁnds that Bp = 1 then the errors have fused into the vacuum. If one ﬁnds that Bp = 0 the errors have fused into a single plaquette error.

If the root qubit is in the state 0 then the errors may have fused back to the vacuum. But as | i for the case described above, in general it is possible that the there is still a type (1, 1) plaquette

error, which is tree in the thickened picture. To check for this one can, again, insert

a vacuum loop, carry out a plaquette swap, and measure the resulting loop states. If the result is

Bp = 1 then the errors in the error tree have fused back into the vacuum, as shown in Fig. 3.32.

However, if the result is Bp = 0 then the result of the error fusion is a single plaquette error. Again, this error will then have to be moved through the lattice in order to attempt to fuse it with other errors so that it returns to the vacuum.

3.5 Summary of Results

In conclusion, in this Chapter we ﬁrst reviewed earlier proposed methods [29] for measuring the

Qv and Bp operators in the Fibonacci Levin-Wen model. We then proposed a new signiﬁcantly improved scheme which not only measures these operators, but also corrects any vertex and plaquette errors found during the process. The new method is achieved basically by moving these errors with the aid of F -moves onto the so-called ancillary qubits we have added. For the vertex errors the ancillary qubits are the string ends, where for the plaquette errors they are the tadpole qubits. Moreover, the quantum circuits which realize such measurements and error corrections are

87 also constructed, with explicit gate counts given to quantify the complexity comparing to the old method and even that in the Abelian version of Levin-Wen models. It is important for the readers to know that all the circuit we proposed here are verified by computer simulations. Additionally, we notice that the plaquette swapping method will reduce to the conventional Abelian plaquette error correction if the Fibonacci anyons are replaced by the Abelian anyons. At the end, in order to proceed with the quantum error correction we illustrated the process of moving the detected vertex errors and plaquette errors. Another type of the basic anyon moves: R-move is required here when we need to move those vertex errors. Again, we notice that when applied to the Abelian lattice model the method of moving the plaquette error is exactly the same procedure that is used in Kitaev’s surface code. We have also discussed several different situations encountered with the final error correction step when we attempt to fuse parts of these errors.

88 CHAPTER 4

SUMMARY OF DISSERTATION AND FUTURE WORK

As a summary, in this Dissertation we have first reviewed the basic concepts of quantum computation and shown that the main difficulty to practically building a quantum computer is overcoming the quantum decoherence caused by the inevitable coupling of qubits to the environment. This coupling destroys the quantum information stored in the qubits and produces unrecoverable errors. The proposed method for fighting these errors and decoherence is known as quantum error correction. The basic idea of quantum error correction is to store quantum information in highly entangled states built out of many physical qubits in such a way that the information is “hidden” from the environment. For the particular kind of quantum error correction discussed in this Dis- sertation, defects or errors in these highly entangled states can be viewed as “excitations” of a lattice Hamiltonian. These excitations obey non-Abelian statistics, and hence we refer to this type of quantum error correction as ”non-Abelian quantum error correction.” The starting premise for the work presented here is that a ”conventional” quantum computer is at hand along with the ability to carry out quantum circuits in order to prepare the desired entangled states and correct any possible errors that may emerge. The main purpose of this Dissertation has been to explicitly develop the error correction protocol and the corresponding quantum circuits for carrying out this error correction. In the second chapter we introduced the ”Fibonacci Levin-Wen model,” the lattice model whose ground states define the highly entangled states used to store quantum information in non-Abelian quantum error correction. The excitations of the Fibonacci Levin-Wen model behave like a type of non-Abelian anyons called the Fibonacci anyons. We thus have first discussed the nature of the Fibonacci anyons by reviewing two of their fundamental mathematical properties, the F -move and the R-move, both of which play key roles in this Dissertation. We then showed that ”errors” corresponds to states in the quantum computer which are not in the Levin-Wen ground state, as determined by measuring local terms in the Levin-Wen Hamiltonian.

89 Chapter 3 then presented a complete set of procedures for measuring and correcting any errors that may exist in this theoretical model. There are two different types of errors that need to be dealt with, vertex errors and plaquette errors. To begin with, we have first reviewed the previous work in [29] which gave an initial approach to try to measure these errors (without giving any procedure for error correction). The measurement for the vertex errors is straightforward while for the plaquette errors it uses a so-called “plaquette reduction” method to reduce the target plaquette to a one-side plaquette which can easily be measured. We then focused on the main contribution of this Dissertation: a new method for both measuring and correcting the vertex and plaquette errors. Specifically, we have developed a new technique which could be called “error relocation.” Here the essential idea is to relocate the vertex and plaquette errors onto some extra qubits in such a way that the error measurement and correction can be done at the same time. For instance, to relocate the plaquette errors we envision adding extra qubits representing a one-sided plaquette in the Levin-Wen ground state to each full plaquette. Then any possible error in the full plaquette can be relocated by carrying out a “plaquette swapping” process, after which the full plaquette will be error free. At the end of this error relocation procedure, with vertex and plaquette errors relocated onto extra qubits, we then want to annihilate these errors from the quantum computer and complete the error correction. With that in mind we have developed the necessary procedures which can be used to move these isolated errors through the lattice so they can be gathered together and fused. We have then given a detailed discussion of how these vertex and plaquette errors can be fused and described the four possible results of their fusion. Many open problems related to the work in this Dissertation remain and are waiting to be solved in future research. For instance, one question is: once we have detected errors, how do we determine which group of these errors we attempt to fuse? And, in particular, what is the possibility that a given group of error can be fused back into vacuum? In other words, an error decoding protocol is needed. Moreover, to perform topological quantum computation using these (Fibonacci) quantum codes we will need to develop a general method for creating logical qubits (non-Abelian excitations) out of the Levin-Wen ground state. Finally, the fundamental question of whether non-Abelian quantum error correction is indeed a feasible route to quantum computation must be addressed. To do so it will be necessary to compare it in detail with the more conventional

90 Abelian error correction procedures, as in, e.g., the Kitaev surface code. A careful weighing of the advantages and disadvantages of the Abelian and non-Abelian codes is necessary before one actually chooses which of them to use to construct a functioning, scalable, quantum computer. This Dissertation represents a signiﬁcant ﬁrst step toward realizing this goal.

91 APPENDIX A

QUANTUM CIRCUITS TO VERIFY THE PENTAGON EQUATION

In this appendix we look back onto the pentagon equation and review a circuit construction method which was introduced in [29]. Originally the pentagon equation was brought out as a fundamental relationship in the anyon theory. Later in the Levin-models, as the F -move becomes an unitary operation that locally redraws the topological lattice configuration, the pentagon equation can be interpreted as the identity consisted of five F -moves acting on the qubits that makes part of the lattice. Given the quantum circuit for the F -move in Fig. 3.4, It would be a good practice to actually build the circuit which carries out the pentagon equation. Fig. A.1(a) shows a slightly reworked pentagon equation borrowed from Sec. 2.1.2. Here instead of two different paths we navigate the initial pattern(lattice) in a way that it goes through a cycle by reverse the F -moves in the bottom path. Note that after the cycle it does not fully go back to the original pattern. Instead, final pattern differs from it with the position of two qubit labeled 5 and 6 swapped. In short, the five consecutive F -moves acting on the qubits 1 to 7 is equivalent to a swap operation on this two qubits, which reflects in the quantum circuit shown in Fig. A.1(b). Again, the precondition for the equation to hold is that the vertex constraint Qv = 1 is satisfied for every vertex 1, 2, 5 , 3, 5, 6 { } { } and 4, 6, 7 as labeled by their associated qubits. { } One may also observe the fact that in the above pentagon equation, the target qubits for every of the F -moves are always qubit 5 and 6, while the other qubits only play the role of control qubits. It is then natural to consider a simplified version of the pentagon equation by setting these five effective control qubits to be in a particular state, for instance the state 1 . In such a way one can | i obtain a two-qubit circuit identity which non-trivially involves the F -rotation from the F -move, as shown below: F F = (A.1)

F F F SWAP

92 a) b) 1 2 3 4 1 c d b 5 6 2 d b c F 7 1 2 3 4 F 1 2 3 4 3 b c d 5 6 4 c d b 6 7 5 = 7 5 a e a e a

6 e a e a e swap F 1 2 3 4 1 2 3 4 F 6 F 6 7 d b c 5 5 7 7

Figure A.1: (a) The pentagon equation. The red qubits mark the target qubits where the F -moves are applied. At the end, the pattern goes back to itself only with qubit 5 and 6 swapped. (b) The quantum circuit which carries out the pentagon equation of (a). We use the box notation introduced in Chapter 3. The equation only holds if the vertex constraints are satisﬁed for all the vertices shown up in (a).

The two-qubit circuit identity is also called the simplified pentagon equation. There are five controlled-F rotation gate on the left hand side of the equation, with the control qubits alternate from one qubit to the other. The overall effect is equivalent to a SWAP gate acting on these two qubits. Similarly the simplified pentagon equation would inherit th same requirement on the vertex constraints. Yet note that even if both of the qubit 5 and 6 are in the state 0 , which violates | i the vertex constraint when the other qubits are in the state 1 , the circuits on both sides of Eq. A.1 | i all execute the identity operation. Therefore, the equation holds for all possible combinations of states of the qubits. In practice, Eq. A.1 can be used to calibrate the F -rotation by carefully tuning the F -moves until five consecutive controlled-F gates with alternating control qubits produce a SWAP gate. The reason is because the simplified pentagon equation in Eq. A.1 is the only non-trivial component of the full equation in Fig. A.1(b). It solely determines the value of the F -matrix. More details about this proof shall be found in Sec. 2.1.2.

93 APPENDIX B

QUANTUM CIRCUITS TO VERIFY THE HEXAGON EQUATION

a) b)

a -1 a b a T b a T : b = T-1 : b = T c c T c c T-1

Figure B.1: (a) The quantum circuit for the R-move deﬁned in Eq. 3.18. Both the qubit a and b undergoes an opposite twist against the qubit c. (b) The circuit for the R−1-move.

Similarly, for the other equation in the anyon theory, it is also possible to build up a quantum circuits that carries out the hexagon equation. In fact, in Sec. 3.3.1 the process where we move a string end among neighbouring plaquettes(Fig. 3.23) can be depicted as an exact application of the hexagon equation. More generally, starting from Fig. 2.11 if we relabel the anyons(qubits) as a α,b 2,c 1,d 3,e β , the original hexagon equation will become another diagram { → → → → → } where we move one “string end” around using F amd R’s, although in this case the “string end” is no longer ﬁxed in the state 1 . Again we reverse the bottom path so the whole process forms a | i cycle. Note that the last two R-moves become R−1-moves since now the string ends swing in the opposite direction due to the reverse, as shown in Fig. B.2(a). The three-qubit R-move we used here can be constructed in a similar way that was used in Fig. 3.21, only this time the state of the “string end” needs to be discussed as well. We follow the full equation of the R-move(Eq. 3.18 and Eq. 3.19) to build the quantum circuits for both the R and R−1-moves as shown in Fig. B.1. The T -rotation and its inverse is deﬁned in Eq. 3.20 as a half twist of the topological spin of the qubits.

94 a) b) 2 1 2 1 R 1 d a b b β F α β α F 3 3 2 a b c b

2 1 2 1 3 c c c a i⋅− diag[]θ β β e α α = α b a d a d a 3 3 α 1 α 1 β e b e c e c R 2 2 R β β 3 F 3

Figure B.2: (a) The hexagon equation on a vertex, which has the same topological structure with that in Eq. 2.11. (b) The quantum circuit identity which reﬂects (a). The box notations for the R and R−1-moves are given in Fig. B.1.

Fig. B.2(b) show the quantum circuit which that carries out the hexagon equation in Fig. B.2(a). Note that the circuit does not exactly equal to an identity matrix, instead it is equivalent to a diagonal matrix with the diagonal elements being of the form e−iθ. In other words, it allows “twists” among the qubits as a result of the R-moves involved. Like the pentagon equation, there is also a simpliﬁed version of the hexagon equation which is obtained by setting four of the qubits: 1, 2, 3,α to the state 1 . The resulting circuit is nothing else but a single-qubit rotation chain { } | i which is −iθ F T −1 F T −1 F T −1 = e (B.1)

Again, one would also envision that this simpliﬁed hexagon equation can be used to calibrate the R-matrix after the F -matrix is stabilized by using Eq. A.1. It is veriﬁed that the one-qubit circuit obtained in Eq. B.1 is indeed the only non-trivial and the simplest form of the hexagon equation as a circuit identity, the solution of which is exactly the R-move(composed by T and T −1) we used in Chapter 3.

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98 BIOGRAPHICAL SKETCH

Weibo Feng

EDUCATION

B.S. in Physics, July, 2008 University of Science and Technology of China, HeFei, China Major: Theoretical Physics

Ph.D. in Physics, expected August, 2015 Florida State University, Tallahassee Dissertation: Non-Abelian Quantum Error Correction Dissertation Advisor: Nicholas E. Bonesteel

PUBLICATION

Weibo Feng, N. E. Bonesteel, and D. P. DiVincenzo. Non-Abelian Errors in the Fibonacci Levin-Wen Model. in preparation.

CONFERENCE PRESENTATION

Weibo Feng, N. E. Bonesteel, and D. P. DiVincenzo. Simulating Anyon Interference to Measure the Levin-Wen Plaquette Operator. APS March Meeting Baltimore, Maryland March 18-22 (2013).

99 EXPERIENCES

Teaching Assistant 2008 - 2010 Florida State University, Tallahassee

Research Assistant 2010 - 2015 National High Magnetic Field Laboratory, Tallahassee

HONORS / AWARDS

1st prize award in the National physics Olympiad 2003 Outstanding award on The College Student Research Program 2007

100