114 Pages·2016·1.81 MB·English

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Florida State University Libraries 2015 Non-Abelian Quantum Error Correction Weibo Feng Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES NON-ABELIAN QUANTUM ERROR CORRECTION By WEIBO FENG A Dissertation submitted to the Department of Physics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy 2015 Copyright c 2015 Weibo Feng. All Rights Reserved. (cid:13) 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 TheGraduateSchoolhasveriﬁedandapprovedtheabove-namedcommitteemembers, andcertiﬁes 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 List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 Introduction 1 1.1 Classical and Quantum Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 From Bits to Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Quantum Gates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Brief Overview of Dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Fibonacci Levin-Wen Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3 Non-Abelian Quantum Error Correction 45 3.1 Previous Work on Measuring Q and B . . . . . . . . . . . . . . . . . . . . . . . . . 45 v p 3.1.1 Vertex Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Vertex Errors on String Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.2 Plaquette Swapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.3 Plaquette Swapping: Abelian Case . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3 Moving Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.1 Moving Vertex Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.3.2 Plaquette Errors Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3.3 Moving Plaquette Errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.4 Fusing Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.5 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 does nothing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 quantumgates. (a) Acontrolled-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 qubitsarebothinthestate 1 , otherwiseitdoesnothing. (b)Acontrolled-controlled- | i NOTgatewhichisalsoknownasaToﬀoligate. (c)Acontrolled-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 ArecursivecalculationshowsthatthetotalquantumdimensionofnFibonaccianyons 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 the original and the rotated wave functions is the so called R-matrix Rab . . . . . . . 18 c 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 θ . . . . . . . . . . . . . . . . . . . . . 21 a 2.6 The pentagon equation which tells the story of two equivalent path of basis changing. Startingfromthebasistotheveryleft,onecanendatthebasistotheveryrighteither 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 everybasiscanbethestartingpointortheendpoint. Therearealwaystwoequivalent 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 symme- tries underneath its structure. The dashed red lines mark the symmetric lines of the equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.9 TheF-moveforFibonaccianyonsinitseverycomponent. Foreachfusionpattern(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 lnǫ 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 AnarrayofFibonaccianyonsthataresuitablefortheLevin-Wenmodels. Ahexagonal lattice is formed by placing edges on each of the particles. We deﬁne the Q operator v on the three qubits that converge at the same vertex v. And the B operator on the p 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 Q . (b) Lattice conﬁgurations that will have Q = 1. (c)Lattice conﬁgu- v v rations that violates the vertex constraints, hence have Q = 0. . . . . . . . . . . . . . 36 v 2.16 (a) An example of the Q ground state, in which the typical conﬁgurations are “loops v that allows branches”. (b) Excited states are created by breaking the branching rules. As indicated by the red mark. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 B onto the lattice is equivalent to adding “vacuum strings”(dashed circles) p around those punctures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.18 Encoding a logical qubit by cutting a defect(a “hole”) from the lattice. In this case one stops to measure the B values of the 10 plaquettes inside the “hole”, as well as p the Q values of the vertices that have been wiped out. In that sense we obtain a v degreeoffreedomfromneglectingtheseoperators. LikethedefectsinKitaev’ssurface 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 Q operator deﬁned on the vertex v 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) are marked out. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 A rebuild version of Fig. 3.3 which better exhibits the lattice deformation caused by the F-move in the Fibonacci codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 is deﬁned in Eq. 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 the same edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 B = 1. The matrix S is given in the text. . . . . . . . . . . . . . . . . . . . . . 55 p 3.8 AquantumcircuitwhichcanbeusedtomeasureB foraoneloopplaquette. Oneex- p trasyndrome qubitinitializedtothe state 0 isneeded toperformthenon-demolition | i measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.9 Quantum circuit which can be used to measure B for the Fibonacci code on a hexag- p onal 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 Q measurements for v each vertex. The remaining string ends represented by thick lines are in the state 1 | i and correspond to vertex errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 B value for the target is unknown and p ix

fundamental Toffoli gate, a gate which is equivalent to a controlled-controlled-NOT gate, shown in Fig. 1.4(b). One of the most promising quantum error correcting codes are the so-called surface codes [7]. 01e′ = F01e.

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