Non-Abelian Quantum Error Correction

114 Pages·2016·1.81 MB·English

by  Weibo Feng

Florida State University Libraries
2015
Non-Abelian Quantum Error Correction
Weibo Feng
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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