Table Of ContentSpringerBriefs in Mathematics
Clarice Dias de Albuquerque
Eduardo Brandani da Silva · Waldir Silva Soares Jr.
Quantum Codes for
Topological Quantum
Computation
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Clarice Dias de Albuquerque •
Eduardo Brandani da Silva • Waldir Silva Soares Jr.
Quantum Codes for
Topological Quantum
Computation
ClariceDiasdeAlbuquerque EduardoBrandanidaSilva
ScienceandTechnologyCenter MathematicsDepartment
FederalUniversityofCariri StateUniversityofMaringa
JuazeirodoNorte,Ceará,Brazil Maringá,Paraná,Brazil
WaldirSilvaSoaresJr.
Mathematics
FederalTechnologicalUniversityofParaná
PatoBranco,Paraná,Brazil
ISSN2191-8198 ISSN2191-8201 (electronic)
SpringerBriefsinMathematics
ISBN978-3-031-06832-4 ISBN978-3-031-06833-1 (eBook)
https://doi.org/10.1007/978-3-031-06833-1
MathematicsSubjectClassification:81P68,81P70,94B05,94B10,94B12,94B15
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022
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Contents
1 Introduction .................................................................. 1
1.1 HistoricalSummary ................................................... 1
1.2 TheQuantumProperties............................................... 4
1.2.1 SuperpositionofStates ...................................... 5
1.2.2 Entanglement................................................. 6
1.3 PrincipleofQuantumErrorCorrection............................... 7
1.3.1 StabilizerCodes.............................................. 9
1.4 QuantumBounds ...................................................... 10
1.5 CodifyingwithTopology.............................................. 11
2 ReviewofMathematicalConcepts ........................................ 15
2.1 ClassicalError-CorrectingCodes..................................... 16
2.1.1 BasicDefinitions............................................. 16
2.2 BlockCodes ........................................................... 19
2.2.1 LinearCodes ................................................. 20
2.3 LinearAlgebra......................................................... 24
2.3.1 QuantumBit.................................................. 24
2.3.2 MatricesandOperators...................................... 26
2.4 QuantumInformationandQuantumComputation................... 33
2.5 AGlimpseofQuantumMechanics................................... 33
2.5.1 Postulates..................................................... 33
2.5.2 QuantumGates............................................... 35
2.6 IntroductiontoQuantumError-CorrectingCodes ................... 36
2.6.1 The3-QubitQuantumCode................................. 37
2.6.2 ShorCode .................................................... 40
2.7 QuantumError-CorrectionCriterion ................................. 42
2.8 CSSCodes ............................................................. 43
2.9 StabilizerQuantumCodes............................................. 44
2.9.1 Anti-commutation............................................ 45
2.9.2 StabilizerGroup.............................................. 46
2.9.3 StabilizerCodeandExamples............................... 47
vii
viii Contents
2.10 HyperbolicGeometry.................................................. 49
2.10.1 IsometriesoftheHyperbolicPlane ......................... 52
2.10.2 RegularTessellations ........................................ 53
3 TopologicalQuantumCodes................................................ 55
3.1 ToricCodes ............................................................ 56
3.1.1 ToricCodesfromtheHomologyPointofView............ 59
3.1.2 CorrectionatthePhysicalLevel ............................ 62
3.2 ProjectivePlaneandQuantumCodes ................................ 63
3.3 OtherToricCodes ..................................................... 64
3.3.1 PolyominoQuantumCodes ................................. 66
3.4 HyperbolicTopologicalQuantumCodes............................. 70
(cid:2)
3.4.1 GenerationofaSurfacefromaPolygonP ................ 71
3.4.2 Constructions of Hyperbolic Topological
QuantumCodes.............................................. 77
4 ColorCodes .................................................................. 87
4.1 QuantumColorCodes................................................. 87
4.2 ColorCodesonCompactSurfaces ................................... 90
4.3 ColorCodesonSurfaceswithBoundary............................. 96
5 TheInterplayBetweenColorCodesandToricCodes................... 103
5.1 Introduction............................................................ 103
5.2 QuantumDoubleModels.............................................. 104
5.2.1 ToricCode.................................................... 105
5.2.2 ColorCodes.................................................. 107
5.3 ColorCodeEquivalencetoTwoCopiesofToricCodes............. 110
Bibliography...................................................................... 113
Chapter 1
Introduction
Thisbookisintendedtobeanintroductorymaterialontopologicalquantumcodes,
with the necessary elements from a mathematical and engineering point of view,
aimed at postgraduate or undergraduate students at more advanced level, in the
coursesofMathematics,Engineering,orComputing.
Asanintroductiontothisbook,thischapterpresents,ingeneralterms,thecentral
concepts of quantum computing and how it was developed through a historical
overview.Then,theerror-correctingquantumcodemodelsarehighlighted,uptothe
topologicalquantumcodes,accordingtothechronology,thusgivinganoverviewof
thisresearchareaandwhatwillbecoveredduringtheotherchaptersofthebook.
A review of the main concepts and properties of quantum mechanics that are
importantforquantumcomputation,aswellasthenecessaryalgebraicstructure,is
giveninChap.2.Alsointhesamechapter,thetheoryoferror-correctingquantum
codesandthemaincodesthatgaverisetothisareaofresearcharepresentedinmore
depth.
Chapter 3 is devoted to topological quantum codes (or surface codes), from
Kitaev’s toric code to generalizations on hyperbolic surfaces, along with the
mathematicalstructuresnecessaryfortheirconstruction.Chapter4introducescolor
codesaswellastheirgeneralizationtohyperbolicsurfaces.Finally,Chap.5aimsto
showaconnectionsbetweentoriccodesandcolorcodes.
1.1 HistoricalSummary
In the firsthalf of the twentieth century, as studies on atoms deepened, the known
physicsuntilthenbecameinefficientinexplainingsomephenomena,givingriseto
theemergenceofquantumphysics.
In1982,Feynmanraisedquestionslike“Canphysicsbesimulatedbyauniversal
computer?”inhislandmarkarticle“Simulatingphysicswithcomputers”[47],and
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 1
C.D.deAlbuquerqueetal.,QuantumCodesforTopologicalQuantumComputation,
SpringerBriefsinMathematics,https://doi.org/10.1007/978-3-031-06833-1_1
2 1 Introduction
thuslaunchedforthefirsttimetheideaofacomputerbasedonquantumproperties.
Hewasinterestedintheexactsimulationproblemofquantumphysics,whichdoes
notseemtobepossibletocarryoutonaclassicalcomputer.
In [42], physicist David Deutsch described a quantum generalization of the
class of Turing machines, the universal quantum computer. His conception of a
quantum computer could, in principle, be constructed and have properties that no
Turing machine would be able to reproduce, such as “quantum parallelism”—the
possibility of evaluating functions f(x) for different values of x simultaneously.
Deutsch’s conjecture that a universal computer is sufficient to efficiently simulate
anarbitraryphysicalsystemhasnotyetbeenprovenordisproved.
Among many important results of David Deutsch in the quantum theory of
computation, the quantum algorithm of Deutsch–Jozsa stands out, together with
Richard Jozsa in [43], as one of the first examples of a quantum algorithm that is
exponentially faster than any possible classical deterministic algorithm. However,
the problem that this algorithm solves is not something so practical that it could
arousemoreinterest.
Some quantum algorithms showed a slight advantage over classical computers,
butaresultobtainedbymathematicianPeterShorin1994gavestrengthtotheidea
that quantum computers would in fact be more efficient than classical computers
and boosted research in this direction. Shor’s algorithm became the most famous
result in terms of the speed of a quantum computer because it solves a practical
problem much faster than previously known algorithms. Shor demonstrated that a
quantumcomputercouldfactoranumberwithndigitsusingquantumpropertiesina
numberofcomputationalstepsthatgrowspolynomiallybyn,[90].Onconventional
computers,thebestknownalgorithmsrequireexponentiallygrowingresources.
The problem of factoring integers into prime components is well known in the
mathematicalworldandhasitsmainpracticalapplicationincryptography,suchas
indataencodingschemesusedbybanksoronwebsites.Thisexplainstheimpactof
Shor’swork.Duetotheenormoussuccessofthisalgorithm,manyresearchershave
launched themselves into the search for other efficient quantum algorithms. The
main discovery after Shor’s algorithm was made in 1996 by mathematician Lov
Grover.ThealgorithmproposedbyGroversearchesinadatabase,findingspecific
items that have predetermined properties, [58]. In the classic case, to specifically
find an item in a list of N items, where N grows exponentially as the problem
sizeincreases,about N operations areneeded.√Grover’s algorithm,usingquantum
properties,solvesthissameproblemwithonly N operations,whichrepresentsa
muchsmallernumberofsteps.Despiteofferingonlyaquadraticspeedgain,thisis
asignificantadvance, especiallysinceitisanalgorithmthatisoftenusedinother
problems. A feature of many quantum algorithms, which can be observed in Shor
andGrover’salgorithms,isquantumparallelism.
Other quantum algorithms based on the quantum Fourier transform, amplitude
amplification, quantum walks, and topological quantum field theory, among other
techniquesandideas,havebeenpresentedoverthelasttwodecades.However,itis
notonlyalongthislinethatresearchinquantumcomputingisdeveloped.
