Quantum Algorithms for Scientific Computing and Approximate Optimization

264 Pages·2017·1.27 MB·English

by  Stuart Andrew Hadfield

Quantum Algorithms for Scientific Computing and
Approximate Optimization
Stuart Andrew Hadfield
Submittedinpartialfulfillmentofthe
requirementsforthedegree
ofDoctorofPhilosophy
intheGraduateSchoolofArtsandSciences
COLUMBIA UNIVERSITY
2018
(cid:13)c 2018
StuartAndrewHadfield
AllRightsReserved
ABSTRACT
Quantum Algorithms for Scientific Computing and
Approximate Optimization
Stuart Andrew Hadfield
Quantum computation appears to offer significant advantages over classical computation and this
hasgeneratedatremendousinterestinthefield. Inthisthesiswestudytheapplicationofquantum
computerstocomputationalproblemsinscienceandengineering,andtocombinatorialoptimization
problems. Weoutlinetheresultsbelow.
Algorithms for scientific computing require modules, i.e., building blocks, implementing ele-
mentary numerical functions that have well-controlled numerical error, are uniformly scalable and
reversible, andthatcanbeimplementedefficiently. Wederivequantumalgorithmsandcircuitsfor
computingsquareroots,logarithms,andarbitraryfractionalpowers,andderiveworst-caseerrorand
cost bounds. We describe a modular approach to quantum algorithm design as a first step towards
numericalstandardsandmathematicallibrariesforquantumscientificcomputing.
A fundamental but computationally hard problem in physics is to solve the time-independent
Schro¨dinger equation. This is accomplished by computing the eigenvalues of the corresponding
Hamiltonianoperator. Theeigenvaluesdescribethedifferentenergylevelsofasystem. Thecostof
classical deterministic algorithms computing these eigenvalues grows exponentially with the num-
ber of system degrees of freedom. The number of degrees of freedom is typically proportional to
thenumberofparticlesinaphysicalsystem. Weshowanefficientquantumalgorithmforapproxi-
matingaconstantnumberoflow-ordereigenvaluesofaHamiltonianusingaperturbationapproach.
WeapplythisalgorithmtoaspecialcaseoftheSchro¨dingerequationandshowthatouralgorithm
succeeds with high probability, and has cost that scales polynomially with the number of degrees
offreedomandthereciprocalofthedesiredaccuracy. Thisimprovesandextendsearlierresultson
quantumalgorithmsforestimatingthegroundstateenergy.
Weconsiderthesimulationofquantummechanicalsystemsonaquantumcomputer. Weshow
anoveldivideandconquerapproachforHamiltoniansimulation. UsingtheHamiltonianstructure,
wecanobtainfastersimulationalgorithms. ConsideringasumofHamiltonianswesplittheminto
groups, simulate each group separately, and combine the partial results. Simulation is customized
totakeadvantageofthepropertiesofeachgroup,andhenceyieldrefinedboundstotheoverallsim-
ulationcost. Weillustrateourresultsusingtheelectronicstructureproblemofquantumchemistry,
whereweobtainsignificantlyimprovedcostestimatesundermildassumptions.
Weturntocombinatorialoptimizationproblems. Animportantopenquestioniswhetherquan-
tum computers provide advantages for the approximation of classically hard combinatorial prob-
lems. A promising recently proposed approach of Farhi et al. is the Quantum Approximate Opti-
mization Algorithm (QAOA). We study the application of QAOA to the Maximum Cut problem,
and derive analytic performance bounds for the lowest circuit-depth realization, for both general
and special classes of graphs. Along the way, we develop a general procedure for analyzing the
performance of QAOA for other problems, and show an example demonstrating the difficulty of
obtainingsimilarresultsforgreaterdepth.
We show a generalization of QAOA and its application to wider classes of combinatorial opti-
mization problems, in particular, problems with feasibility constraints. We introduce the Quantum
AlternatingOperatorAnsatz,whichutilizesmoregeneralunitaryoperatorsthantheoriginalQAOA
proposal. Our framework facilitates low-resource implementations for many applications which
maybeparticularlysuitableforearlyquantumcomputers. Wespecifydesigncriteria, anddevelop
asetofresultsandtoolsformappingdiverseproblemstoexplicitquantumcircuits. Wederivecon-
structionsforseveralimportantprototypicalproblemsincludingMaximumIndependentSet,Graph
Coloring,andtheTravelingSalesmanproblem,andshowappealingresourcecostestimatesfortheir
implementations.
Table of Contents
ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction 1
1.1 QuantumAlgorithmsandCircuitsforScientificComputing . . . . . . . . . . . . . 2
1.2 ApproximatingGroundandExcitedStateEnergiesonaQuantumComputer . . . . 3
1.3 DivideandConquerHamiltonianSimulation . . . . . . . . . . . . . . . . . . . . 5
1.4 QuantumApproximateOptimization . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 QuantumAlgorithmsandCircuitsforScientificComputing 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Reciprocal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 SquareRoot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 2k-Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.4 Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.5 FractionalPower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 NumericalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 ApproximatingGroundandExcitedStateEnergiesonaQuantumComputer 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 ProblemDefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
i
3.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.1 PreliminaryAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.2 AlgorithmDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Application: Time-IndependentSchro¨dingerEquation . . . . . . . . . . . . . . . . 49
3.5.1 SetofTrialEigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5.2 FiniteDifferenceDiscretization . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.3 AlgorithmforExcitedStateEnergies . . . . . . . . . . . . . . . . . . . . 54
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 DivideandConquerApproachtoHamiltonianSimulation 64
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.1 ProblemDefinitionandBackground . . . . . . . . . . . . . . . . . . . . . 66
4.1.2 DivideandConquerSimulation . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.3 OverviewofMainResults . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 PreliminaryAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.1 DiscardingSmallHamiltonians . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.2 RecursiveLie-TrotterFormulas . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.3 CombiningDifferentSimulationMethods . . . . . . . . . . . . . . . . . . 77
4.3 DivideandConquerSplittingFormulas . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1 Algorithm1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.2 Algorithm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3.3 SelectingtheOrderoftheSplittingFormulas . . . . . . . . . . . . . . . . 85
4.3.4 Speedup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4 ApplicationtoQuantumChemistry . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.4.1 ElectronicHamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.4.2 SimulationCost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5 QuantifyingthePerformanceofLow-DepthQAOA 99
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
ii
5.2.1 ApproximateOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2.2 QuantumApproximateOptimizationAlgorithm(QAOA) . . . . . . . . . . 104
5.3 UnconstrainedOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.1 QuadraticUnconstrainedBinaryOptimization . . . . . . . . . . . . . . . 110
5.4 MaximumCut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.4.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4.2 PauliSolverAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4.3 Triangle-FreeGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4.4 GeneralGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.5 Depth-TwoQAOAfortheRingofDisagrees . . . . . . . . . . . . . . . . 122
5.4.6 WeightedMaximumCut . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4.7 DirectedMaximumCut . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6 QuantumApproximateOptimizationwithHardandSoftConstraints 129
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.2 TheQuantumAlternatingOperatorAnsatz . . . . . . . . . . . . . . . . . . . . . . 132
6.2.1 DesignCriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2.2 SimultaneousandSequentialMixers . . . . . . . . . . . . . . . . . . . . . 137
6.2.3 ConstraintSatisfactionviaCommutingOperators . . . . . . . . . . . . . . 139
6.3 ToolkitforQuantumOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.4 MappingsonBits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.4.1 MaxIndependentSet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.5 Mappingsonk-Dits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.5.1 Maxk-Colorability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.5.2 Maxk-ColorableInducedSubgraph . . . . . . . . . . . . . . . . . . . . . 155
6.5.3 MinChromaticNumber . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.6 MappingsonPermutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.6.1 TravelingSalesmanProblem . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.6.2 SingleMachineScheduling . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.7 DiscussionandConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . 164
iii
7 ConclusionsandFutureWork 168
Bibliography 171
AppendixA ABriefOverviewofQuantumComputation 194
A.1 QuantumMechanicsofQuantumComputation . . . . . . . . . . . . . . . . . . . 195
A.2 QuantumComputationalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
AppendixB ToolkitforQuantumOptimization 203
B.1 Representingn-bitFunctionsasDiagonalHamiltonians . . . . . . . . . . . . . . . 203
B.1.1 BooleanFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
B.1.2 Pseudo-BooleanFunctionsandConstraintSatisfactionProblems . . . . . . 209
B.2 ControlledHamiltonianEvolution . . . . . . . . . . . . . . . . . . . . . . . . . . 210
AppendixC QuantumAlgorithmsforScientificComputing 213
AppendixD DivideandConquerHamiltonianSimulation 227
D.1 ReviewofHigh-OrderSplittingFormulas . . . . . . . . . . . . . . . . . . . . . . 227
D.2 DivideandConquerHamiltonianSimulation . . . . . . . . . . . . . . . . . . . . 230
AppendixE QuantumApproximateOptimization 242
iv
List of Figures
2.1 Ann-qubitfixed-precisionrepresentationofanumberw ≥ 0onaquantumregister. 13
2.2 Elementary module using fixed-precision arithmetic to implement exactly res ←
xy+z forx,y,andz. Notethatregistersizes,ancillaregisters,andtheirvaluesare
notindicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
√
2.3 Block diagram of the overall circuit computing w. Two stages of Newton’s iter-
ation using the functions g and g are applied s and s times respectively. The
1 2 1 2
first stage outputs xˆ ≈ 1, which is then used by the second stage to compute
s1 w
√
yˆ ≈ √1 ≈ w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
s2 xˆs1
2.4 Aquantumcircuitcomputingtheinitialstate|xˆ (cid:105) = |2−p(cid:105),for|w(cid:105)givenbynbits
0
of which m are for its integer part, where p ∈ N and 2p > w ≥ 2p−1. Here we
have taken b ≥ n−m. This circuit is used in step 4 of Algorithm 1 SQRT. Each
horizontal line (“wire”) represents a qubit. This circuit consists of controlled-not
(CNOT)andcontrolled-controlled-not(Toffoli)gates. Eachcrossedcircleindicates
atargetqubit,controlledbythequbit(s)indicatedbyblackdots. Whitedotsindicate
invertedcontrol. SeeAppendixAforareviewofsomeimportantquantumgates. . 20
(cid:12) (cid:69)
2.5 A quantum circuit computing the state |yˆ0(cid:105) = (cid:12)(cid:12)2(cid:98)q−21(cid:99) , for 0 < xˆs < 1 given
by b bits, where q ∈ N and 21−q > xˆ ≥ 2−q. This circuit is used in step 10 of
s
Algorithm1SQRT.Itisforthecaseofevenb;asimilarcircuitfollowsforoddb. . 21
2.6 Overall circuit schematic for approximating lnw. The gate f(tˆ ) outputs yˆ =
p p
(tˆ −1)− 1(tˆ −1)2. Oncez isobtainedtheapproximationoflnw iscomputed
p 2 p p
bytheexpressionz +(p−1)r,whererapproximatesln2withhighaccuracyand
p
p−1isobtainedfromaquantumcircuitastheoneshowninFig.2.8. . . . . . . . 24
v
2.7 For w ≥ 1 this quantum circuit computes |x(cid:105) where x is an m bit number x ∈
[21−m,1]. Forw ≥ 2wesetx = 21−p,wherep−1 = (cid:98)log w(cid:99) ≥ 1. For1 ≤ w < 2
2
we set x = 1. Thus m bits are needed for the representation of x, with the first bit
x(0)denotingitsintegerpartandalltheremainingbitsx(−1),...,x−(m−1)denoting
its fractional part. This circuit is used in steps 6 – 10 of Algorithm 3 LN to derive
x = 21−p soonecanimplementtheshiftofwintermsofmultiplicationbetweenw
andx,i.e.,w = wx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
p
2.8 Example of a quantum circuit computing p − 1 ≥ 0 required in the last step of
Algorithm3LN.Theinputtothiscircuitisthestate|x(cid:105)computedinFig.2.7where
x = 2−(p−1). Recallthatmbitsareusedtostorex,andclearly(cid:100)log m(cid:101)bitssuffice
2
to store p−1 exactly. In this example, m = 9. It is straightforward to generalize
thiscircuittoanarbitrarynumberm. . . . . . . . . . . . . . . . . . . . . . . . . 27
2.9 Example of a quantum circuit computing the positive integer k such that 2kw ≥
1 > 2k−1w,for0 < w < 1withn−mbitsafterthedecimalpoint. Inthisexample,
n−m = 8. It is straightforward to generalize this circuit to arbitrary numbers of
bits. ThiscircuitisusedinAlgorithm5FractionalPower2. . . . . . . . . . . . . . 31
2.10 Semilog plot showing the error of Algorithm 1 SQRT versus the number of preci-
sion bits b. The solid blue line is a plot of the worst-case error of Theorem 1, for
n = 2m = 64. The three data sets represent the absolute value of the difference
√
between Matlab’s floating point calculation of w and our algorithm’s result for
threedifferentvaluesofw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.11 Semilog plot showing the error of Algorithm 3 LN versus the parameter (cid:96) which
controls the accuracy of the result. The solid blue line shows the worst-case error
of Theorem 3, for n = 2m = 64, b = max{5(cid:96),25}. The three data sets represent
the absolute value of the difference between Matlab’s floating point calculation of
ln(w)andouralgorithm’sresultforthreedifferentvaluesofw. . . . . . . . . . . . 34
vi