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Arnab Das Bikas K. Chakrabarti (Eds.)
Quantum Annealing and
Related Optimization Methods
ABC
Editors
ArnabDas
BikasK.Chakrabarti
SahaInstituteofNuclearPhysics
CentreforAppliedMathematics
andComputationalScience
Bidhannagar1/AF
700064Kolkata,India
E-mail:[email protected]
[email protected]
ArnabDas,BikasK.Chakrabarti,QuantumAnnealingandRelatedOptimizationMethods,
Lect.NotesPhys.679(Springer,BerlinHeidelberg2005),DOI10.1007/b135699
LibraryofCongressControlNumber:2005930442
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Preface
Quantum annealing employs quantum fluctuations in frustrated systems or
networks to anneal the system down to its ground state or to its minimum
coststate,tuningthequantumfluctuationdowntozeroeventually.Oftenthis
can be more effective in multivariable optimization problems, over classical
annealing performed utilizing tunable thermal fluctuations. The effectiveness
comesfromthefactthatunlikeinclassicalannealing,wherethesystemscales
the individual barrier heights by utilizing thermal fluctuations, in quantum
annealing, fluctuations can help tunneling through these (even infinite but
narrow) barriers. Apart from the recent theoretical demonstrations, this has
been demonstrated experimentally.
In this book, we discuss the problems and the recent achievements in de-
tail. This book grew out of an international workshop on quantum annealing,
held in March 2004 in Kolkata under the auspices of the Centre for Applied
Mathematics and Computational Science, Saha Institute of Nuclear Physics,
India. With contributions from all the leading scientists/groups involved in
its development so far, this first ever book on quantum annealing is expected
to become an invaluable primer and also a guidebook for all researchers in
this important field.
Thebookisdividedintothreeparts.Inthefirstpart,tutorialmaterialsare
introduced.B.K.ChakrabartiandA.DasintroducethetransverseIsingmodel
andquantumMonteCarlotechniques,followingwhichmostofthetheoretical
studies on quantum annealing have been made so far. The decomposition
of exponential operators used for the Suzuki–Trotter classical mapping in
quantum Monte Carlo techniques is discussed in detail by N. Hatano and M.
Suzuki. Latest quantum Monte Carlo and other numerical investigations and
developmentsinquantumspinglassesarereviewedbyH.Rieger.Thequestion
of ergodicity and consequent replica symmetry restoration in quantum spin
glassesandferroelectricglasses,experimentalindicationsincluded,isreviewed
by J.-J. Kim. A. Fisher reviewes the theory of quantum systems coupled
to noisy condensed-phase environments and describes how to tailor response
functions so as to optimize the coherent evolution of the system.
VI Preface
In the next part, quantum annealing techniques are developed and em-
ployed. G. Aeppli and T.F. Rosenbaum describe the experimental realization
where the ground state of a glassy sample can be reached faster by tun-
ing the external field (inducing changes in the tunneling field) rather than
by tuning the temperature. D. Battaglia, L. Stella, O. Zagordi, G. Santoro,
and E. Tosatti discuss the effectiveness of quantum annealing algorithms in
solving hard computational problems such as the traveling salesman problem
or a satisfiability problem and also in solving some very simple illustrative
problems for a basic comparative study with thermal annealing. S. Suzuki
and M. Okada investigate the prospect of adiabatic quantum annealing us-
ing real-time quantum evolution. A. Das and B.K. Chakrabarti discuss the
application of quantum annealing in a kinetically constrained system and in
an infinite range quantum spin glass. J.-I. Inoue reviewes the applicability
of quantum annealing techniques in restoring informations and images after
transportation through corrupted channels.
In the last part some of the classical optimization studies are reviewed
and discussed. H. Rieger reviewes the classical algorithms for solving various
combinatorial optimization problems. P. Sen and P.K. Das discuss classical
annealing in the context of the ANNNI model and make a comparative study
with quantum annealing in the same system. V. Martin-Mayor reviewes the
problem of annealing and relaxation in the context of classical glasses and
supercooled liquids.
Withthesefirsthandanddetailedreviewsbythepoineersinthisfield,this
bookonananalogversionofquantumcomputation,wehope,willimmediately
inspire further research and development.
We are extremely grateful to all the contributors for excellent support
and cooperation. We are also grateful to J. Zittartz for his encouragement
regarding the publication of this lecture note volume.
Kolkata Arnab Das
May, 2005 Bikas K. Chakrabarti
Contents
Part I Tutorial: Introductory Material
Transverse Ising Model, Glass and Quantum Annealing
Bikas K. Chakrabarti, Arnab Das .................................. 3
1 Introduction ................................................. 3
2 Transverse Ising Model (TIM) ................................. 4
3 Mean Field Theory (MFT) .................................... 5
4 Dynamic Mode-Softening Picture............................... 8
5 Suzuki-Trotter Formalism ..................................... 9
6 Classical Spin Glasses: A Summary ............................. 12
7 Quantum Spin Glasses ........................................ 14
7.1 Models ................................................. 15
7.2 Replica Symmetry in Quantum Spin Glasses ................ 18
8 Quantum Annealing .......................................... 21
8.1 Multivariable Optimization and Simulated Annealing ........ 21
8.2 Ergodicity of Quantum Spin Glasses and Quantum Annealing . 22
8.3 Quantum Annealing in Kinetically Constrained Systems ...... 24
9 Summary and Discussions ..................................... 25
10 Appendix ................................................... 26
References ...................................................... 35
Finding Exponential Product Formulas
of Higher Orders
Naomichi Hatano, Masuo Suzuki................................... 37
1 Introduction ................................................. 37
2 Why Do We Need the Exponential Product Formula?............. 38
3 Why is the Exponential Product Formula
a Good Approximant? ........................................ 40
3.1 Example: Spin Precession................................. 42
3.2 Example: Symplectic Integrator ........................... 43
4 Fractal Decomposition ........................................ 45
VIII Contents
5 Time-Ordered Exponential .................................... 47
6 Quantum Analysis – Towards the Construction
of General Decompositions .................................... 50
6.1 Operator Differential ..................................... 51
6.2 Inner Derivation......................................... 52
6.3 Differential of Exponential Operators....................... 54
6.4 Example: Baker-Campbell-Hausdorff Formula ............... 55
6.5 Example: Ruth’s Formula................................. 57
6.6 Example: Perturbational Composition ...................... 59
7 Summary.................................................... 61
8 Appendix ................................................... 62
References ...................................................... 67
Quantum Spin Glasses
Heiko Rieger .................................................... 69
1 Introduction ................................................. 69
2 Random Transverse Ising Models in Finite Dimensions ............ 70
2.1 Random Transverse Ising Chain
and the Infinite Randomness Fixed Point ................... 71
2.2 Diluted Ising Ferromagnet in a Transverse Field ............. 75
2.3 Higher Dimensional Random Bond Ferromagnets
in a Transverse Field ..................................... 76
2.4 Quantum Ising Spin Glass in a Transverse Field ............. 78
3 Mean-Field Theory for Quantum Ising Spin Glasses............... 79
3.1 Quantum Phase Transition................................ 79
3.2 Dissipative Effects ....................................... 83
3.3 Off Equilibrium Dynamics ................................ 86
4 Heisenberg Quantum Spin Glasses.............................. 88
4.1 Finite Dimensions ....................................... 88
4.2 Mean-Field Model ....................................... 95
References ...................................................... 97
Ergodicity, Replica Symmetry, Spin Glass
and Quantum Phase Transition
Jong-Jean Kim ..................................................101
1 Introduction .................................................101
2 Overview of Spin Glass........................................102
3 Ergodicity...................................................107
4 Replica Symmetry............................................111
5 Glass Transition..............................................116
6 Quantum Phase Transition ....................................121
7 Quantum Spin Glass..........................................123
References ......................................................126
Contents IX
Decoherence and Quantum Couplings
in a Noisy Environment
Andrew Fisher...................................................131
1 Qubits Coupled to a Bath .....................................131
1.1 Quantum Operations.....................................131
1.2 Examples...............................................133
1.3 The Lindblad Equation...................................134
1.4 The Markovian Weak-Coupling Limit ......................137
1.5 Good Qubits – the Rotating Wave Approximation ...........140
1.6 The Quantum Optical Master Equation ....................142
1.7 Bad Qubits–Quantum Brownian Motion ....................144
1.8 Simplifications for a Harmonic Environment.................145
1.9 Brownian Motion with Ohmic Dissipation...................147
1.10 The Fluctuation-Dissipation Theorem and the Link
Between Coherent and Incoherent Evolution ................149
1.11 Irreducible Decoherence and Decoherence-Free Subspaces .....151
2 Scaling Transformations
for Partially Coherent Dynamics ...............................151
2.1 Scaling for Thermodynamic Properties .....................151
2.2 Scaling the Liouvillian....................................152
3 Quantum Gates via Optical Excitation..........................153
3.1 Advantages of Localised States ............................153
3.2 The UCL Project ........................................153
4 Conclusions..................................................154
References ......................................................154
Part II Quantum Annealing: Basics and Applications
Experiments on Quantum Annealing
Gabriel Aeppli, Thomas F. Rosenbaum..............................159
1 Introduction .................................................159
2 System with a Complex Free Energy Surface
and Tuneable Quantum Fluctuations ...........................160
3 Demonstration of Domain Wall Tunnelling
as the Dominant Mechanism
for Low Temperature Magnetic Relaxation ......................163
4 Comparing Quantum and Thermal ‘Computations’ ...............165
5 Conclusions..................................................168
References ......................................................169
Deterministic and Stochastic Quantum Annealing Approaches
Demian Battaglia, Lorenzo Stella, Osvaldo Zagordi,
Giuseppe E. Santoro and Erio Tosatti ..............................171
1 Introduction .................................................171
X Contents
2 Deterministic Approaches on the Continuum.....................173
2.1 The Simplest Barrier: A Double-Well Potential ..............175
2.2 Other Simple One-Dimensional Potentials with Many Minima .181
3 Role of Disorder, and Landau-Zener Tunneling ...................183
4 Path Integral Monte Carlo Quantum Annealing ..................184
4.1 Path Integral Monte Carlo: Introduction....................184
4.2 PIMC-QA Applied to Combinatorial Optimization Problems ..186
4.3 PIMC-QA and 3-SAT: Lessons from a Hard Case ............192
4.4 PIMC-QA of a Double-Well: Lessons from a Simple Case .....199
5 Beyond Naive Local Search ....................................201
5.1 Focusing in 3-SAT and GFMC Quantum Annealing ..........201
5.2 Message-Passing Optimization.............................202
6 Summary and Conclusions.....................................203
References ......................................................204
Simulated Quantum Annealing
by the Real-time Evolution
Sei Suzuki, Masato Okada.........................................207
1 Introduction .................................................207
2 Formulation and Mechanism of Quantum Annealing ..............210
2.1 Formulation of Quantum Annealing ........................210
2.2 Adiabatic Evolution of Quantum States ....................212
3 Residual Energies ............................................223
3.1 Simulations for Small-Sized Problems ......................223
3.2 Analytic Considerations ..................................228
3.3 Discussion ..............................................229
4 A method of Simulation for Large-Sized Problems ................231
4.1 Real-Time Evolution by Means of DMRG...................232
4.2 Results of Simulation.....................................234
4.3 Comments ..............................................236
5 Conclusion ..................................................236
References ......................................................237
Quantum Annealing of a ±J Spin Glass
and a Kinetically Constrained System
Arnab Das, Bikas K. Chakrabarti ..................................239
1 Introduction .................................................239
2 Quantum Annealing of ±J Ising Spin Glass
at Infinite Dimension .........................................241
2.1 Model..................................................241
2.2 The Zero Temperature Quantum Monte Carlo Method Used ..242
2.3 Results and Discussions ..................................247
3 Quantum Annealing
in a Kinetically Constrained System ............................248
3.1 Model..................................................250
