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云计算-绝热量子计算与量子退火中的能级差值最优化问题.pdf
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云计算-绝热量子计算与量子退火中的能级差值最优化问题.pdf
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Abstract
Quantum annealing and adiabatic quantum computing have drawn much atten tion nowadays since the classical approaches
to optimization and computation problems become possible toobtainadvantageinquantumarea.Quantumannealingand
adiabatic quantum computing are probabilistic in nature, and from the related literature, increasing the minimum gap
between the ground state and the first excited state of the system during evolution can monotonically raise the success
probability. In this thesis, we introduce additional Hamiltonians to modify the evolution path between the initial and final
Hamiltonian and the refore the incr e ase in ga p becomes possible. We mainly concentrate on the gap maximizing problem
searching for the optimal schedule path, through which we expect to raise the minimum gap to one of the end points of
the evolution and achieve the best performance.
We investigate an op timization proble m related to recent quantum annealing implementations and provide strong
numerical evidence that there exists at least one local intermediate Hamiltonian that can always achieve best case perfor-
mance in terms of pushing the minimum gap to one of the end points of the evolution, and the optimal adiabatic schedule
also raise the success probability. Then we study the perfo r mance of random intermediate Hamiltonians on the minimum
gap and success probability, and our numerical results reveal that random intermediate Hamiltonians can increase the
success probability with a significant probability, but onlybyamodestamount.
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Contents
1Introduction 3
2Backgroundtheory 5
2.1 The representation of quantum mechanical system . . . . . . ........................ 6
2.2 Adiabatic Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............. 6
2.3 Perform computation using adiabatic evolution . . . . . . . ........................ 10
3Preliminariesofschedulepathoptimization 13
3.1 Problem specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................ 16
3.2 Details in solving minimax problem . . . . . . . . . . . . . . . . . . ................... 18
4Optimaladiabaticschedulessearchingfromconvexoptimization 21
4.1 Approximate continuous optimization formulation . . . . .......................... 22
4.2 Approximate discrete convex optimization problem . . . . ......................... 24
4.3 The proof of the convex adiabatic schedules optimizationproblem . . . . . . . . . . . . . . . . . . . . . 26
4.4 Optimal adiabatic schedules . . . . . . . . . . . . . . . . . . . . . . . .................. 31
4.5 Effect of optimization parameter on SPO . . . . . . . . . . . . . . ..................... 34
4.6 The investigation of the time complexity of schedule pathoptimizationalgorithm . . . . . . . . . . . . . 35
4.7 Relation to AQC success probability . . . . . . . . . . . . . . . . . .................... 36
4.8 Investigation on the matrix couplings under SPO interpolation . . . . . . . . . . . . . . . . . . . . . . . 37
4.9 Hard QUBO instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. 41
5RandomizedintermediateHamiltonians 51
5.1 Average case behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............... 52
5.2 Hard QUBO instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. 53
6Conclusion 59
APreliminariesoftheproofofourconvexoptimization 63
A.1 Introduction of some general convex sets . . . . . . . . . . . . . ...................... 63
A.2 The introduction of convex an d concave function . . . . . . . ....................... 67
A.3 Convex optimization problem . . . . . . . . . . . . . . . . . . . . . . . .................. 68
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Chapter 1
Introduction
For decades the rapid growth in computer hardware has been achieved by decreasing the size of electronic devices. Con-
vention approaches to the fabrication of computer technology are approaching fundamental limits due to quantum effects.
One possible solution to the problem is to turn to a d ifferent computing paradigm provided by quantum inform ation pr o -
cessing. Quantum information pro cessing (Nielsen and Chuang, 2010) explores an amazing field at the foundations of
computer science and quantum mechanics. Quantum comp uters promise an exponential improvement in speed compared
to their classical counterparts, especially in integer factorization. In particular, the properties of quantum bits can be used
to achieve secure communication. Now quantum information processing appears as a mixture of wide applications (e.g.
quantum computing, cryptography and telecommunication) and challenges (e.g. coming up with quantum algorithms
better then exiting classical ones and extending classical information theoretic results to quantum information).
Recently a novel idea, quantum adiabatic evolution, was proposed for the design of quantum algorithms by Farhi et
al (Farhi et al., 2001). This alg o r ithm concentrates on the evolution of the quantum state governed by a time dependent
Hamiltonian, which performs an interpolation between an initial Hamiltonian and a final Hamiltonian. To describe it
in details, it starts from an initial disordered state and slowly turns into a desired state as the parameter ’s’issmoothly
varied from 0 to 1. In order to make sure that the system evolvestothetargetedstate,adiabatictheoremrequiresthat
the evolution time should be long enough. The evolution time mainly depends on the minimum gap between the two
lowest energy of the time varying Hamiltonian. From the literature (Cullimore et al., 2012), we believe that the success
probability of evolving into the required ground state of thefinalHamiltonianismonotonouslyrelatedtothevalueofthe
minimum gap.
Such problem are also known as quantum annealing, which has similar process as adiabatic quantum computation
mentioned above. This gap plays an important role in quantum adiabatic computation. Our main purpose is to m ake
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4 Introduction
the gap as big as possible by changing the structure of interpolating Hamiltonian through optimal control. Our method
creatively introduces the intermediate Hamiltonian, localorsingle-qubitHamiltonianthatisexperimentallyfeasible in
physics, to change the eigenvalue structure of the time varying Hamiltonian during the evolution. We use the minimum
gap which is much more easier to access, to be a proxy for success probability, which is physically hard to obtain. Our goal
is to maximize the minimum gap, therefore, we formulate an minimax optimization problem and figure out an effective
algorithm to obtain the optimal solution in this thesis.
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