Local Hamiltonians in quantum computation
Name
317945098-MIT.pdf
Description
Full printable version
Size
2.54 MB
Format
Adobe PDF
Checksum (MD5)
f8e1d61e744436c00aae0f59b8586e59
Author(s)
Nagaj, Daniel
Advisor(s)
Edward H. Farhi.
Date Issued
2008
Publisher
Massachusetts Institute of Technology
Abstract
In this thesis, I investigate aspects of local Hamiltonians in quantum computing. First, I focus on the Adiabatic Quantum Computing model, based on evolution with a time- dependent Hamiltonian. I show that to succeed using AQC, the Hamiltonian involved must have local structure, which leads to a result about eigenvalue gaps from information theory. I also improve results about simulating quantum circuits with AQC. Second, I look at classically simulating time evolution with local Hamiltonians and finding their ground state properties. I give a numerical method for finding the ground state of translationally invariant Hamiltonians on an infinite tree. This method is based on imaginary time evolution within the Matrix Product State ansatz, and uses a new method for bringing the state back to the ansatz after each imaginary time step. I then use it to investigate the phase transition in the transverse field Ising model on the Bethe lattice. Third, I focus on locally constrained quantum problems Local Hamiltonian and Quantum Satisfiability and prove several new results about their complexity. Finally, I define a Hamiltonian Quantum Cellular Automaton, a continuous-time model of computation which doesn't require control during the computation process, only preparation of product initial states. I construct two of these, showing that time evolution with a simple, local, translationally invariant and time-independent Hamiltonian can be used to simulate quantum circuits.
Description
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2008.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 169-176).
Subjects
Physics.
MIT Department
Massachusetts Institute of Technology. Department of Physics
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