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Quantum walks and ground state problems
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Quantum walks and ground state problems
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Richter
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Peter C. (Peter Courtland)
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Peter C. Richter
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Szegedy
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Mario
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Advisory Committee
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Mario Szegedy
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chair
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Eric
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Eric Allender
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Kilian
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Joe
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Joe Kilian
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Kendon
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Viv
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Advisory Committee
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Viv Kendon
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Rutgers University
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Graduate School - New Brunswick
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theses
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2007
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2007
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English
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electronic
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viii, 101 pages
Abstract
Since the appearance of Shor's factoring algorithm in 1994, the search for novel quantum computer algorithms has proved surprisingly difficult. Two design approaches that have yielded some progress are quantum walks and adiabatic computing. The former has been shown to speed up algorithms whose complexity is related to the classical hitting time of a symmetric Markov chain, and there is evidence that the latter speeds up simulated annealing algorithms for computing ground states of classical Hamiltonians.
In this thesis, we look into the possibility of obtaining a quantum speedup for the mixing time of a symmetric Markov chain. We prove that by subjecting a quantum walk to a small amount of decoherence (typically the adversary of a quantum computer), it can be forced to mix to the correct stationary distribution, often considerably faster than its classical counterpart. A more general theorem to this effect would imply quantum speedups for a variety of approximation algorithms for #P-complete problems.
We conclude with some observations on adiabatic computing -- a time-dependent generalization of the quantum walk framework -- and the problem of estimating the ground state energy of a quantum Hamiltonian with local spin interactions.
In this thesis, we look into the possibility of obtaining a quantum speedup for the mixing time of a symmetric Markov chain. We prove that by subjecting a quantum walk to a small amount of decoherence (typically the adversary of a quantum computer), it can be forced to mix to the correct stationary distribution, often considerably faster than its classical counterpart. A more general theorem to this effect would imply quantum speedups for a variety of approximation algorithms for #P-complete problems.
We conclude with some observations on adiabatic computing -- a time-dependent generalization of the quantum walk framework -- and the problem of estimating the ground state energy of a quantum Hamiltonian with local spin interactions.
Note
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Ph.D.
Note
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Includes bibliographical references (p. 94-100).
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Computer Science
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Computer algorithms
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Quantum field theory
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Topic
Random walks (Mathematics)
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.16768
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doi:10.7282/T3W37WQ1
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ETD doctoral
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Peter Richter
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Rutgers University. Graduate School - New Brunswick
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Author Agreement License
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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