Quantum computer simulations of time-dependent hamiltonians

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dc.contributor.advisorSanders, Barry C.
dc.contributor.advisorHøyer, Peter
dc.contributor.authorWiebe, Nathan
dc.date.accessioned2017-12-18T22:17:36Z
dc.date.available2017-12-18T22:17:36Z
dc.date.issued2011
dc.descriptionBibliography: p. 153-163en
dc.description.abstractIn this thesis, we present three innovations that can be used to simulate quantum dynamics on both quantum and classical computers. Our primary innovation is an efficient quantum algorithm for simulating time-dependent Hamiltonian evolution of general input states on a quantum computer. Given conditions on the smoothness of the Hamiltonian, the complexity of the algorithm is close to linear in the evolution time, and therefore is comparable to algorithms for time-independent Hamiltonians. In addition, we show how the complexity can be reduced by optimizing the time steps. The complexity of the algorithm is quantified by calls to an oracle, which yields information about the Hamiltonian. In contrast to previous work, which allowed an oracle query to yield an arbitrary number of bits or qubits, we assign a cost for each bit or qubit accessed, revealing hitherto unnoticed simulation costs. We also account for discretization errors in the time and the representation of the Hamiltonian. This work is enabled by our development of a decomposition scheme based on Lie-Trotter- Suzuki product formulae for ordered operator exponentials. We show, by counterexample, that Lie-Trotter-Suzuki approximations may be lower order than expected when applied to problems that have singularities or discontinuous derivatives of appropriate order. We address this problem by presenting a set of criteria that is sufficient for the validity of these approximations, prove convergence and provide upper bounds for the approximation error. We also provide a new analysis of the evolution of quantum adiabatic systems in this thesis. Our work contains the tightest upper bounds that are currently known, and the first tight lower bounds, for the error in the adiabatic approximation applied to fixed Hamiltonians in the limit of slow evolution. We use our results to address the Marzlin- Sanders counterexample Hamiltonian. Specifically, we provide sufficiency conditions for the validity of the adiabatic approximation and demonstrate that the Marzlin-Sanders counterexample fails to satisfy them. We also find a new class of time-dependent Hamiltonians that are related to the Marzlin0Sanders counterexample Hamiltonian, but nonetheless obey the adiabatic approximation for sufficiently slow evolutions.
dc.format.extentviii, 163 leaves : ill. ; 30 cm.en
dc.identifier.citationWiebe, N. (2011). Quantum computer simulations of time-dependent hamiltonians (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/4047en_US
dc.identifier.doihttp://dx.doi.org/10.11575/PRISM/4047
dc.identifier.urihttp://hdl.handle.net/1880/105048
dc.language.isoeng
dc.publisher.institutionUniversity of Calgaryen
dc.publisher.placeCalgaryen
dc.rightsUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.
dc.titleQuantum computer simulations of time-dependent hamiltonians
dc.typedoctoral thesis
thesis.degree.disciplineInstitute for Quantum Information Science
thesis.degree.disciplinePhysics and Astronomy
thesis.degree.grantorUniversity of Calgary
thesis.degree.nameDoctor of Philosophy (PhD)
ucalgary.item.requestcopytrue
ucalgary.thesis.accessionTheses Collection 58.002:Box 2054 627942896
ucalgary.thesis.notesUARCen
ucalgary.thesis.uarcreleaseyen
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