Graduation date: 2003

It is known that almost all 2-qubit gates are universal for quantum computing (Lloyd 1995; Deutsch, Barenco, Eckert 1995). However, an explicit characterization of non-universal 2-qubit gates is not known. We consider a closely related problem of characterizing the set of non-universal 2-qubit Hamiltonians. We call a 2-qubit Hamiltonian n-universal if, when applied on different pairs of qubits, it can be used to approximate any unitary operation on n qubits. It follows directly from the results of Lloyd and Deutsch, Barenco, Eckert, that almost any 2-qubit Hamiltonian is 2-universal. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. There are three cases when a 2-qubit Hamiltonian H is not universal: (1) H shares an eigenvector with the gate that swaps two qubits; (2) H acts on the two qubits independently (in any of a certain family of bases); (3) H has zero trace. The last condition rules out the Hamiltonians that generate SU(4)---it can be omitted if the global phase is not important. A Hamiltonian that is not 2-universal can still be 3-universal. We give a (possibly incomplete) list of 2-qubit Hamiltonians that are not 3-universal. If this list happens to be complete, it actually gives a classification of n-universal 2-qubit Hamiltonians for all n >= 3.

It is known that almost all 2-qubit gates are universal for quantum computing (Lloyd 1995; Deutsch, Barenco, Eckert 1995). However, an explicit characterization of non-universal 2-qubit gates is not known. We consider a closely related problem of characterizing the set of non-universal 2-qubit Hamiltonians. We call a 2-qubit Hamiltonian n-universal if, when applied on different pairs of qubits, it can be used to approximate any unitary operation on n qubits. It follows directly from the results of Lloyd and Deutsch, Barenco, Eckert, that almost any 2-qubit Hamiltonian is 2-universal. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. There are three cases when a 2-qubit Hamiltonian H is not universal: (1) H shares an eigenvector with the gate that swaps two qubits; (2) H acts on the two qubits independently (in any of a certain family of bases); (3) H has zero trace. The last condition rules out the Hamiltonians that generate SU(4)---it can be omitted if the global phase is not important. A Hamiltonian that is not 2-universal can still be 3-universal. We give a (possibly incomplete) list of 2-qubit Hamiltonians that are not 3-universal. If this list happens to be complete, it actually gives a classification of n-universal 2-qubit Hamiltonians for all n >= 3.

The problem considered in this thesis is the following: We are given a Hamiltonian H and time t, and our goal is to approximately implement the unitary operator e^{-iHt} with an efficient quantum algorithm. We present an efficient algorithm for simulating sparse Hamiltonians. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian acts, this algorithm uses (d^2(d+log^* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d^4(log^* N)||Ht||)^{1+o(1)}. In terms of the parameter t, these algorithms are essentially optimal due to a no--fast-forwarding theorem. In the second part of this thesis, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, and rule out generic simulations taking time o(||Ht||), even though ||H|| is not a unique measure of the size of a dense Hamiltonian H. We also present a stronger limitation ruling out the possibility of generic simulations taking time poly(||Ht||,log N), showing that known simulations based on discrete-time quantum walks cannot be dramatically improved in general. We also show some positive results about simulating structured Hamiltonians efficiently.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains vi, 52 p. : ill. Includes abstract. Includes bibliographical references (p. 50-52).

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The problem considered in this thesis is the following: We are given a Hamiltonian H and time t, and our goal is to approximately implement the unitary operator e^{-iHt} with an efficient quantum algorithm. We present an efficient algorithm for simulating sparse Hamiltonians. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian acts, this algorithm uses (d^2(d+log^* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d^4(log^* N)||Ht||)^{1+o(1)}. In terms of the parameter t, these algorithms are essentially optimal due to a no--fast-forwarding theorem. In the second part of this thesis, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, and rule out generic simulations taking time o(||Ht||), even though ||H|| is not a unique measure of the size of a dense Hamiltonian H. We also present a stronger limitation ruling out the possibility of generic simulations taking time poly(||Ht||,log N), showing that known simulations based on discrete-time quantum walks cannot be dramatically improved in general. We also show some positive results about simulating structured Hamiltonians efficiently.