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Characterization of non-universal two-qubit Hamiltonians

Laura Mančinska

Published Dec 16, 2009• 0 citations
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Abstract

I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii 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

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Suppose we can apply a given 2-qubit Hamiltonian H to any (ordered) pair of qubits.We say H is n-universal if it can be used to approximate any unitary operation onn qubits. While it is well known that almost any 2-qubit Hamiltonian is 2-universal(Deutsch, Barenco, Ekert 1995; Lloyd 1995), an explicit characterization of the set ofnon-universal 2-qubit Hamiltonians has been elusive. Our main result is a completecharacterization of 2-non-universal 2-qubit Hamiltonians. In particular, there are threeways that a 2-qubit Hamiltonian H can fail to be universal: (1) H shares an eigenvectorwith the gate that swaps two qubits, (2) H acts on the two qubits independently (inany of a certain family of bases), or (3) H has zero trace (with the third conditionrelevant only when the global phase of the unitary matters). A 2-non-universal 2-qubitHamiltonian can still be n-universal for some n ≥ 3. We give some partial results on3-universality.

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Suppose we can apply a given 2-qubit Hamiltonian H to any (ordered) pair of qubits. We say H is n-universal if it can be used to approximate any unitary operation on n qubits. While it is well known that almost any 2-qubit Hamiltonian is 2-universal, an explicit characterization of the set of non-universal 2-qubit Hamiltonians has been elusive. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. In particular, there are three ways that a 2-qubit Hamiltonian $H$ can fail to be 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), or (3) H has zero trace (with the third condition relevant only when the global phase of the unitary matters). A 2-non-universal 2-qubit Hamiltonian can still be n-universal for some n \geq 3. We give some partial results on 3-universality.

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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii 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
Suppose we can apply a given 2-qubit Hamiltonian H to any (ordered) pair of qubits.We say H is n-universal if it can be used to approximate any unitary operation onn qubits. While it is well known that almost any 2-qubit Hamiltonian is 2-universal(Deutsch, Barenco, Ekert 1995; Lloyd 1995), an explicit characterization of the set ofnon-universal 2-qubit Hamiltonians has been elusive. Our main result is a completecharacterization of 2-non-universal 2-qubit Hamiltonians. In particular, there are threeways that a 2-qubit Hamiltonian H can fail to be universal: (1) H shares an eigenvectorwith the gate that swaps two qubits, (2) H acts on the two qubits independently (inany of a certain family of bases), or (3) H has zero trace (with the third conditionrelevant only when the global phase of the unitary matters). A 2-non-universal 2-qubitHamiltonian can still be n-universal for some n ≥ 3. We give some partial results on3-universality.
Suppose we can apply a given 2-qubit Hamiltonian H to any (ordered) pair of qubits. We say H is n-universal if it can be used to approximate any unitary operation on n qubits. While it is well known that almost any 2-qubit Hamiltonian is 2-universal, an explicit characterization of the set of non-universal 2-qubit Hamiltonians has been elusive. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. In particular, there are three ways that a 2-qubit Hamiltonian HHH can fail to be 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), or (3) H has zero trace (with the third condition relevant only when the global phase of the unitary matters). A 2-non-universal 2-qubit Hamiltonian can still be n-universal for some n \geq 3. We give some partial results on 3-universality.
A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers.        The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected.       This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate.       Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. G. Passante ii Quantum information processors have the ability to drastically change our world. By manipulating bits of information ruled by the laws of quantum mechanics, computational devices can perform some computations that are classically intractable. Most quantum algorithms rely on pure qubits as inputs and require entanglement throughout the compu-tation. In this thesis, we explore a model of computation that uses mixed qubits without en-tanglement known as DQC1 (deterministic quantum computation with one quantum bit), using the physical system of liquid-state Nuclear Magnetic Resonance (NMR). Throughout our research, we experimentally implement an algorithm that completely encapsulates the DQC1 model, and take a close look at the quantum nature of DQC1-states as given by the
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii This thesis investigates a novel optoelectronic platform based on the integration of super-conductive structures, such as thin films and micro-constrictions, with optical waveguides for ultra-fast and ultra-sensitive devices with applications including high-speed optical communications, quantum optical information processing, and terahertz (THz) devices and systems. The kinetic-inductive photoresponse of superconducting thin films will be studied as the basic optoelectronic process underlying the operation of these novel devices. Analytical formulation for the non-bolometric response is presented, and experimental photodetection in YBCO meander-line structures will be demonstrated. A set of superconducting coplanar waveguides (CPW) are designed and characterized,