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14
A quantum computing primer for operator theorists
 Linear Algebra Appl
, 2005
"... Abstract. This is an exposition of some of the aspects of quantum computation and quantum information that have connections with operator theory. After a brief introduction, we discuss quantum algorithms. We outline basic properties of quantum channels, or equivalently, completely positive trace pre ..."
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Abstract. This is an exposition of some of the aspects of quantum computation and quantum information that have connections with operator theory. After a brief introduction, we discuss quantum algorithms. We outline basic properties of quantum channels, or equivalently, completely positive trace preserving maps. The main theorems for quantum error detection and correction are presented and we conclude with a description of a particular passive method of quantum error correction. 1.
Wavelet representations and Fock space on positive matrices
 J. Funct. Anal
, 2003
"... Abstract. We show that every biorthogonal wavelet determines a representation by operators on Hilbert space satisfying simple identities, which captures the established relationship between orthogonal wavelets and Cuntzalgebra representations in that special case. Each of these representations is s ..."
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Abstract. We show that every biorthogonal wavelet determines a representation by operators on Hilbert space satisfying simple identities, which captures the established relationship between orthogonal wavelets and Cuntzalgebra representations in that special case. Each of these representations is shown to have tractable finitedimensional coinvariant doublycyclic subspaces. Further, motivated by these representations, we introduce a general Fockspace Hilbert space construction which yields creation operators containing the Cuntz–Toeplitz isometries as a special case. In this paper, we wish to establish a connection between biorthogonal wavelets on the one hand [16], and representation theory for operators on Hilbert space on the other [9, 18]. This is accomplished by showing that each of these wavelets yields a collection of operators acting on Hilbert space which satisfy simple identities, and which contain the Cuntz relations [15] as a special case. In fact, this new relationship collapses to the now wellknown connection between orthogonal wavelets and representations of the Cuntz C ∗algebra in that special case [10]. Our second goal is to develop a framework for studying this new class of representations. Toward this end, we introduce a general Fock space Hilbert space construction which reduces to unrestricted Fock space in the familiar cases. Indeed, the natural creation operators we get can be thought of as an analogue of the Cuntz–Toeplitz creation operators to this more general setting. We regard this construction and the creation operators determined by it as interesting objects of study in their own right. Finally, our hope is that this paper will lead to further study of the relationships and objects introduced here.
Noiseless subsystems for collective rotation channels in quantum information theory
, 2004
"... Collective rotation channels are a fundamental class of channels in quantum computing and quantum information theory. The commutant of the noise operators for such a channel is a C ∗algebra which is equal to the set of fixed points for the channel. Finding the precise spatial structure of the comm ..."
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Cited by 5 (2 self)
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Collective rotation channels are a fundamental class of channels in quantum computing and quantum information theory. The commutant of the noise operators for such a channel is a C ∗algebra which is equal to the set of fixed points for the channel. Finding the precise spatial structure of the commutant algebra for a set of noise operators associated with a channel is a core problem in quantum error prevention. We draw on methods of operator algebras, quantum mechanics and combinatorics to explicitly determine the structure of the commutant for the class of collective rotation channels.
UNIVERSAL COLLECTIVE ROTATION CHANNELS AND QUANTUM ERROR CORRECTION
, 2004
"... Abstract. We present and investigate a new class of quantum channels, what we call ‘universal collective rotation channels’, that includes the class of collective rotation channels as a special case. The fixed point set and noise commutant coincide for a channel in this class. Computing the precise ..."
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Abstract. We present and investigate a new class of quantum channels, what we call ‘universal collective rotation channels’, that includes the class of collective rotation channels as a special case. The fixed point set and noise commutant coincide for a channel in this class. Computing the precise structure of this C ∗algebra is a core problem in a particular noiseless subsystem method of quantum error correction. We prove that there is an abundance of noiseless subsystems for every channel in this class and that the Young tableaux combinatorial machine may be used to explicitly compute these subsystems. 1.
On bilateral weighted shifts in noncommutative multivariable operator theory
 Indiana Univ. Math. J
"... Abstract. We present a generalization of bilateral weighted shift operators for the noncommutative multivariable setting. We discover a notion of periodicity for these shifts, which has an appealing diagramatic interpretation in terms of an infinite tree structure associated with the underlying Hilb ..."
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Abstract. We present a generalization of bilateral weighted shift operators for the noncommutative multivariable setting. We discover a notion of periodicity for these shifts, which has an appealing diagramatic interpretation in terms of an infinite tree structure associated with the underlying Hilbert space. These shifts arise naturally through weighted versions of certain representations of the Cuntz C ∗algebras On. It is convenient, and equivalent, to consider the weak operator topology closed algebras generated by these operators when investigating their joint reducing subspace structure. We prove these algebras have nontrivial reducing subspaces exactly when the shifts are doublyperiodic; that is, the weights for the shift have periodic behaviour, and the corresponding representation of On has a certain spatial periodicity. This generalizes Nikolskii’s Theorem for the single variable case. In [24] and [25], we began studying versions of unilateral weighted shift operators in noncommutative multivariable operator theory. We called them weighted shifts on Fock space since they act naturally on the full Fock space Hilbert space. These shifts and the algebras they generate were first studied by Arias and Popescu [3] from the perspective of weighted Fock spaces. The basic goals of this program are to extend results from the commutative (single variable) setting and, at the same time, expose new noncommutative phenomena. In the current paper, we continue this line of investigation by presenting versions of bilateral weighted shift operators for the noncommutative multivariable setting. Our analysis is chiefly spatial in nature: we examine the joint reducing subspace structure for these shifts, and consider reducibility questions for the weak operator topology closed (nonselfadjoint) algebras they generate. In particular, we give a complete characterization 2000 Mathematics Subject Classification. 47L75, 47B37, 47L55. key words and phrases. Hilbert space, operator, bilateral weighted shift, periodicity, reducing subspaces, infinite word, noncommutative multivariable operator theory, nonselfadjoint operator algebras, Fock space. 1 partially supported by a Canadian NSERC Postdoctoral Fellowship.
Quantum error correction and Young tableaux
"... Abstract. A quantum channel is a completely positive trace preserving map which acts on the set of operators for the Hilbert space associated with a given quantum system. Analysis of such channels is central to quantum computing and quantum information theory. We present and investigate a new class ..."
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Abstract. A quantum channel is a completely positive trace preserving map which acts on the set of operators for the Hilbert space associated with a given quantum system. Analysis of such channels is central to quantum computing and quantum information theory. We present and investigate a new class of quantum channels that includes the class of collective rotation channels as a special case. We use the phrase ‘universal collective rotation channels ’ for this class. The fixed point set and noise commutant coincide for a channel in this class. Computing the precise structure of this operator algebra is a core problem in a particular noiseless subsystem method of quantum error correction. We apply classical representation theory of the symmetric group via Young tableaux and give a computationally amenable method for explicitly finding this structure for the class of universal collective rotation channels. 1.
A Class of Completely Positive Maps ∗
, 908
"... Abstract. Let H be a complex Hilbert space, B(H) be the set of bounded linear operators on H, E(H) be the set of {A ∈ B(H) : 0 ≤ A ≤ I}, 1 ≤ n ≤ ∞. In this paper, we show that if A = {Ei} n i=1 ⊆ E(H) is commutative, and ΦA is the completely positive map which is defined by ΦA: B(H) − → B(H) : B ↦− ..."
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Abstract. Let H be a complex Hilbert space, B(H) be the set of bounded linear operators on H, E(H) be the set of {A ∈ B(H) : 0 ≤ A ≤ I}, 1 ≤ n ≤ ∞. In this paper, we show that if A = {Ei} n i=1 ⊆ E(H) is commutative, and ΦA is the completely positive map which is defined by ΦA: B(H) − → B(H) : B ↦− → ∑ EiBEi, then we have:
A Theory of Physical Quantum Computation: The Quantum Computer Condition
, 2005
"... In this paper we present a new unified theoretical framework that describes the full dynamics of quantum computation. Our formulation allows any questions pertaining to the physical behavior of a quantum computer to be framed, and in principle, answered. We refer to the central organizing principle ..."
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In this paper we present a new unified theoretical framework that describes the full dynamics of quantum computation. Our formulation allows any questions pertaining to the physical behavior of a quantum computer to be framed, and in principle, answered. We refer to the central organizing principle developed in this paper, on which our theoretical structure is based, as the Quantum Computer Condition (QCC), a rigorous mathematical statement that connects the irreversible dynamics of the quantum computing machine, with the reversible operations that comprise the quantum computation intended to be carried out by the quantum computing machine. Armed with the QCC, we derive a powerful result that we call the Encoding NoGo Theorem. This theorem gives a precise mathematical statement of the conditions under which faulttolerant quantum computation becomes impossible in the presence of dissipation and/or decoherence. In connection with this theorem, we explicitly calculate a universal critical damping value for faulttolerant quantum computation. In addition we show that the recentlydiscovered approach
Information Amidst Noise: Preserved Codes, Error Correction, and Fault Tolerance in a Quantum World
, 2010
"... For my dad, whose secret ambition has always been to be a physicist, for my mum, for being the strong woman that she is, and of course, for my husband, for always being here. ivv Acknowledgments Looking back at the five years I have spent in Caltech as a part of the Institute of Quantum Information ..."
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For my dad, whose secret ambition has always been to be a physicist, for my mum, for being the strong woman that she is, and of course, for my husband, for always being here. ivv Acknowledgments Looking back at the five years I have spent in Caltech as a part of the Institute of Quantum Information (IQI), I cannot help but consider myself incredibly fortunate to have had guidance from so many people. The first person I have to thank is of course my advisor John Preskill. He has put together the amazing place that is IQI, with its constant flux of visitors, postdocs and students, offering me an invaluable chance to meet so many talented people in the field. Along with sharing his insights in physics and always extending a willing hand in times of difficulty, he taught me a most valuable