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Generalized Flow and Determinism in Measurementbased Quantum Computation
 New J. Physics
, 2007
"... Abstract. We extend the notion of quantum information flow defined by Danos and Kashefi [1] for the oneway model [2] and present a necessary and sufficient condition for the deterministic computation in this model. The generalized flow also applied in the extended model with measurements in the (X, ..."
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Abstract. We extend the notion of quantum information flow defined by Danos and Kashefi [1] for the oneway model [2] and present a necessary and sufficient condition for the deterministic computation in this model. The generalized flow also applied in the extended model with measurements in the (X, Y), (X, Z) and (Y, Z) planes. We apply both measurement calculus and the stabiliser formalism to derive our main theorem which for the first time gives a full characterization of the deterministic computation in the oneway model. We present several examples to show how our result improves over the traditional notion of flow, such as geometries (entanglement graph with input and output) with no flow but having generalized flow and we discuss how they lead to an optimal implementation of the unitaries. More importantly one can also obtain a better quantum computation depth with the generalized flow rather than with flow. We believe our characterization result is particularly essential for the study of the algorithms and complexity in the oneway model. Generalized Flow and Determinism 2 1.
Quantum information and computation
 arXiv:quantph/0512125. Forthcoming in Butterfield and Earman (eds.) Handbook of Philosophy of Physics
, 2005
"... This Chapter deals with theoretical developments in the subject of quantum information and quantum computation, and includes an overview of classical information and some relevant quantum mechanics. The discussion covers topics in quantum communication, quantum cryptography, and quantum computation, ..."
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This Chapter deals with theoretical developments in the subject of quantum information and quantum computation, and includes an overview of classical information and some relevant quantum mechanics. The discussion covers topics in quantum communication, quantum cryptography, and quantum computation, and concludes by considering whether a perspective in terms of quantum information
Improved gap estimates for simulating quantum circuits by adiabatic evolution
 Quantum Information Processing
"... We use elementary variational arguments to prove, and improve on, gap estimates which arise in simulating quantum circuits by adiabatic evolution. There are several models for quantum computation [5]. The quantum Turing machine model and the quantum circuit model, are equivalent in the sense that an ..."
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Cited by 3 (0 self)
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We use elementary variational arguments to prove, and improve on, gap estimates which arise in simulating quantum circuits by adiabatic evolution. There are several models for quantum computation [5]. The quantum Turing machine model and the quantum circuit model, are equivalent in the sense that any algorithm that runs in polynomial time in one requires only polynomial time in the other. There are also several “oneway ” measurementbased models [3], such as the cluster state model [6], which can simulate any polynomial time quantum circuit in polynomial time. c ○ by authors. Reproduction permitted for noncommercial purposes.
Computational depth complexity of measurementbased quantum computation
, 909
"... Abstract. We prove that oneway quantum computations have the same computational power as quantum circuits with unbounded fanout. It demonstrates that the oneway model is not only one of the most promising models of physical realisation, but also a very powerful model of quantum computation. It co ..."
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Abstract. We prove that oneway quantum computations have the same computational power as quantum circuits with unbounded fanout. It demonstrates that the oneway model is not only one of the most promising models of physical realisation, but also a very powerful model of quantum computation. It confirms and completes previous results which have pointed out, for some specific problems, a depth separation between the oneway model and the quantum circuit model. Since oneway model has the same computational power as unbounded quantum fanout circuits, the quantum Fourier transform can be approximated in constant depth in the oneway model, and thus the factorisation can be done by a polytime probabilistic classical algorithm which has access to a constantdepth oneway quantum computer. The extra power of the oneway model, comparing with the quantum circuit model, comes from its classicalquantum hybrid nature. We show that this extra power is reduced to the capability to perform unbounded classical parity gates in constant depth. 1
Generalised Clifford groups and simulation of associated quantum circuits
, 2007
"... Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the GottesmanKnill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the ai ..."
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Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the GottesmanKnill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup G of U(d). In particular we seek G such that G ⊗ G has an entangling normaliser. Via a generalisation of the GottesmanKnill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d = 2 we exhaustively treat all finite subgroups of U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the n th root of Z for n ∈ N. 1
Computational Distinguishability of Quantum Channels
, 909
"... c ○ William Rosgen 2009I 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. The computational problem of disti ..."
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c ○ William Rosgen 2009I 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. The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the wellknown satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that have quantum interactive proof systems, which implies that it is hard for the class PSPACE of problems solvable by a classical computation in polynomial space. Several restrictions of distinguishability are also shown to be hard. It is no easier when restricted to quantum computations of logarithmic depth, to mixedunitary channels, to degradable channels, or to antidegradable channels. These hardness results are demonstrated by finding reductions between these classes of quantum channels. These techniques have applications outside the distinguishability problem, as the construction for mixedunitary channels is used to prove that the additivity problem for the classical capacity of quantum channels can be equivalently restricted to the mixed unitary channels. iii Acknowledgements I would like to thank my supervisor John Watrous for years of guidance, support, and insight. Without his help this would not have been possible. I would also like to thank the rest of my committee, Richard Cleve, Stephen Fenner, Achim Kempf, and Ben Reichardt, for providing helpful comments on an earlier draft of this thesis. I would also like to thank Lana for putting up with me during the writing of this thesis and supporting me throughout the process. v 4 The Close Images Problem 77 4.1 Logdepth mixedstate quantum circuits................... 78 4.2 QIP completeness of close images...................... 79
Projecting onto Qubit Irreps of Young Diagrams
, 2006
"... Let K be the diagonal subgroup of U(2) ⊗n. For the onequbit state spaceH 1 = C{0〉} ⊕ C{1〉}, we may view H 1 as a standard representation of U(2) and the nqubit state spaceH n = (H 1) ⊗n as the nfold tensor product of standard representations. Representation theory then decomposesH n into irredu ..."
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Let K be the diagonal subgroup of U(2) ⊗n. For the onequbit state spaceH 1 = C{0〉} ⊕ C{1〉}, we may view H 1 as a standard representation of U(2) and the nqubit state spaceH n = (H 1) ⊗n as the nfold tensor product of standard representations. Representation theory then decomposesH n into irreducible subrepresentations of K parametrized by combinatorial objects known as Young diagrams. We argue that n − 1 classically controlled measurement circuits, each a Fredkin interferometer, may be used to form a projection operator onto a random Young diagram irrep withinH n. ForH 2, the two irreps happen to be orthogonal and correspond to the symmetric and wedge product. The latter is spanned by Ψ − 〉, and the standard twoqubit swap interferometer requiring a single Fredkin gate suffices in this case. In the nqubit case, it is possible to extract many copies of Ψ − 〉. Thus applying this process using nondestructive Fredkin interferometers allows for the creation of entangled bits (ebits) using fully mixed states and von Neumann measurements.
unknown title
, 2006
"... The basic principles to construct a generalized statelocking pulse field and simulate efficiently the reversible and unitary halting protocol of a universal quantum computer ..."
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The basic principles to construct a generalized statelocking pulse field and simulate efficiently the reversible and unitary halting protocol of a universal quantum computer