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

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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 well-known 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 mixed-unitary 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 mixed-unitary 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 Log-depth mixed-state quantum circuits................... 78 4.2 QIP completeness of close images...................... 79

Abstract. We prove that one-way quantum computations have the same computational power as quantum circuits with unbounded fan-out. It demonstrates that the one-way 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 one-way model and the quantum circuit model. Since one-way model has the same computational power as unbounded quantum fanout circuits, the quantum Fourier transform can be approximated in constant depth in the one-way model, and thus the factorisation can be done by a polytime probabilistic classical algorithm which has access to a constant-depth one-way quantum computer. The extra power of the one-way model, comparing with the quantum circuit model, comes from its classical-quantum hybrid nature. We show that this extra power is reduced to the capability to perform unbounded classical parity gates in constant depth. 1

Twisted graph states for ancilla-driven quantum computation

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 n-qubit state spaceH n = (H 1) ⊗n as the n-fold 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 two-qubit swap interferometer requiring a single Fredkin gate suffices in this case. In the n-qubit 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 (e-bits) using fully mixed states and von Neumann measurements.

The basic principles to construct a generalized state-locking pulse field and simulate efficiently the reversible and unitary halting protocol of a universal quantum computer

The basic principles to construct a generalized state-locking pulse field and to simulate efficiently the reversible and unitary halting protocol of a universal quantum computer

The basic principles to construct a generalized state-locking pulse field and simulate efficiently the reversible and unitary halting protocol of a universal quantum computer

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 Gottesman-Knill 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 Gottesman-Knill 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