5,952 research outputs found

    Decoherence in quantum walks - a review

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    The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, led rapidly to several new quantum algorithms. These all follow unitary quantum evolution, apart from the final measurement. Since logical qubits in a quantum computer must be protected from decoherence by error correction, there is no need to consider decoherence at the level of algorithms. Nonetheless, enlarging the range of quantum dynamics to include non-unitary evolution provides a wider range of possibilities for tuning the properties of quantum walks. For example, small amounts of decoherence in a quantum walk on the line can produce more uniform spreading (a top-hat distribution), without losing the quantum speed up. This paper reviews the work on decoherence, and more generally on non-unitary evolution, in quantum walks and suggests what future questions might prove interesting to pursue in this area.Comment: 52 pages, invited review, v2 & v3 updates to include significant work since first posted and corrections from comments received; some non-trivial typos fixed. Comments now limited to changes that can be applied at proof stag

    Decoherence can be useful in quantum walks

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    We present a study of the effects of decoherence in the operation of a discrete quantum walk on a line, cycle and hypercube. We find high sensitivity to decoherence, increasing with the number of steps in the walk, as the particle is becoming more delocalised with each step. However, the effect of a small amount of decoherence is to enhance the properties of the quantum walk that are desirable for the development of quantum algorithms. Specifically, we observe a highly uniform distribution on the line, a very fast mixing time on the cycle, and more reliable hitting times across the hypercube.Comment: (Imperial College London) 6 (+epsilon) pages, 6 embedded eps figures, RevTex4. v2 minor changes to correct typos and refs, submitted version. v3 expanded into article format, extra figure, updated refs, Note on "glued trees" adde

    Long path and cycle decompositions of even hypercubes

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    We consider edge decompositions of the nn-dimensional hypercube QnQ_n into isomorphic copies of a given graph HH. While a number of results are known about decomposing QnQ_n into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if nn is even, â„“<2n\ell < 2^n and â„“\ell divides the number of edges of QnQ_n, then the path of length â„“\ell decomposes QnQ_n. Tapadia et al.\ proved that any path of length 2mn2^mn, where 2m<n2^m<n, satisfying these conditions decomposes QnQ_n. Here, we make progress toward resolving Erde's conjecture by showing that cycles of certain lengths up to 2n+1/n2^{n+1}/n decompose QnQ_n. As a consequence, we show that QnQ_n can be decomposed into copies of any path of length at most 2n/n2^{n}/n dividing the number of edges of QnQ_n, thereby settling Erde's conjecture up to a linear factor

    Parity balance of the ii-th dimension edges in Hamiltonian cycles of the hypercube

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    Let n≥2n\geq 2 be an integer, and let i∈{0,...,n−1}i\in\{0,...,n-1\}. An ii-th dimension edge in the nn-dimensional hypercube QnQ_n is an edge v1v2{v_1}{v_2} such that v1,v2v_1,v_2 differ just at their ii-th entries. The parity of an ii-th dimension edge \edg{v_1}{v_2} is the number of 1's modulus 2 of any of its vertex ignoring the ii-th entry. We prove that the number of ii-th dimension edges appearing in a given Hamiltonian cycle of QnQ_n with parity zero coincides with the number of edges with parity one. As an application of this result it is introduced and explored the conjecture of the inscribed squares in Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in QnQ_n contains two opposite edges in a 4-cycle. We prove this conjecture for n≤7n \le 7, and for any Hamiltonian cycle containing more than 2n−22^{n-2} edges in the same dimension. This bound is finally improved considering the equi-independence number of Qn−1Q_{n-1}, which is a concept introduced in this paper for bipartite graphs

    An extremal theorem in the hypercube

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    The hypercube Q_n is the graph whose vertex set is {0,1}^n and where two vertices are adjacent if they differ in exactly one coordinate. For any subgraph H of the cube, let ex(Q_n, H) be the maximum number of edges in a subgraph of Q_n which does not contain a copy of H. We find a wide class of subgraphs H, including all previously known examples, for which ex(Q_n, H) = o(e(Q_n)). In particular, our method gives a unified approach to proving that ex(Q_n, C_{2t}) = o(e(Q_n)) for all t >= 4 other than 5.Comment: 6 page

    Query Complexity of Approximate Nash Equilibria

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    We study the query complexity of approximate notions of Nash equilibrium in games with a large number of players nn. Our main result states that for nn-player binary-action games and for constant ε\varepsilon, the query complexity of an ε\varepsilon-well-supported Nash equilibrium is exponential in nn. One of the consequences of this result is an exponential lower bound on the rate of convergence of adaptive dynamics to approxiamte Nash equilibrium
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