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55
LowRank Tensor Krylov Subspace Methods for Parametrized Linear Systems
, 2010
"... We consider linear systems A(α)x(α) = b(α) depending on possibly many parameters α = (α1,...,αp). Solving these systems simultaneously for a standard discretization of the parameter space would require a computational effort growing exponentially in the number of parameters. We show that this curse ..."
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Cited by 25 (3 self)
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We consider linear systems A(α)x(α) = b(α) depending on possibly many parameters α = (α1,...,αp). Solving these systems simultaneously for a standard discretization of the parameter space would require a computational effort growing exponentially in the number of parameters. We show that this curse of dimensionality can be avoided for sufficiently smooth parameter dependencies. For this purpose, computational methods are developed that benefit from the fact that x(α) can be well approximated by a tensor of low rank. In particular, lowrank tensor variants of shortrecurrence Krylov subspace methods are presented. Numerical experiments for deterministic PDEs with parametrized coefficients and stochastic elliptic PDEs demonstrate the effectiveness of our approach.
LiebRobinson bounds in quantum manybody physics, In: Entropy and the Quantum
 Contemp. Math
, 2010
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The complexity of the consistency and Nrepresentability problems for quantum states
"... QMA (Quantum MerlinArthur) is the quantum analogue of the class NP. There are a few QMAcomplete problems, most of which are variants of the “Local Hamiltonian” problem introduced by Kitaev. In this dissertation we show some new QMAcomplete problems which are very different from those known previo ..."
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QMA (Quantum MerlinArthur) is the quantum analogue of the class NP. There are a few QMAcomplete problems, most of which are variants of the “Local Hamiltonian” problem introduced by Kitaev. In this dissertation we show some new QMAcomplete problems which are very different from those known previously, and have applications in quantum chemistry. The first one is “Consistency of Local Density Matrices”: given a collection of density matrices describing different subsets of an nqubit system (where each subset has constant size), decide whether these are consistent with some global state of all n qubits. This problem was first suggested by Aharonov. We show that it is QMAcomplete, via an oracle reduction from Local Hamiltonian. Our reduction is based on algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, “Fermionic Local Hamiltonian” and “Nrepresentability, ” are QMAcomplete. These problems involve systems of fermions, rather than qubits; they arise in calculating the ground state energies of molecular systems. Nrepresentability is particularly interesting, as it is a key component
Spreading of correlations and entanglement after a quench in the BoseHubbard model
, 2008
"... We investigate the spreading of information in a BoseHubbard system after a sudden parameter change. In particular, we study the timeevolution of correlations and entanglement following a quench. The investigated quantities show a lightcone like evolution, i.e. the spreading with a finite veloci ..."
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We investigate the spreading of information in a BoseHubbard system after a sudden parameter change. In particular, we study the timeevolution of correlations and entanglement following a quench. The investigated quantities show a lightcone like evolution, i.e. the spreading with a finite velocity. We discuss the relation of this veloctiy to other characteristic velocities of the system, like the sound velocity. The entanglement is investigated using two different measures, the vonNeuman entropy and mutual information. Whereas the vonNeumann entropy grows rapidly with time the mutual information can as well decrease after an initial increase. Additionally we show that the static von Neuman entropy characterises the location of the quantum phase transition.
Completegraph tensor network states: a new fermionic wave function ansatz for molecules
 New J. Phys
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Classification of quantum phases and topology of logical operators in an exactly solved model . . .
, 2011
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Ground State Entanglement in One Dimensional Translationally Invariant Quantum Systems
, 2009
"... We examine whether it is possible for onedimensional translationallyinvariant Hamiltonians to have ground states with a high degree of entanglement. We present a family of translationally invariant Hamiltonians {Hn} for the infinite chain. The spectral gap of Hn is Ω(1/poly(n)). Moreover, for any ..."
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Cited by 4 (1 self)
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We examine whether it is possible for onedimensional translationallyinvariant Hamiltonians to have ground states with a high degree of entanglement. We present a family of translationally invariant Hamiltonians {Hn} for the infinite chain. The spectral gap of Hn is Ω(1/poly(n)). Moreover, for any state in the ground space of Hn and any m, there are regions of size m with entanglement entropy Ω(min{m, n}). A similar construction yields translationallyinvariant Hamiltonians for finite chains that have unique ground states exhibiting high entanglement. The area law proven by Hastings (Has07) gives a constant upper bound on the entanglement entropy for 1D ground states that is independent of the size of the region but exponentially dependent on 1/∆, where ∆ is the spectral gap. This paper provides a lower bound, showing a family of Hamiltonians for which the entanglement entropy scales polynomially with 1/∆. Previously, the best known such bound was logarithmic in 1/∆. 1 I.
The Local Consistency Problem for Stoquastic and 1D Quantum Systems. ArXiv eprints
, 2007
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10 Dynamical Transition in the Openboundary Totally Asymmetric Exclusion Process
, 2010
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Engineered open systems and quantum simulations with atoms and ions
 Advances In Atomic, Molecular, and Optical Physics
, 2012
"... The enormous experimental progress in atomic, molecular and optical (AMO) physics during the last decades allows us nowadays to isolate single, a few or even manybody ensembles of microscopic particles, and to manipulate their quantum properties at a level of precision, which still seemed unthin ..."
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Cited by 3 (1 self)
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The enormous experimental progress in atomic, molecular and optical (AMO) physics during the last decades allows us nowadays to isolate single, a few or even manybody ensembles of microscopic particles, and to manipulate their quantum properties at a level of precision, which still seemed unthinkable some years ago. This versatile set of tools has enabled the development of the wellestablished concept of engineering of manybody Hamiltonians in various physical platforms. These available tools, however, can also be harnessed to extend the scenario of Hamiltonian engineering to a more general Liouvillian setting, which in addition to coherent dynamics also includes controlled dissipation in manybody quantum systems. Here, we review recent theoretical and experimental progress in different directions along these lines, with a particular focus on physical realizations with systems