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Aw Harrow - One of the best experts on this subject based on the ideXlab platform.

  • Product-State Approximations to Quantum States
    Communications in Mathematical Physics, 2016
    Co-Authors: Fernando G. S. L. Brandão, Aw Harrow
    Abstract:

    We show that for any many-body Quantum state there exists an unentangled Quantum state such that most of the two-body reduced density matrices are close to those of the original state. This is a statement about the monogamy of entanglement, which cannot be shared without limit in the same way as classical correlation. Our main application is to Hamiltonians that are sums of two-body terms. For such Hamiltonians we show that there exist product states with energy that is close to the ground-state energy whenever the interaction graph of the Hamiltonian has high degree. This proves the validity of mean-field theory and gives an explicitly bounded approximation error. If we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the Hamiltonian satisfies one or more of the following properties: (1) high degree, (2) small expansion, or (3) a ground state where the blocks in the partition have sublinear entanglement. Previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero. Our approximations allow an extensive error in energy, which is the scale considered by the Quantum PCP (probabilistically checkable proof) and NLTS (no low-energy trivial-state) conjectures. Thus our results put restrictions on the possible Hamiltonians that could be used for a possible proof of the qPCP or NLTS conjectures. By contrast the classical PCP constructions are often based on constraint graphs with high degree. Likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for Quantum Hamiltonians, unless qPCP is false. The main technical tool behind our results is a collection of new classical and Quantum de Finetti theorems which do not make any symmetry assumptions on the underlying states.

  • Product-state approximations to Quantum ground states
    Proceedings of the Annual ACM Symposium on Theory of Computing, 2013
    Co-Authors: Aw Harrow
    Abstract:

    The local Hamiltonian problem consists of estimating the ground-state energy (given by the minimum eigenvalue) of a local Quantum Hamiltonian. It can be considered as a Quantum generalization of constraint satisfaction problems (CSPs) and has a key role in Quantum complexity theory, being the first and most natural QMA-complete problem known. An interesting regime for the local Hamiltonian problem is that of extensive error, where one is interested in estimating the mean ground-state energy to constant accuracy. The problem is NP-hard by the PCP theorem, but whether it is QMA-hard is an important open question in Quantum complexity theory. A positive solution would represent a Quantum analogue of the PCP theorem. A key feature that distinguishes Quantum Hamiltonians from classical CSPs is that the solutions may involve complicated entangled states. In this paper, we demonstrate several large classes of Hamiltonians for which product (i.e. unentangled) states can approximate the ground state energy to within a small extensive error. First, we show the mere existence of a good product-state approximation for the ground-state energy of 2-local Hamiltonians with one or more of the following properties: (1) super-constant degree, (2) small expansion, or (3) a ground state with sublinear entanglement with respect to some partition into small pieces. The approximation based on degree is a new and surprising difference between Quantum Hamiltonians and classical CSPs, since in the classical setting, higher degree is usually associated with harder CSPs. The approximation based on expansion is not new, but the approximation based on low entanglement was previously known only in the regime where the entanglement was close to zero. Since the existence of a low-energy product state can be checked in NP, this implies that any Hamiltonian used for a Quantum PCP theorem should have: (1) constant degree, (2) constant expansion, (3) a "volume law" for entanglement with respect to any partition into small parts. Second, we show that in several cases, good product-state approximations not only exist, but can be found in deterministic polynomial time: (1) 2-local Hamiltonians on any planar graph, solving an open problem of Bansal, Bravyi, and Terhal, (2) dense k-local Hamiltonians for any constant k, solving an open problem of Gharibian and Kempe, and (3) 2-local Hamiltonians on graphs with low threshold rank, via a Quantum generalization of a recent result of Barak, Raghavendra and Steurer. Our work involves two new tools which may be of independent interest. First, we prove a new Quantum version of the de Finetti theorem which does not require the usual assumption of symmetry. Second, we describe a way to analyze the application of the Lasserre/Parrilo SDP hierarchy to local Quantum Hamiltonians. Copyright 2013 ACM.

Christiane Quesne - One of the best experts on this subject based on the ideXlab platform.

  • superintegrability of the tremblay turbiner winternitz Quantum Hamiltonians on a plane for odd k
    Journal of Physics A, 2010
    Co-Authors: Christiane Quesne
    Abstract:

    In a recent communication paper by Tremblay et al (2009 J. Phys. A: Math. Theor. 42 205206), it has been conjectured that for any integer value of k, some novel exactly solvable and integrable Quantum Hamiltonian Hk on a plane is superintegrable and that the additional integral of motion is a 2kth-order differential operator Y2k. Here we demonstrate the conjecture for the infinite family of Hamiltonians Hk with odd k ? 3, whose first member corresponds to the three-body Calogero?Marchioro?Wolfes model after elimination of the centre-of-mass motion. Our approach is based on the construction of some D2k-extended and invariant Hamiltonian , which can be interpreted as a modified boson oscillator Hamiltonian. The latter is then shown to possess a D2k-invariant integral of motion , from which Y2k can be obtained by projection in the D2k identity representation space.

  • Superintegrability of the Tremblay?Turbiner?Winternitz Quantum Hamiltonians on a plane for odd k
    Journal of Physics A, 2010
    Co-Authors: Christiane Quesne
    Abstract:

    In a recent communication paper by Tremblay et al (2009 J. Phys. A: Math. Theor. 42 205206), it has been conjectured that for any integer value of k, some novel exactly solvable and integrable Quantum Hamiltonian Hk on a plane is superintegrable and that the additional integral of motion is a 2kth-order differential operator Y2k. Here we demonstrate the conjecture for the infinite family of Hamiltonians Hk with odd k ? 3, whose first member corresponds to the three-body Calogero?Marchioro?Wolfes model after elimination of the centre-of-mass motion. Our approach is based on the construction of some D2k-extended and invariant Hamiltonian , which can be interpreted as a modified boson oscillator Hamiltonian. The latter is then shown to possess a D2k-invariant integral of motion , from which Y2k can be obtained by projection in the D2k identity representation space.

Herschel Rabitz - One of the best experts on this subject based on the ideXlab platform.

  • estimation of many body Quantum Hamiltonians via compressive sensing
    Physical Review A, 2011
    Co-Authors: Alireza Shabani, Masoud Mohseni, Seth Lloyd, Robert L Kosut, Herschel Rabitz
    Abstract:

    We develop an efficient and robust approach for Quantum measurement of nearly sparse many-body Quantum Hamiltonians based on the method of compressive sensing. This work demonstrates that with only O(sln(d)) experimental configurations, consisting of random local preparations and measurements, one can estimate the Hamiltonian of a d-dimensional system, provided that the Hamiltonian is nearly s sparse in a known basis. The classical postprocessing is a convex optimization problem on the total Hilbert space which is generally not scalable. We numerically simulate the performance of this algorithm for three- and four-body interactions in spin-coupled Quantum dots and atoms in optical lattices. Furthermore, we apply the algorithm to characterize Hamiltonian fine structure and unknown system-bath interactions.

  • estimation of many body Quantum Hamiltonians via compressive sensing
    APS, 2011
    Co-Authors: Alireza Shabani, Masoud Mohseni, Seth Lloyd, Robert L Kosut, Herschel Rabitz
    Abstract:

    A. Shabani,1 M. Mohseni,2 S. Lloyd,2,3 R. L. Kosut,4 and H. Rabitz1 1Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA 2Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 4SC Solutions, Sunnyvale, California 94085, USA (Received 1 March 2010; revised manuscript received 21 February 2011; published 11 July 2011)

  • Efficient extraction of Quantum Hamiltonians from optimal laboratory data
    Physical Review A, 2004
    Co-Authors: J. M. Geremia, Herschel Rabitz
    Abstract:

    Optimal identification (OI) is a recently developed procedure for extracting information about Quantum Hamiltonians from experimental data. It employs techniques from coherent learning control to drive the Quantum system such that dynamical measurements provide maximal information about its Hamiltonian. OI is an optimal procedure as initially presented; however, the data inversion component is computationally expensive. Here, we demonstrate that highly efficient global, nonlinear, map-facilitated inversion procedures can be combined with the OI concept to make it more suitable for laboratory implementation. A simulation of map-facilitated OI illustrates how the input-output maps can greatly accelerate the data inversion process.

  • Optimal Hamiltonian identification: The synthesis of Quantum optimal control and Quantum inversion
    Journal of Chemical Physics, 2003
    Co-Authors: J. M. Geremia, Herschel Rabitz
    Abstract:

    We introduce optimal identification (OI), a collaborative laboratory/computational algorithm for extracting Quantum Hamiltonians from experimental data specifically sought to minimize the inversion error. OI incorporates the components of Quantum control and inversion by combining ultrafast pulse shaping technology and high throughput experiments with global inversion techniques to actively identify Quantum Hamiltonians from tailored observations. The OI concept rests on the general notion that optimal data can be measured under the influence of suitable controls to minimize uncertainty in the extracted Hamiltonian despite data limitations such as finite resolution and noise. As an illustration of the operating principles of OI, the transition dipole moments of a multilevel Quantum Hamiltonian are extracted from simulated population transfer experiments. The OI algorithm revealed a simple optimal experiment that determined the Hamiltonian matrix elements to an accuracy two orders of magnitude better than ...

Alireza Shabani - One of the best experts on this subject based on the ideXlab platform.

  • estimation of many body Quantum Hamiltonians via compressive sensing
    Physical Review A, 2011
    Co-Authors: Alireza Shabani, Masoud Mohseni, Seth Lloyd, Robert L Kosut, Herschel Rabitz
    Abstract:

    We develop an efficient and robust approach for Quantum measurement of nearly sparse many-body Quantum Hamiltonians based on the method of compressive sensing. This work demonstrates that with only O(sln(d)) experimental configurations, consisting of random local preparations and measurements, one can estimate the Hamiltonian of a d-dimensional system, provided that the Hamiltonian is nearly s sparse in a known basis. The classical postprocessing is a convex optimization problem on the total Hilbert space which is generally not scalable. We numerically simulate the performance of this algorithm for three- and four-body interactions in spin-coupled Quantum dots and atoms in optical lattices. Furthermore, we apply the algorithm to characterize Hamiltonian fine structure and unknown system-bath interactions.

  • estimation of many body Quantum Hamiltonians via compressive sensing
    APS, 2011
    Co-Authors: Alireza Shabani, Masoud Mohseni, Seth Lloyd, Robert L Kosut, Herschel Rabitz
    Abstract:

    A. Shabani,1 M. Mohseni,2 S. Lloyd,2,3 R. L. Kosut,4 and H. Rabitz1 1Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA 2Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 4SC Solutions, Sunnyvale, California 94085, USA (Received 1 March 2010; revised manuscript received 21 February 2011; published 11 July 2011)

Andrei Iftimovici - One of the best experts on this subject based on the ideXlab platform.