The Complexity of the Separable Hamiltonian Problem

In this paper, we study variants of the canonical Local-Hamiltonian problem where, in addition, the witness is promised to be separable. We define two variants of the Local-Hamiltonian problem. The input for the Separable-Local-Hamiltonian problem is the same as the Local-Hamiltonian problem, i.e. a local Hamiltonian and two energies a and b, but the question is somewhat different: the answer is YES if there is a separable quantum state with energy at most a, and the answer is NO if all separable quantum states have energy at least b. The Separable-Sparse-Hamiltonian problem is defined similarly, but the Hamiltonian is not necessarily local, but rather sparse. We show that the Separable-Sparse-Hamiltonian problem is QMA(2)-Complete, while Separable-Local-Hamiltonian is in QMA. This should be compared to the Local-Hamiltonian problem, and the Sparse-Hamiltonian problem which are both QMA-Complete. To the best of our knowledge, Separable-SPARSE-Hamiltonian is the first non-trivial problem shown to be QMA(2)-Complete

Analysing multiparticle quantum states

The analysis of multiparticle quantum states is a central problem in quantum information processing. This task poses several challenges for experimenters and theoreticians. We give an overview over current problems and possible solutions concerning systematic errors of quantum devices, the reconstruction of quantum states, and the analysis of correlations and complexity in multiparticle density matrices.Comment: 20 pages, 4 figures, prepared for proceedings of the "Quantum [Un]speakables II" conference (Vienna, 2014

Effective interactions and large-scale diagonalization for quantum dots

The widely used large-scale diagonalization method using harmonic oscillator basis functions (an instance of the Rayleigh-Ritz method, also called a spectral method, configuration-interaction method, or ``exact diagonalization'' method) is systematically analyzed using results for the convergence of Hermite function series. We apply this theory to a Hamiltonian for a one-dimensional model of a quantum dot. The method is shown to converge slowly, and the non-smooth character of the interaction potential is identified as the main problem with the chosen basis, while on the other hand its important advantages are pointed out. An effective interaction obtained by a similarity transformation is proposed for improving the convergence of the diagonalization scheme, and numerical experiments are performed to demonstrate the improvement. Generalizations to more particles and dimensions are discussed.Comment: 7 figures, submitted to Physical Review B Single reference error fixe

Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach

This paper introduces and analyses the new grid-based tensor approach to approximate solution of the elliptic eigenvalue problem for the 3D lattice-structured systems. We consider the linearized Hartree-Fock equation over a spatial $L_1\times L_2\times L_3$ lattice for both periodic and non-periodic problem setting, discretized in the localized Gaussian-type orbitals basis. In the periodic case, the Galerkin system matrix obeys a three-level block-circulant structure that allows the FFT-based diagonalization, while for the finite extended systems in a box (Dirichlet boundary conditions) we arrive at the perturbed block-Toeplitz representation providing fast matrix-vector multiplication and low storage size. The proposed grid-based tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) in the periodic case the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems in a box with Dirichlet boundary conditions are treated numerically by our previous tensor solver for single molecules, which makes possible calculations on rather large $L_1\times L_2\times L_3$ lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic $L\times 1\times 1$ lattice chain in a 3D rectangular "tube" with $L$ up to several hundred confirm the theoretical complexity bounds for the block-structured eigenvalue solvers in the limit of large $L$.Comment: 30 pages, 12 figures. arXiv admin note: substantial text overlap with arXiv:1408.383

Many body physics from a quantum information perspective

The quantum information approach to many body physics has been very successful in giving new insight and novel numerical methods. In these lecture notes we take a vertical view of the subject, starting from general concepts and at each step delving into applications or consequences of a particular topic. We first review some general quantum information concepts like entanglement and entanglement measures, which leads us to entanglement area laws. We then continue with one of the most famous examples of area-law abiding states: matrix product states, and tensor product states in general. Of these, we choose one example (classical superposition states) to introduce recent developments on a novel quantum many body approach: quantum kinetic Ising models. We conclude with a brief outlook of the field.Comment: Lectures from the Les Houches School on "Modern theories of correlated electron systems". Improved version new references adde