140 research outputs found
Approximations of Schatten Norms via Taylor Expansions
In this paper we consider symmetric, positive semidefinite (SPSD) matrix
and present two algorithms for computing the -Schatten norm . The
first algorithm works for any SPSD matrix . The second algorithm works for
non-singular SPSD matrices and runs in time that depends on , where is the -th
eigenvalue of . Our methods are simple and easy to implement and can be
extended to general matrices. Our algorithms improve, for a range of
parameters, recent results of Musco, Netrapalli, Sidford, Ubaru and Woodruff
(ITCS 2018) and match the running time of the methods by Han, Malioutov, Avron,
and Shin (SISC 2017) while avoiding computations of coefficients of Chebyshev
polynomials
Taylor approximations of operator functions
This survey on approximations of perturbed operator functions addresses
recent advances and some of the successful methods.Comment: 12 page
Faster quantum and classical SDP approximations for quadratic binary optimization
We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. The class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms. We also provide an efficient randomized rounding procedure that converts approximately optimal SDP solutions into constant factor approximations of the original quadratic optimization problem
Entanglement entropies of an interval in the free Schrödinger field theory on the half line
We study the entanglement entropies of an interval adjacent to the boundary of the half line for the free fermionic spinless Schrodinger field theory at finite density and zero temperature, with either Neumann or Dirichlet boundary conditions. They are finite functions of the dimensionless parameter given by the product of the Fermi momentum and the length of the interval. The entanglement entropy displays an oscillatory behaviour, differently from the case of the interval on the whole line. This behaviour is related to the Friedel oscillations of the mean particle density on the half line at the entangling point. We find analytic expressions for the expansions of the entanglement entropies in the regimes of small and large values of the dimensionless parameter. They display a remarkable agreement with the curves obtained numerically. The analysis is extended to a family of free fermionic Lifshitz models labelled by their integer Lifshitz exponent, whose parity determines the properties of the entanglement entropies. The cumulants of the local charge operator and the Schatten norms of the underlying kernels are also explored
A Milstein scheme for SPDEs
This article studies an infinite dimensional analog of Milstein's scheme for
finite dimensional stochastic ordinary differential equations (SODEs). The
Milstein scheme is known to be impressively efficient for SODEs which fulfill a
certain commutativity type condition. This article introduces the infinite
dimensional analog of this commutativity type condition and observes that a
certain class of semilinear stochastic partial differential equation (SPDEs)
with multiplicative trace class noise naturally fulfills the resulting infinite
dimensional commutativity condition. In particular, a suitable infinite
dimensional analog of Milstein's algorithm can be simulated efficiently for
such SPDEs and requires less computational operations and random variables than
previously considered algorithms for simulating such SPDEs. The analysis is
supported by numerical results for a stochastic heat equation and stochastic
reaction diffusion equations showing signifficant computational savings.Comment: The article is slightly revised and shortened. In particular, some
numerical simulations are remove
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