How to decompose arbitrary continuous-variable quantum operations

We present a general, systematic, and efficient method for decomposing any given exponential operator of bosonic mode operators, describing an arbitrary multi-mode Hamiltonian evolution, into a set of universal unitary gates. Although our approach is mainly oriented towards continuous-variable quantum computation, it may be used more generally whenever quantum states are to be transformed deterministically, e.g. in quantum control, discrete-variable quantum computation, or Hamiltonian simulation. We illustrate our scheme by presenting decompositions for various nonlinear Hamiltonians including quartic Kerr interactions. Finally, we conclude with two potential experiments utilizing offline-prepared optical cubic states and homodyne detections, in which quantum information is processed optically or in an atomic memory using quadratic light-atom interactions.Comment: Ver. 3: published version with supplementary materia

Modular Networks: Learning to Decompose Neural Computation

Scaling model capacity has been vital in the success of deep learning. For a typical network, necessary compute resources and training time grow dramatically with model size. Conditional computation is a promising way to increase the number of parameters with a relatively small increase in resources. We propose a training algorithm that flexibly chooses neural modules based on the data to be processed. Both the decomposition and modules are learned end-to-end. In contrast to existing approaches, training does not rely on regularization to enforce diversity in module use. We apply modular networks both to image recognition and language modeling tasks, where we achieve superior performance compared to several baselines. Introspection reveals that modules specialize in interpretable contexts.Comment: NIPS 201

Using Wavelets to decompose time-frequency economic relations

Economic agents simultaneously operate at different horizons. Many economic processes are the result of the actions of several agents with different term objectives. Therefore, economic time-series is a combination of components operating on different frequencies. Several questions about the data are connected to the understanding of the time-series behavior at different frequencies. While Fourier analysis is not appropriate to study the cyclical nature of economic time-series, because these are rarely stationary, wavelet analysis performs the estimation of the spectral characteristics of a time-series as a function of time. In spite of all its advantages, wavelets are hardly ever used in economics. The purpose of this paper is to show that cross wavelet analysis can be used to directly study the interactions different time-series in the time-frequency domain. We use wavelets to analyze the impact of interest rate price changes on some macroeconomic variables: Industrial Production, Inflation and the monetary aggregates M1 and M2. Specifically, three tools are utilized: the wavelet power spectrum, wavelet coherency and wavelet phase-difference. These instruments illustrate how the use of wavelets may help to unravel economic time-frequency relations that would otherwise remain hidden.Monetary policy, time-frequency analysis, non-stationary time series, wavelets, cross wavelets, wavelet coherency.

Symplectic analogs of polar decomposition and their applications to bosonic Gaussian channels

We obtain several analogs of real polar decomposition for even dimensional matrices. In particular, we decompose a non-degenerate matrix as a product of a Hamiltonian and an anti-symplectic matrix and under additional requirements we decompose a matrix as a skew-Hamiltonian and a symplectic matrix. We apply our results to study bosonic Gaussian channels up to inhomogeneous symplectic transforms

On a question of Drinfeld on the Weil representation I: the finite field case

Let F be a finite field of odd cardinality, and let G= GL2(F). The group G \times G \times G acts on F^2 \otimes F^2 \otimes F^2 via symplectic similitudes, and has a natural Weil representation. Answering a question rasised by V. Drinfeld, we decompose that representation into irreducibles. We also decompose the analogous representation of GL2(A), where A is a cubic algebra over F.Comment: 29 pages. Comments welcom

On multiplicity-free skew characters and the Schubert Calculus

In this paper we classify the multiplicity-free skew characters of the symmetric group. Furthermore we show that the Schubert calculus is equivalent to that of skew characters in the following sense: If we decompose the product of two Schubert classes we get the same as if we decompose a skew character and replace the irreducible characters by Schubert classes of the `inverse' partitions (Theorem 4.2).Comment: 14 pages, to appear in Annals. Comb. minor changes from v1 to v2 as suggested by the referees, Example 3.4 inserted so numeration changed in section

Using Red Clump Stars to Decompose the Galactic Magnetic Field with Distance

A new method for measuring the large-scale structure of the Galactic magnetic field is presented. The Galactic magnetic field has been probed through the Galactic disk with near-infrared starlight polarimetry, however the distance to each background star is unknown. Using red clump stars as near-infrared standard candles, this work presents the first attempt to decompose the line of sight structure of the sky-projected Galactic magnetic field. Two example lines-of-sight are decomposed: toward a field with many red clump stars and toward a field with few red clump stars. A continuous estimate of magnetic field orientation over several kiloparsecs of distance is possible in the field with many red clump stars, while only discrete estimates are possible in the sparse example. toward the Outer Galaxy, there is a continuous field orientation with distance that shows evidence of perturbation by the Galactic warp. toward the Inner Galaxy, evidence for a large-scale change in the magnetic field geometry is consistent with models of magnetic field reversals, independently derived from Faraday rotation studies. A photo-polarimetric method for identifying candidate intrinsically polarized stars is also presented. The future application of this method to large regions of the sky will begin the process of mapping the Galactic magnetic field in a way never before possible.Comment: 11 pages, 8 figures, 2 tables, accepted for publication in The Astronomical Journa