Complete flavor decompositions of the matrix elements of the scalar, axial, and tensor currents in the proton, deuteron, diproton, and ^{3}He at SU(3)-symmetric values of the quark masses corresponding to a pion mass m_{π}∼806  MeV are determined using lattice quantum chromodynamics. At the physical quark masses, the scalar interactions constrain mean-field models of nuclei and the low-energy interactions of nuclei with potential dark matter candidates. The axial and tensor interactions of nuclei constrain their spin content, integrated transversity, and the quark contributions to their electric dipole moments. External fields are used to directly access the quark-line connected matrix elements of quark bilinear operators, and a combination of stochastic estimation techniques is used to determine the disconnected sea-quark contributions. The calculated matrix elements differ from, and are typically smaller than, naive single-nucleon estimates. Given the particularly large, O(10%), size of nuclear effects in the scalar matrix elements, contributions from correlated multinucleon effects should be quantified in the analysis of dark matter direct-detection experiments using nuclear targets.

Many fields require computing the trace of the inverse of a large, sparse matrix. Since dense matrix methods are not practical, the typical method used for such computations is the Hutchinson method which is a Monte Carlo (MC) averaging over matrix quadratures. To improve its slow convergence, several variance reductions techniques have been proposed. In this paper, we study the effects of deflating the near null singular value space. We make two main contributions: One theoretical and one by engineering a solution to a real world application.First, we analyze the variance of the Hutchinson method as a function of the deflated singular values and vectors. By assuming additionally that the singular vectors are random unitary matrices, we arrive at concise formulas for the deflated variance that include only the variance and the mean of the singular values. We make the remarkable observation that deflation may increase variance for Hermitian matrices but not for non-Hermitian ones. The theory can be used as a model for predicting the benefits of deflation. Experimentation shows that the model is robust even when the singular vectors are not random.Second, we use deflation in the context of a large scale application of "disconnected diagrams" in Lattice QCD. On lattices, Hierarchical Probing (HP) has previously provided significant variance reduction over MC by removing "error" from neighboring nodes of increasing distance in the lattice. Although deflation used directly on MC yields a limited improvement of 30% in our problem, when combined with HP they reduce variance by a factor of about 60 over MC. We explain this synergy theoretically and provide a thorough experimental analysis. One of the important steps of our solution is the pre-computation of 1000 smallest singular values of an ill-conditioned matrix of size 25 million. Using the state-of-the-art packages PRIMME and a domain-specific Algebraic Multigrid preconditioner, we solve this large eigenvalue computation on 32 nodes of Cray Edison in about 1.5 hours and at a fraction of the cost of our trace computation.

The role of gluons in the structure of the nucleon and light nuclei is investigated using lattice quantum chromodynamics (QCD) calculations. The first moment of the unpolarized gluon distribution is studied in nuclei up to atomic number A ¼ 3 at quark masses corresponding to pion masses of m π ∼ 450 and 806 MeV. Nuclear modification of this quantity defines a gluonic analogue of the EMC effect and is constrained to be less than ∼10% in these nuclei. This is consistent with expectations from phenomenological quark distributions and the momentum sum rule. In the deuteron, the combination of gluon distributions corresponding to the b 1 structure function is found to have a small first moment compared with the corresponding momentum fraction. The first moment of the gluon transversity structure function is also investigated in the spin-1 deuteron, where a nonzero signal is observed at m π ∼ 806 MeV. This is the first indication of gluon contributions to nuclear structure that can not be associated with an individual nucleon.

Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of 10 −8 .