qTorch: The Quantum Tensor Contraction Handler

Classical simulation of quantum computation is necessary for studying the numerical behavior of quantum algorithms, as there does not yet exist a large viable quantum computer on which to perform numerical tests. Tensor network (TN) contraction is an algorithmic method that can efficiently simulate some quantum circuits, often greatly reducing the computational cost over methods that simulate the full Hilbert space. In this study we implement a tensor network contraction program for simulating quantum circuits using multi-core compute nodes. We show simulation results for the Max-Cut problem on 3- through 7-regular graphs using the quantum approximate optimization algorithm (QAOA), successfully simulating up to 100 qubits. We test two different methods for generating the ordering of tensor index contractions: one is based on the tree decomposition of the line graph, while the other generates ordering using a straight-forward stochastic scheme. Through studying instances of QAOA circuits, we show the expected result that as the treewidth of the quantum circuit's line graph decreases, TN contraction becomes significantly more efficient than simulating the whole Hilbert space. The results in this work suggest that tensor contraction methods are superior only when simulating Max-Cut/QAOA with graphs of regularities approximately five and below. Insight into this point of equal computational cost helps one determine which simulation method will be more efficient for a given quantum circuit. The stochastic contraction method outperforms the line graph based method only when the time to calculate a reasonable tree decomposition is prohibitively expensive. Finally, we release our software package, qTorch (Quantum TensOR Contraction Handler), intended for general quantum circuit simulation.Comment: 21 pages, 8 figure

Relativistic MHD and black hole excision: Formulation and initial tests

A new algorithm for solving the general relativistic MHD equations is described in this paper. We design our scheme to incorporate black hole excision with smooth boundaries, and to simplify solving the combined Einstein and MHD equations with AMR. The fluid equations are solved using a finite difference Convex ENO method. Excision is implemented using overlapping grids. Elliptic and hyperbolic divergence cleaning techniques allow for maximum flexibility in choosing coordinate systems, and we compare both methods for a standard problem. Numerical results of standard test problems are presented in two-dimensional flat space using excision, overlapping grids, and elliptic and hyperbolic divergence cleaning.Comment: 22 pages, 8 figure

A Tensor-Based Dictionary Learning Approach to Tomographic Image Reconstruction

We consider tomographic reconstruction using priors in the form of a dictionary learned from training images. The reconstruction has two stages: first we construct a tensor dictionary prior from our training data, and then we pose the reconstruction problem in terms of recovering the expansion coefficients in that dictionary. Our approach differs from past approaches in that a) we use a third-order tensor representation for our images and b) we recast the reconstruction problem using the tensor formulation. The dictionary learning problem is presented as a non-negative tensor factorization problem with sparsity constraints. The reconstruction problem is formulated in a convex optimization framework by looking for a solution with a sparse representation in the tensor dictionary. Numerical results show that our tensor formulation leads to very sparse representations of both the training images and the reconstructions due to the ability of representing repeated features compactly in the dictionary.Comment: 29 page

Hamiltonian Relaxation

Due to the complexity of the required numerical codes, many of the new formulations for the evolution of the gravitational fields in numerical relativity are not tested on binary evolutions. We introduce in this paper a new testing ground for numerical methods based on the simulation of binary neutron stars. This numerical setup is used to develop a new technique, the Hamiltonian relaxation (HR), that is benchmarked against the currently most stable simulations based on the BSSN method. We show that, while the length of the HR run is somewhat shorter than the equivalent BSSN simulation, the HR technique improves the overall quality of the simulation, not only regarding the satisfaction of the Hamiltonian constraint, but also the behavior of the total angular momentum of the binary. The latest quantity agrees well with post-Newtonian estimations for point-mass binaries in circular orbits.Comment: More detailed description of the numerical implementation added and some typos corrected. Version accepted for publication in Class. and Quantum Gravit

Veamy: an extensible object-oriented C++ library for the virtual element method

This paper summarizes the development of Veamy, an object-oriented C++ library for the virtual element method (VEM) on general polygonal meshes, whose modular design is focused on its extensibility. The linear elastostatic and Poisson problems in two dimensions have been chosen as the starting stage for the development of this library. The theory of the VEM, upon which Veamy is built, is presented using a notation and a terminology that resemble the language of the finite element method (FEM) in engineering analysis. Several examples are provided to demonstrate the usage of Veamy, and in particular, one of them features the interaction between Veamy and the polygonal mesh generator PolyMesher. A computational performance comparison between VEM and FEM is also conducted. Veamy is free and open source software