Efficient Distribution Analysis via Graph Contraction

Alignment and distribution of data by an optimizing compiler is a dream of both manufacturers and users of parallel computers. The distribution problem has been formulated as an NP-complete graph optimization problem. The graphs arising in applications are large, and the optimization problem does not lend itself to traditional heuristic optimization techniques. In this paper, we improve some earlier results on methods that use graph contraction to reduce the size of a distribution problem. We report on an experiment using seven example programs that show these contraction operations to be effective in practice; we obtain from 60 to 99 percent reductions in problem size, the larger number being more typical, without loss of solution quality

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

On the Geometry of Message Passing Algorithms for Gaussian Reciprocal Processes

Reciprocal processes are acausal generalizations of Markov processes introduced by Bernstein in 1932. In the literature, a significant amount of attention has been focused on developing dynamical models for reciprocal processes. Recently, probabilistic graphical models for reciprocal processes have been provided. This opens the way to the application of efficient inference algorithms in the machine learning literature to solve the smoothing problem for reciprocal processes. Such algorithms are known to converge if the underlying graph is a tree. This is not the case for a reciprocal process, whose associated graphical model is a single loop network. The contribution of this paper is twofold. First, we introduce belief propagation for Gaussian reciprocal processes. Second, we establish a link between convergence analysis of belief propagation for Gaussian reciprocal processes and stability theory for differentially positive systems.Comment: 15 pages; Typos corrected; This paper introduces belief propagation for Gaussian reciprocal processes and extends the convergence analysis in arXiv:1603.04419 to the Gaussian cas

Relaxed Schedulers Can Efficiently Parallelize Iterative Algorithms

There has been significant progress in understanding the parallelism inherent to iterative sequential algorithms: for many classic algorithms, the depth of the dependence structure is now well understood, and scheduling techniques have been developed to exploit this shallow dependence structure for efficient parallel implementations. A related, applied research strand has studied methods by which certain iterative task-based algorithms can be efficiently parallelized via relaxed concurrent priority schedulers. These allow for high concurrency when inserting and removing tasks, at the cost of executing superfluous work due to the relaxed semantics of the scheduler. In this work, we take a step towards unifying these two research directions, by showing that there exists a family of relaxed priority schedulers that can efficiently and deterministically execute classic iterative algorithms such as greedy maximal independent set (MIS) and matching. Our primary result shows that, given a randomized scheduler with an expected relaxation factor of $k$ in terms of the maximum allowed priority inversions on a task, and any graph on $n$ vertices, the scheduler is able to execute greedy MIS with only an additive factor of poly($k$) expected additional iterations compared to an exact (but not scalable) scheduler. This counter-intuitive result demonstrates that the overhead of relaxation when computing MIS is not dependent on the input size or structure of the input graph. Experimental results show that this overhead can be clearly offset by the gain in performance due to the highly scalable scheduler. In sum, we present an efficient method to deterministically parallelize iterative sequential algorithms, with provable runtime guarantees in terms of the number of executed tasks to completion.Comment: PODC 2018, pages 377-386 in proceeding

Classical Ising model test for quantum circuits

We exploit a recently constructed mapping between quantum circuits and graphs in order to prove that circuits corresponding to certain planar graphs can be efficiently simulated classically. The proof uses an expression for the Ising model partition function in terms of quadratically signed weight enumerators (QWGTs), which are polynomials that arise naturally in an expansion of quantum circuits in terms of rotations involving Pauli matrices. We combine this expression with a known efficient classical algorithm for the Ising partition function of any planar graph in the absence of an external magnetic field, and the Robertson-Seymour theorem from graph theory. We give as an example a set of quantum circuits with a small number of non-nearest neighbor gates which admit an efficient classical simulation.Comment: 17 pages, 2 figures. v2: main result strengthened by removing oracular settin

Tensor network states in time-bin quantum optics

The current shift in the quantum optics community towards large-size experiments -- with many modes and photons -- necessitates new classical simulation techniques that go beyond the usual phase space formulation of quantum mechanics. To address this pressing demand we formulate linear quantum optics in the language of tensor network states. As a toy model, we extensively analyze the quantum and classical correlations of time-bin interference in a single fiber loop. We then generalize our results to more complex time-bin quantum setups and identify different classes of architectures for high-complexity and low-overhead boson sampling experiments