Automated searching for quantum subsystem codes

Quantum error correction allows for faulty quantum systems to behave in an effectively error free manner. One important class of techniques for quantum error correction is the class of quantum subsystem codes, which are relevant both to active quantum error correcting schemes as well as to the design of self-correcting quantum memories. Previous approaches for investigating these codes have focused on applying theoretical analysis to look for interesting codes and to investigate their properties. In this paper we present an alternative approach that uses computational analysis to accomplish the same goals. Specifically, we present an algorithm that computes the optimal quantum subsystem code that can be implemented given an arbitrary set of measurement operators that are tensor products of Pauli operators. We then demonstrate the utility of this algorithm by performing a systematic investigation of the quantum subsystem codes that exist in the setting where the interactions are limited to 2-body interactions between neighbors on lattices derived from the convex uniform tilings of the plane.Comment: 38 pages, 15 figure, 10 tables. The algorithm described in this paper is available as both library and a command line program (including full source code) that can be downloaded from http://github.com/gcross/CodeQuest/downloads. The source code used to apply the algorithm to scan the lattices is available upon request. Please feel free to contact the authors with question

Sampling from Stochastic Finite Automata with Applications to CTC Decoding

Stochastic finite automata arise naturally in many language and speech processing tasks. They include stochastic acceptors, which represent certain probability distributions over random strings. We consider the problem of efficient sampling: drawing random string variates from the probability distribution represented by stochastic automata and transformations of those. We show that path-sampling is effective and can be efficient if the epsilon-graph of a finite automaton is acyclic. We provide an algorithm that ensures this by conflating epsilon-cycles within strongly connected components. Sampling is also effective in the presence of non-injective transformations of strings. We illustrate this in the context of decoding for Connectionist Temporal Classification (CTC), where the predictive probabilities yield auxiliary sequences which are transformed into shorter labeling strings. We can sample efficiently from the transformed labeling distribution and use this in two different strategies for finding the most probable CTC labeling

Area-Universal Rectangular Layouts

A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. Rectangular layouts appear in various applications: as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it might hence be desirable if one rectangular layout can represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. More generally, given any rectangular layout L and any assignment of areas to its regions, we show that there can be at most one layout (up to horizontal and vertical scaling) which is combinatorially equivalent to L and achieves a given area assignment. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists.Comment: 19 pages, 16 figure

Parsimonious Labeling

We propose a new family of discrete energy minimization problems, which we call parsimonious labeling. Specifically, our energy functional consists of unary potentials and high-order clique potentials. While the unary potentials are arbitrary, the clique potentials are proportional to the {\em diversity} of set of the unique labels assigned to the clique. Intuitively, our energy functional encourages the labeling to be parsimonious, that is, use as few labels as possible. This in turn allows us to capture useful cues for important computer vision applications such as stereo correspondence and image denoising. Furthermore, we propose an efficient graph-cuts based algorithm for the parsimonious labeling problem that provides strong theoretical guarantees on the quality of the solution. Our algorithm consists of three steps. First, we approximate a given diversity using a mixture of a novel hierarchical $P^n$ Potts model. Second, we use a divide-and-conquer approach for each mixture component, where each subproblem is solved using an effficient $\alpha$-expansion algorithm. This provides us with a small number of putative labelings, one for each mixture component. Third, we choose the best putative labeling in terms of the energy value. Using both sythetic and standard real datasets, we show that our algorithm significantly outperforms other graph-cuts based approaches

Universal Communication, Universal Graphs, and Graph Labeling

We introduce a communication model called universal SMP, in which Alice and Bob receive a function f belonging to a family ?, and inputs x and y. Alice and Bob use shared randomness to send a message to a third party who cannot see f, x, y, or the shared randomness, and must decide f(x,y). Our main application of universal SMP is to relate communication complexity to graph labeling, where the goal is to give a short label to each vertex in a graph, so that adjacency or other functions of two vertices x and y can be determined from the labels ?(x), ?(y). We give a universal SMP protocol using O(k^2) bits of communication for deciding whether two vertices have distance at most k in distributive lattices (generalizing the k-Hamming Distance problem in communication complexity), and explain how this implies a O(k^2 log n) labeling scheme for deciding dist(x,y) ? k on distributive lattices with size n; in contrast, we show that a universal SMP protocol for determining dist(x,y) ? 2 in modular lattices (a superset of distributive lattices) has super-constant ?(n^{1/4}) communication cost. On the other hand, we demonstrate that many graph families known to have efficient adjacency labeling schemes, such as trees, low-arboricity graphs, and planar graphs, admit constant-cost communication protocols for adjacency. Trees also have an O(k) protocol for deciding dist(x,y) ? k and planar graphs have an O(1) protocol for dist(x,y) ? 2, which implies a new O(log n) labeling scheme for the same problem on planar graphs

Worst-case Optimal Submodular Extensions for Marginal Estimation

Submodular extensions of an energy function can be used to efficiently compute approximate marginals via variational inference. The accuracy of the marginals depends crucially on the quality of the submodular extension. To identify the best possible extension, we show an equivalence between the submodular extensions of the energy and the objective functions of linear programming (LP) relaxations for the corresponding MAP estimation problem. This allows us to (i) establish the worst-case optimality of the submodular extension for Potts model used in the literature; (ii) identify the worst-case optimal submodular extension for the more general class of metric labeling; and (iii) efficiently compute the marginals for the widely used dense CRF model with the help of a recently proposed Gaussian filtering method. Using synthetic and real data, we show that our approach provides comparable upper bounds on the log-partition function to those obtained using tree-reweighted message passing (TRW) in cases where the latter is computationally feasible. Importantly, unlike TRW, our approach provides the first practical algorithm to compute an upper bound on the dense CRF model.Comment: Accepted to AISTATS 201