arXiv subjects

Joseph Fluegemann

Publications and source records attributed to Joseph Fluegemann.

5 recordsLinked to original sources

arXiv2026

Singular Points on Positroid Varieties and Planar N=4 Supersymmetric Yang-Mills Theory

Positroid varieties $\Pi_f$ provide a decomposition of the Grassmannian $Gr(k,n)$; they can be enumerated using bounded affine permutations ($f$) which have bijections with a number of interesting combinatorial objects. Furthermore, (the nonnegative part of) positroid varieties parameterize the space that is integrated over when calculating amplitudes in planar N=4 supersymmetric Yang-Mills theory. The main question we answer in this thesis is whether a positroid variety $\Pi_f$ has any geometric singularities. We show that it is sufficient to check singularity at the $T$-fixed points ($\lambda$) and we can obtain the multiplicity at these points $\lambda$ by calculating the equivariant cohomology of $\Pi_f$ restricted to the point. We give 2 ways of doing this: (1) A diagrammatic way using affine pipe dreams and (2) A computational method. For (2), we have written code that does the computation and outputs the multiplicity (files at josephflueg.github.io). We have included tables listing the multiplicities of all the points on positroid varieties up to $n=6$. We also describe an ordering on pairs $(\Pi_f,\lambda)$ given by deletion/contraction that interacts nicely with smoothness, and describe the relationship between affine pipe dreams and finite pipe dreams. In Part II of this thesis connects with physics of planar N=4 SYM. We briefly introduce quantum field theory, leading singularities, and positroids in N=4 SYM. We then explain Britto-Cachazo-Feng-Witten (BCFW) recursion and the BCFW bridge decomposition of an on-shell diagram. We describe how to build an on-shell diagram on a pipe dream using BCFW bridges. Finally, we work out some combinatorics related to inverse soft factors for on-shell diagrams and explore whether singularities in positroid varieties have relevance to amplitudes. This is a revised version of my thesis originally written in August 2024.

math.AG↗

arXiv2024

Smooth Points on Positroid Varieties

In the Grassmannian $Gr_{\mathbb{C}}(k,n)$ we have positroid varieties $\Pi_f$, each indexed by a bounded affine permutation $f$ and containing torus-fixed points $\lambda \in \Pi_f$. In this paper we consider the partially ordered set consisting of quadruples $(k,n,\Pi_f,\lambda)$ (or \textit{(positroid) pairs} $(\Pi_f,\lambda)$ for short). The partial order is the ordering given by the covering relation $\lessdot$ where $(\Pi_f',\lambda') \lessdot (\Pi_f,\lambda)$ if $\Pi_f'$ is obtained by $\Pi_f$ by \textit{deletion} or \textit{contraction.} Using the results of Snider [2010], we know that positroid varieties can be studied in a neighborhood of each of these points by \textit{affine pipe dreams.} Our main theorem provides a quick test of when a positroid variety is smooth at one of these given points. It is sufficient to test smoothness of a positroid variety by using the main result to test smoothness at each of these points. These results can also be applied to the question of whether Schubert varieties in flag manifolds are smooth at points given by 321-avoiding permutations, as studied in Graham/Kreimer [2020]. We have a secondary result, which describes the minimal singular positroid pairs in our ordering - these are the positroid pairs where any deletion or contraction causes it to become smooth.

math.CO↗

arXiv2015

Determination and correction of persistent biases in quantum annealers

Calibration of quantum computing technologies is essential to the effective utilization of their quantum resources. Specifically, the performance of quantum annealers is likely to be significantly impaired by noise in their programmable parameters, effectively misspecification of the computational problem to be solved, often resulting in spurious suboptimal solutions. We developed a strategy to determine and correct persistent, systematic biases between the actual values of the programmable parameters and their user-specified values. We applied the recalibration strategy to two D-Wave Two quantum annealers, one at NASA Ames Research Center in Moffett Field, California, and another at D-Wave Systems in Burnaby, Canada. We show that the recalibration procedure not only reduces the magnitudes of the biases in the programmable parameters but also enhances the performance of the device on a set of random benchmark instances.

quant-ph↗

arXiv2015

A Performance Estimator for Quantum Annealers: Gauge selection and Parameter Setting

With the advent of large-scale quantum annealing devices, several challenges have emerged. For example, it has been shown that the performance of a device can be significantly affected by several degrees of freedom when programming the device; a common example being gauge selection. To date, no experimentally-tested strategy exists to select the best programming specifications. We developed a score function that can be calculated from a number of readouts much smaller than the number of readouts required to find the desired solution. We show how this performance estimator can be used to guide, for example, the selection of the optimal gauges out of a pool of random gauge candidates and how to select the values of parameters for which we have no a priori knowledge of the optimal value. For the latter, we illustrate the concept by applying the score function to set the strength of the parameter intended to enforce the embedding of the logical graph into the hardware architecture, a challenge frequently encountered in the implementation of real-world problem instances. Since the harder the problem instances, the more useful the strategies proposed in this work are, we expect the programming strategies proposed to significantly reduce the time of future benchmark studies and in help finding the solution of hard-to-solve real-world applications implemented in the next generation of quantum annealing devices.

quant-ph↗

arXiv2014

A Quantum Annealing Approach for Fault Detection and Diagnosis of Graph-Based Systems

Diagnosing the minimal set of faults capable of explaining a set of given observations, e.g., from sensor readouts, is a hard combinatorial optimization problem usually tackled with artificial intelligence techniques. We present the mapping of this combinatorial problem to quadratic unconstrained binary optimization (QUBO), and the experimental results of instances embedded onto a quantum annealing device with 509 quantum bits. Besides being the first time a quantum approach has been proposed for problems in the advanced diagnostics community, to the best of our knowledge this work is also the first research utilizing the route Problem $\rightarrow$ QUBO $\rightarrow$ Direct embedding into quantum hardware, where we are able to implement and tackle problem instances with sizes that go beyond previously reported toy-model proof-of-principle quantum annealing implementations; this is a significant leap in the solution of problems via direct-embedding adiabatic quantum optimization. We discuss some of the programmability challenges in the current generation of the quantum device as well as a few possible ways to extend this work to more complex arbitrary network graphs.

quant-ph↗