IA Scholar Query: How hard is deciding trivial versus nontrivial in the dihedral coset problem?
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Internet Archive Scholar query results feedeninfo@archive.orgTue, 30 Aug 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Straightening Out the Frobenius-Schur Indicator
https://scholar.archive.org/work/voswa6a34jeafhtlephnl6gojq
The Frobenius-Schur indicator is a parameter κ_a=± 1 assigned to each self-dual particle a in a TQFT. If κ_a is negative then straightening out a timelike zig-zag in the worldline of a particle of type a can incur a minus sign and in this case the amplitude associated with the diagram is not invariant under deformation. This has caused some confusion about the topological invariance of even simple theories to space-time deformations. We clarify that, given a TQFT with negative Frobenius-Schur indicators, there are two distinct conventions commonly used to interpret a spacetime diagram as a physical amplitude, only one of which is isotopy invariant. We clarify in what sense TQFTs based on Chern-Simons theory with negative Frobenius-Schur indicators are isotopy invariant, and we explain how the Frobenius-Schur indicator is intimately linked with the need to frame world-lines in Chern-Simons theory. Further, in the non-isotopy-invariant interpretation of the diagram algebra we show how a trick of bookkeeping can usually be invoked to push minus signs onto the diagrammatic value of a loop (the "loop weight"), such that most of the evaluation of a diagram does not incur minus signs from straightening zig-zags, and only at the last step minus signs are added. We explain the conditions required for this to be possible. We then further examine what is required in order for a theory to have full isotopy invariance of planar spacetime diagrams, and discover that, if we have successfully pushed the signs from zig-zags onto the loop weight, the only possible obstruction to this is given by an object related to vertices, known as the "third Frobenius-Schur indicator". We finally discuss the extent to which this gives us full isotopy invariance for braided theories.Steven H. Simon, Joost K. Slingerlandwork_voswa6a34jeafhtlephnl6gojqTue, 30 Aug 2022 00:00:00 GMTTaming Genus 0 (or 1) components on variables-separated equations
https://scholar.archive.org/work/b7b62h6m7rfstjoklhroxxwyme
To figure properties of a curve of form C_f,g = (x,y)| f(x) - g(y)= 0 you must address the genus 0 and 1 components of its projective normalization C̃_f,g. For f and g polynomials with f indecomposable, [Fr73a] distinguished C̃_f,g with u=1 versus u > 1 components (Schinzel's problem). For u = 1, [Prop. 1, Fr73b] gave a direct genus formula. To complete u > 1 required an adhoc genus computation. [Pak22] dropped the indecomposable and polynomial restrictions but added C̃_f,g is irreducible (u = 1). He showed - for fixed f - unless the Galois closure of the cover for f has genus 0 or 1, the genus grows linearly in deg(g). Method I and Method II extend [Prop. 1, Fr73b] using Nielsen classes to generalize Pakovich's formulation for u > 1. Method I plays on the covers f and g to the z-line, P^1_z, from which we compute the fiber product. Method II uses the projection to the y-line, P^1_y, based on explicitly computing branch cycles for this cover. Hurwitz families track the significance of these components. Expanding on [Prop. 2, Fr73a] shows how to approach Pakovich's problem. With no loss, start with (f^*,g^*) which have the same Galois closures, and for which their canonical representations are entangled. They, therefore, produce more than one component on the fiber product. Then, we classify the possible component types, W, that appear on C̃_f^*,g^* using the branch cycles for W that come from Method II. The result is a Nielsen class formulation telling explicitly what g_1s to avoid to assure the growth of the component genuses of C̃_f*,g*og_1 as deg(g_1) increases. Of particular note: using and expanding on Nielsen classes and the solution of the genus 0 problem (classifying the monodromy groups of indecomposable rational functions).Michael D. Friedwork_b7b62h6m7rfstjoklhroxxwymeFri, 19 Aug 2022 00:00:00 GMTGeneral Framework for Randomized Benchmarking
https://scholar.archive.org/work/yttxrwgyzvdfrg6syevhshfhni
Randomized benchmarking refers to a collection of protocols that in the past decade have become central methods for characterizing quantum gates. These protocols aim at efficiently estimating the quality of a set of quantum gates in a way that is resistant to state preparation and measurement errors. Over the years many versions have been developed, however a comprehensive theoretical treatment of randomized benchmarking has been missing. In this work, we develop a rigorous framework of randomized benchmarking general enough to encompass virtually all known protocols as well as novel, more flexible extensions. Overcoming previous limitations on error models and gate sets, this framework allows us, for the first time, to formulate realistic conditions under which we can rigorously guarantee that the output of any randomized benchmarking experiment is well described by a linear combination of matrix exponential decays. We complement this with a detailed analysis of the fitting problem associated with randomized benchmarking data. We introduce modern signal processing techniques to randomized benchmarking, prove analytical sample complexity bounds, and numerically evaluate performance and limitations. In order to reduce the resource demands of this fitting problem, we introduce novel, scalable postprocessing techniques to isolate exponential decays, significantly improving the practical feasibility of a large set of randomized benchmarking protocols. These postprocessing techniques overcome shortcomings in efficiency of several previously proposed methods such as character benchmarking and linear-cross entropy benchmarking. Finally, we discuss, in full generality, how and when randomized benchmarking decay rates can be used to infer quality measures like the average fidelity. On the technical side, our work substantially extends the recently developed Fourier-theoretic perspective on randomized benchmarking by making use of the perturbation theory of invariant subspaces, as well as ideas from signal processing.J. Helsen, I. Roth, E. Onorati, A.H. Werner, J. Eisertwork_yttxrwgyzvdfrg6syevhshfhniThu, 16 Jun 2022 00:00:00 GMTA suite of quantum algorithms for the shortestvector problem
https://scholar.archive.org/work/ej6v2xo54bfu3lbigheg6n577e
Crytography has come to be an essential part of the cybersecurity infrastructure that provides a safe environment for communications in an increasingly connected world. The advent of quantum computing poses a threat to the foundations of the current widely-used cryptographic model, due to the breaking of most of the cryptographic algorithms used to provide confidentiality, authenticity, and more. Consequently a new set of cryptographic protocols have been designed to be secure against quantum computers, and are collectively known as post-quantum cryptography (PQC). A forerunner among PQC is lattice-based cryptography, whose security relies upon the hardness of a number of closely related mathematical problems, one of which is known as the shortest vector problem (SVP). In this thesis I describe a suite of quantum algorithms that utilize the energy minimization principle to attack the shortest vector problem. The algorithms outlined span the gate-model and continuous time quantum computing, and explore methods of parameter optimization via variational methods, which are thought to be effective on near-term quantum computers. The performance of the algorithms are analyzed numerically, analytically, and on quantum hardware where possible. I explain how the results obtained in the pursuit of solving SVP apply more broadly to quantum algorithms seeking to solve general real-world problems; minimize the effect of noise on imperfect hardware; and improve efficiency of parameter optimization.David Joseph, Cong Ling, Academic Centres Of Excellence In Cyber Security Researchwork_ej6v2xo54bfu3lbigheg6n577eWed, 04 May 2022 00:00:00 GMT