718 research outputs found

    With a Few Square Roots, Quantum Computing Is as Easy as Pi

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    Rig groupoids provide a semantic model of Π, a universal classical reversible programming language over finite types. We prove that extending rig groupoids with just two maps and three equations about them results in a model of quantum computing that is computationally universal and equationally sound and complete for a variety of gate sets. The first map corresponds to an 8th root of the identity morphism on the unit 1. The second map corresponds to a square root of the symmetry on 1+1. As square roots are generally not unique and can sometimes even be trivial, the maps are constrained to satisfy a nondegeneracy axiom, which we relate to the Euler decomposition of the Hadamard gate. The semantic construction is turned into an extension of Π, called √Π, that is a computationally universal quantum programming language equipped with an equational theory that is sound and complete with respect to the Clifford gate set, the standard gate set of Clifford+T restricted to ≀2 qubits, and the computationally universal Gaussian Clifford+T gate set

    Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries

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    We study the symmetries of closed Majorana chains in 1+1d, including the translation, fermion parity, spatial parity, and time-reversal symmetries. The algebra of the symmetry operators is realized projectively on the Hilbert space, signaling anomalies on the lattice, and constraining the long-distance behavior. In the special case of the free Hamiltonian (and small deformations thereof), the continuum limit is the 1+1d free Majorana CFT. Its continuum chiral fermion parity (−1)FL(-1)^{F_\text{L}} emanates from the lattice translation symmetry. We find a lattice precursor of its mod 8 't Hooft anomaly. Using a Jordan-Wigner transformation, we sum over the spin structures of the lattice model (a procedure known as the GSO projection), while carefully tracking the global symmetries. In the resulting bosonic model of Ising spins, the Majorana translation operator leads to a non-invertible lattice translation symmetry at the critical point. The non-invertible Kramers-Wannier duality operator of the continuum Ising CFT emanates from this non-invertible lattice translation of the transverse-field Ising model.Comment: 78 pages, 1 figure, 1 table. v2: added clarifications and reference

    The Quantum Monadology

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    The modern theory of functional programming languages uses monads for encoding computational side-effects and side-contexts, beyond bare-bone program logic. Even though quantum computing is intrinsically side-effectful (as in quantum measurement) and context-dependent (as on mixed ancillary states), little of this monadic paradigm has previously been brought to bear on quantum programming languages. Here we systematically analyze the (co)monads on categories of parameterized module spectra which are induced by Grothendieck's "motivic yoga of operations" -- for the present purpose specialized to HC-modules and further to set-indexed complex vector spaces. Interpreting an indexed vector space as a collection of alternative possible quantum state spaces parameterized by quantum measurement results, as familiar from Proto-Quipper-semantics, we find that these (co)monads provide a comprehensive natural language for functional quantum programming with classical control and with "dynamic lifting" of quantum measurement results back into classical contexts. We close by indicating a domain-specific quantum programming language (QS) expressing these monadic quantum effects in transparent do-notation, embeddable into the recently constructed Linear Homotopy Type Theory (LHoTT) which interprets into parameterized module spectra. Once embedded into LHoTT, this should make for formally verifiable universal quantum programming with linear quantum types, classical control, dynamic lifting, and notably also with topological effects.Comment: 120 pages, various figure

    With a Few Square Roots, Quantum Computing is as Easy as {\Pi}

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    Rig groupoids provide a semantic model of \PiLang, a universal classical reversible programming language over finite types. We prove that extending rig groupoids with just two maps and three equations about them results in a model of quantum computing that is computationally universal and equationally sound and complete for a variety of gate sets. The first map corresponds to an 8th8^{\text{th}} root of the identity morphism on the unit 11. The second map corresponds to a square root of the symmetry on 1+11+1. As square roots are generally not unique and can sometimes even be trivial, the maps are constrained to satisfy a nondegeneracy axiom, which we relate to the Euler decomposition of the Hadamard gate. The semantic construction is turned into an extension of \PiLang, called \SPiLang, that is a computationally universal quantum programming language equipped with an equational theory that is sound and complete with respect to the Clifford gate set, the standard gate set of Clifford+T restricted to ≀2\le 2 qubits, and the computationally universal Gaussian Clifford+T gate set

    SG-classes, singular symplectic geoemtry, and order preserving isomorphisms

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    Die geometrischen KalkĂŒle von Pseudo-differenzial- und Fourier-Integraloperatoren beruhen auf den symplektischen Eigenschaften des KotangentialbĂŒndels. Um neue KalkĂŒle zu entwickeln, die an eine besondere Geometrie angepasst sind, ist es nötig, singulĂ€r-symplektische Mannigfaltigkeiten zu betrachten. Diese mĂŒssen zuerst verstanden werden, bevor man die zugehorigen OperatorkalkĂŒle konstruieren kann. In dieser Dissertation geben wir neue Einblicke in die singulĂ€r-symplektischen Strukturen, die aus asymptotisch-Euklidischen Mannigfaltigkeiten entstehen. Insbesondere rechnen wir aus, wie die Poisson-Klammer auf SG-Pseudo-Differenzialoperatoren wirkt, und definieren eine neue Klasse symplektischer Abbildungen, die an die geometrischen Besonderheiten angepasst sind. Wir betrachten außerdem die ordnungserhaltenden Isomorphismen der SG-Algebra und zeigen, dass unser Konzept von singulĂ€r-symplektischen Abbildungen natĂŒrlich in diesem Zusammenhang auftaucht. Wir benutzen es, um diese Isomorphismen als Konjugation mit einem SG-Fourier-Integraloperator zu charakterisieren.The geometric theory of pseudo-differential and Fourier Integral Operators relies on the symplectic structure of cotangent bundles. If one is to study calculi with some specific feature adapted to a geometric situation, the corresponding notion of cotangent bundle needs to be adapted as well and leads to spaces with a singular symplectic structure. Analysing these singularities is a necessary step in order to construct the calculus itself. In this thesis we provide some new insights into the symplectic structures arising from asymptotically Euclidean manifolds. In particular, we study the action of the Poisson bracket on SG-pseudo-differential operators and define a new class of singular symplectomorphisms, taking into account the geometric picture. We then consider this notion in the context of the characterisation of order-preserving isomorphisms of the SG-algebra, and show that these are in fact given by conjugation with a Fourier Integral Operator of SG-type

    Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism

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    This note describes a local Poisson structure (up to homotopy) associated with corners in four-dimensional gravity in the coframe (Palatini–Cartan) formalism. This is achieved through the use of the BFV formalism. The corner structure contains in particular an Atiyah algebroid that couples the internal symmetries to diffeomorphisms. The relation with BF theory is also described

    From telescopes to frames and simple groups

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    We introduce the notion of a telescope of groups. Very roughly a telescope is a directed system of groups that contains various commuting images of some fixed group BB. Telescopes are inspired from the theory of groups acting on rooted trees. Imitating known constructions of branch groups, we obtain a number of examples of BB-telescopes and discuss several applications. We give examples of 22-generated infinite amenable simple groups. We show that every finitely generated residually finite (amenable) group embeds into a finitely generated (amenable) LEF simple group. We construct 22-generated frames in products of finite simple groups and show that there are Grothendieck pairs consisting of amenable groups and groups with property (τ)(\tau). We give examples of automorphisms of finitely generated, residually finite, amenable groups that are not inner, but become inner in the profinite completion. We describe non-elementary amenable examples of finitely generated, residually finite groups all of whose finitely generated subnormal subgroups are direct factors.Comment: 41 pages, comments welcom

    Topological Manipulations of Quantum Field Theories

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    In this thesis we study some topological aspects of Quantum Field Theories (QFTs). In particular, we study the way in which an arbitrary QFT can be separated into “local” and “global” data by means of a “symmetry Topological Field Theory” (symmetry TFT). We also study how various “topological manipulations” of the global data correspond to various well-known operations that previously existed in the literature, and how the symmetry TFT perspective provides a systematic tool for studying these topological manipulations. We start by reviewing the bijection between G-symmetric d-dimensional QFTs and boundary conditions for G-gauge theories in (d+1)-dimensions, which effectively defines the symmetry TFT. We use this relationship to study the “orbifold groupoids” which control the composition of “topological manipulations,” relating theories with the same local data but different global data. Particular attention is paid to examples in d = 2 dimensions. We also discuss the extension to fermionic symmetry groups and find that the familiar “Jordan-Wigner transformation” (fermionization) and “GSO projection” (bosonization) appear as examples of topological manipulations. We also study applications to fusion categorical symmetries and constraining RG flows in WZW models as well. After this, we present a short chapter showcasing an application of this symmetry TFT framework to the study of minimal models in 2d CFT. In particular, we complete the classification of 2d fermionic unitary minimal models. Finally, we discuss how the symmetry TFT intuition can be used to classify duality defects in QFTs. In particular, we focus on Zm duality defects in holomorphic Vertex Operator Algebras (VOAs) (and especially the E8 lattice VOA), where we use symmetry TFT intuition to conjecture, and then rigorously prove, a formula relating (duality-)defected partition functions to Z2 twists of invariant sub-VOAs

    Para-fusion Category and Topological Defect Lines in ZN\mathbb Z_N-parafermionic CFTs

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    We study topological defect lines (TDLs) in two-dimensional ZN\mathbb Z_N-parafermoinic CFTs. Different from the bosonic case, in the 2d parafermionic CFTs, there exist parafermionic defect operators that can live on the TDLs and satisfy interesting fractional statistics. We propose a categorical description for these TDLs, dubbed as ``para-fusion category", which contains various novel features, including ZM\mathbb Z_M qq-type objects for M∣NM\vert N, and parafermoinic defect operators as a type of specialized 1-morphisms of the TDLs. The para-fusion category in parafermionic CFTs can be regarded as a natural generalization of the super-fusion category for the description of TDLs in 2d fermionic CFTs. We investigate these distinguishing features in para-fusion category from both a 2d pure CFT perspective, and also a 3d anyon condensation viewpoint. In the latter approach, we introduce a generalized parafermionic anyon condensation, and use it to establish a functor from the parent fusion category for TDLs in bosonic CFTs to the para-fusion category for TDLs in the parafermionized ones. At last, we provide many examples to illustrate the properties of the proposed para-fusion category, and also give a full classification for a universal para-fusion category obtained from parafermionic condensation of Tambara-Yamagami ZN\mathbb Z_N fusion category.Comment: 45+4 page

    Topological symmetry in quantum field theory

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    We introduce a framework for internal topological symmetries in quantum field theory, including "noninvertible symmetries" and "categorical symmetries". This leads to a calculus of topological defects which takes full advantage of well-developed theorems and techniques in topological field theory. Our discussion focuses on finite symmetries, and we give indications for a generalization to other symmetries. We treat quotients and quotient defects (often called "gauging" and "condensation defects"), finite electromagnetic duality, and duality defects, among other topics. We include an appendix on finite homotopy theories, which are often used to encode finite symmetries and for which computations can be carried out using methods of algebraic topology. Throughout we emphasize exposition and examples over a detailed technical treatment.Comment: 65 pages, 37 figures. v2 adds references and corrects topological misstatements in section 4.4.2. v3 is a major revision with improved exposition throughout, a new section 2.5 on the passage from local to global defects, substantially expanded sections 4.4 and A.
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