718 research outputs found

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

Rig groupoids provide a semantic model of Î , 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 Î , called âÎ , 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 â€2 qubits, and the computationally universal Gaussian Clifford+T gate set

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

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 $(-1)^{F_\text{L}}$ 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

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}

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
$8^{\text{th}}$ 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 \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 $\le 2$ qubits, and the computationally universal
Gaussian Clifford+T gate set

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

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 nĂ¶tig, 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 auĂerdem 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

This note describes a local Poisson structure (up to homotopy) associated with corners in four-dimensional gravity in the coframe (PalatiniâCartan) 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

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 $B$. 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 $B$-telescopes and discuss several applications. We give
examples of $2$-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 $2$-generated frames in
products of finite simple groups and show that there are Grothendieck pairs
consisting of amenable groups and groups with property $(\tau)$. 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

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 âlocalâ and âglobalâ data by means of a âsymmetry Topological Field Theoryâ (symmetry TFT). We also study how various âtopological manipulationsâ 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 âorbifold groupoidsâ which control the composition of âtopological manipulations,â 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 âJordan-Wigner transformationâ (fermionization) and âGSO projectionâ (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 $\mathbb Z_N$-parafermionic CFTs

We study topological defect lines (TDLs) in two-dimensional $\mathbb
Z_N$-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 $\mathbb Z_M$ $q$-type objects
for $M\vert N$, 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 $\mathbb Z_N$ fusion
category.Comment: 45+4 page

### Topological symmetry in quantum field theory

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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