8,985 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

    Symmetries of Riemann surfaces and magnetic monopoles

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    This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions

    Gibbs–Wilbraham phenomenon on Lagrange interpolation based on analytic weights on the unit circle

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    This paper is devoted to study Lagrange interpolation based on nodal systems constituted by the roots of para-orthogonal polynomials with respect to analytic weights on the unit circle. The presented results address, in addition to algorithmic and convergence questions for continuous and discontinuous functions, a detailed study of the Gibbs-Wilbraham phenomenon.Ministerio de EconomĂ­a y Competitividad | Ref. AGL2014-60412-

    Techniques for high-multiplicity scattering amplitudes and applications to precision collider physics

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    In this thesis, we present state-of-the-art techniques for the computation of scattering amplitudes in Quantum Field Theories. Following an introduction to the topic, we describe a robust framework that enables the calculation of multi-scale two-loop amplitudes directly relevant to modern particle physics phenomenology at the Large Hadron Collider and beyond. We discuss in detail the use of finite fields to bypass the algebraic complexity of such computations, as well as the method of integration-by-parts relations and differential equations. We apply our framework to calculate the two-loop amplitudes contributing to three process: Higgs boson production in association with a bottom-quark pair, W boson production with a photon and a jet, as well as lepton-pair scattering with an off-shell and an on-shell photon. Finally, we draw our conclusions and discuss directions for future progress of amplitude computations

    Quantum-Classical hybrid systems and their quasifree transformations

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    The focus of this work is the description of a framework for quantum-classical hybrid systems. The main emphasis lies on continuous variable systems described by canonical commutation relations and, more precisely, the quasifree case. Here, we are going to solve two main tasks: The first is to rigorously define spaces of states and observables, which are naturally connected within the general structure. Secondly, we want to describe quasifree channels for which both the Schrödinger picture and the Heisenberg picture are well defined. We start with a general introduction to operator algebras and algebraic quantum theory. Thereby, we highlight some of the mathematical details that are often taken for granted while working with purely quantum systems. Consequently, we discuss several possibilities and their advantages respectively disadvantages in describing classical systems analogously to the quantum formalism. The key takeaway is that there is no candidate for a classical state space or observable algebra that can be put easily alongside a quantum system to form a hybrid and simultaneously fulfills all of our requirements for such a partially quantum and partially classical system. Although these straightforward hybrid systems are not sufficient enough to represent a general approach, we use one of the candidates to prove an intermediate result, which showcases the advantages of a consequent hybrid ansatz: We provide a hybrid generalization of classical diffusion generators where the exchange of information between the classical and the quantum side is controlled by the induced noise on the quantum system. Then, we present solutions for our initial tasks. We start with a CCR-algebra where some variables may commute with all others and hence generate a classical subsystem. After clarifying the necessary representations, our hybrid states are given by continuous characteristic functions, and the according state space is equal to the state space of a non-unital C*-algebra. While this C*-algebra is not a suitable candidate for an observable algebra itself, we describe several possible subsets in its bidual which can serve this purpose. They can be more easily characterized and will also allow for a straightforward definition of a proper Heisenberg picture. The subsets are given by operator-valued functions on the classical phase space with varying degrees of regularity, such as universal measurability or strong*-continuity. We describe quasifree channels and their properties, including a state-channel correspondence, a factorization theorem, and some basic physical operations. All this works solely on the assumption of a quasifree system, but we also show that the more famous subclass of Gaussian systems fits well within this formulation and behaves as expected

    Language integrated relational lenses

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    Relational databases are ubiquitous. Such monolithic databases accumulate large amounts of data, yet applications typically only work on small portions of the data at a time. A subset of the database defined as a computation on the underlying tables is called a view. Querying views is helpful, but it is also desirable to update them and have these changes be applied to the underlying database. This view update problem has been the subject of much previous work before, but support by database servers is limited and only rarely available. Lenses are a popular approach to bidirectional transformations, a generalization of the view update problem in databases to arbitrary data. However, perhaps surprisingly, lenses have seldom actually been used to implement updatable views in databases. Bohannon, Pierce and Vaughan propose an approach to updatable views called relational lenses. However, to the best of our knowledge this proposal has not been implemented or evaluated prior to the work reported in this thesis. This thesis proposes programming language support for relational lenses. Language integrated relational lenses support expressive and efficient view updates, without relying on updatable view support from the database server. By integrating relational lenses into the programming language, application development becomes easier and less error-prone, avoiding the impedance mismatch of having two programming languages. Integrating relational lenses into the language poses additional challenges. As defined by Bohannon et al. relational lenses completely recompute the database, making them inefficient as the database scales. The other challenge is that some parts of the well-formedness conditions are too general for implementation. Bohannon et al. specify predicates using possibly infinite abstract sets and define the type checking rules using relational algebra. Incremental relational lenses equip relational lenses with change-propagating semantics that map small changes to the view into (potentially) small changes to the source tables. We prove that our incremental semantics are functionally equivalent to the non-incremental semantics, and our experimental results show orders of magnitude improvement over the non-incremental approach. This thesis introduces a concrete predicate syntax and shows how the required checks are performed on these predicates and show that they satisfy the abstract predicate specifications. We discuss trade-offs between static predicates that are fully known at compile time vs dynamic predicates that are only known during execution and introduce hybrid predicates taking inspiration from both approaches. This thesis adapts the typing rules for relational lenses from sequential composition to a functional style of sub-expressions. We prove that any well-typed functional relational lens expression can derive a well-typed sequential lens. We use these additions to relational lenses as the foundation for two practical implementations: an extension of the Links functional language and a library written in Haskell. The second implementation demonstrates how type-level computation can be used to implement relational lenses without changes to the compiler. These two implementations attest to the possibility of turning relational lenses into a practical language feature

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources

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    Our aim is twofold: First, we rigorously analyse the generators of quantum-dynamical semigroups of thermodynamic processes. We characterise a wide class of GKSL-generators for quantum maps within thermal operations and argue that every infinitesimal generator of (a one-parameter semigroup of) Markovian thermal operations belongs to this class. We completely classify and visualise them and their non-Markovian counterparts for the case of a single qubit. Second, we use this description in the framework of bilinear control systems to characterise reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath. The core problem reduces to studying a hybrid control system ("toy model") on the standard simplex allowing for two types of evolution: (i) instantaneous permutations and (ii) a one-parameter semigroup of dd-stochastic maps. We generalise upper bounds of the reachable set of this toy model invoking new results on thermomajorisation. Using tools of control theory we fully characterise these reachable sets as well as the set of stabilisable states as exemplified by exact results in qutrit systems.Comment: 46 pages mai

    Unitarity estimation for quantum channels

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    The unitarity is a measure giving information on how much a quantum channel is unitary. Learning the unitarity of an unknown quantum channel E\mathcal{E} is a basic and important task in quantum device certification and benchmarking. Generally, this task can be performed with either coherent or incoherent access. For coherent access, there are no restrictions on learning algorithms; while for incoherent access, at each time, after preparing the input state and applying E\mathcal{E}, the output is measured such that no coherent quantum information can survive or be acted upon by E\mathcal{E} again. Quantum algorithms with only incoherent access allow practical implementations without the use of persistent quantum memory, and thus is more suitable for near-term devices. In this paper, we study unitarity estimation in both settings. For coherent access, we provide an ancilla-efficient algorithm that uses O(ϔ−2)O(\epsilon^{-2}) calls to E\mathcal{E} where Ï”\epsilon is the required precision; we show that this algorithm is query-optimal, giving a matching lower bound Ω(ϔ−2)\Omega(\epsilon^{-2}). For incoherent access, we provide a non-adaptive, non-ancilla-assisted algorithm that uses O(d⋅ϔ−2)O(\sqrt{d}\cdot \epsilon^{-2}) calls to E\mathcal{E}, where dd is the dimension of the system that E\mathcal{E} acts on; we show that this algorithm cannot be substantially improved, giving an Ω(d+ϔ−2)\Omega(\sqrt{d}+\epsilon^{-2}) lower bound, even if adaptive strategies and ancilla systems are allowed. As part of our results, we settle the query complexity of the distinguishing problem for depolarizing and unitary channels with incoherent access by giving a matching lower bound Ω(d)\Omega(\sqrt{d}), improving the prior best lower bound Ω(d3)\Omega(\sqrt[3]{d}) by Aharonov, Cotler, and Qi (Nat. Commun. 2022) and Chen, Cotler, Huang, and Li (FOCS 2021).Comment: 35 page

    On conditional densities of partially observed jump diffusions

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    In this thesis, we study the fi ltering problem for a partially observed jump diffusion (Zₜ)ₜɛ[ₒ,T] = (Xₜ, Yₜ)tɛ[ₒ,T] driven by Wiener processes and Poisson martingale measures, such that the signal and observation noises are correlated. We derive the fi ltering equations, describing the time evolution of the normalised conditional distribution (Pₜ(dx))tɛ[ₒ,T] and the unnormalised conditional distribution of the unobservable signal Xₜ given the observations (Yₛ)ₛɛ[ₒ,T]. We prove that if the coefficients satisfy linear growth and Lipschitz conditions in space, as well as some additional assumptions on the jump coefficients, then, if E|πₒ|á”–Lρ < ∞ for some p ≄ 2, the conditional density π = (πₜ)tɛ[ₒ,T], where πₜ = dPₜ/dx, exists and is a weakly cadlag Lp-valued process. Moreover, for an integer m ≄ 0 and p ≄ 2, we show that if we additionally impose m + 1 continuous and bounded spatial derivatives on the coefficients and if the initial conditional density E|πₒ|á”–Wρᔐ < ∞, then π is weakly cadlag as a Wρᔐ-valued process and strongly cadlag as a WÏËą - valued process for s ɛ [0;m)
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