arXiv subjects

Lorenzo Perticone

Publications and source records attributed to Lorenzo Perticone.

3 recordsLinked to original sources

arXiv2025

Effect Algebras as Omega-categories

We show how an effect algebra $\mathcal{X}$ can be regarded as a category, where the morphisms $x \rightarrow y$ are the elements $f$ such that $x \leq f \leq y$. This gives an embedding $\mathbf{EA} \rightarrow \mathbf{Cat}$. The interval $[x,y]$ proves to be an effect algebra in its own right, so $\mathcal{X}$ is an $\mathbf{EA}$-enriched category. The construction can therefore be repeated, meaning that every effect algebra can be identified with a strict $ω$-category. We describe explicitly the strict $ω$-category structure for two classes of operators on a Hilbert space.

cs.LO↗

arXiv2025

A Graded Modal Type Theory for Pulse Schedules

The operations to be performed by a quantum computer are almost invariably given in the form of a quantum circuit. In the final stage of compilation, a quantum circuit must be translated into the input signals accepted by the quantum hardware itself. For a quantum computer based on superconducting qubits, this will be a sequence of microwave control pulses to be sent to the various input channels. A pulse schedule gives a full specification for which pulse should be applied to which channel at what time. There is as yet no language for these pulse schedules that is very amenable to formal semantics. In this paper, we propose such a language called GRAMPUS (GRAded Modal type theory for PUlse Schedules). It is a graded modal type theory, where the grades represent timing information: a variable $x :^{50} Q_1$ will represent a state of qubit $Q_1$ that will exist 50 nanoseconds in the future, and a variable $y :^{-75} Q_2$ will represent a state of qubit $Q_2$ that existed 75 nanoseconds in the past. We give the syntax for two type theories, one with grades (the annotated language) and one without (the plain language). We prove some metatheoretic properties, and describe the semantics in terms of category theory. We show that the input signals to a quantum chip forms a model of the annotated language. We also give a syntatic model, prove that it is initial, and hence prove soundness and completeness theorems.

cs.LO↗

arXiv2025

Parametric Distributive Laws: uniform monad composition

Monads play an important role in both the syntax and semantics of modern functional programming languages. The problem of combining them has been of profound interest at least since the 90s, and different approaches have been employed to tackle it: currently, the most prevalent notion is that of monad transformers. We provide a novel abstract framework to describe such "compositions" which we call parametric distributive laws. Our description of this framework hinges on the theory of 2-categories and relies strongly on the construction of a left (1-) adjoint to the construction of monads internal to a 2-category: since distributive laws are monads in monads, we obtain a semi-strict "walking distributive law" by Gray-tensoring the walking monad with itself. By Gray-tensoring again, we can then construct a parametric variant thereof. We demonstrate the applicability of such a framework by providing two concrete examples (involving the Writer and Either monads), explicitly describing morphisms of such parametric distributive laws, and showing how (an iterated version of) our construction can be employed to describe (parametric) iterated distributive laws, motivating the appearance of the Yang-Baxter equations as coherences between the distributive laws involved.

math.CT↗