Related papers: Circuits via topoi
The data for many useful bidirectional constructions in applied category theory (optics, learners, games, quantum combs) can be expressed in terms of diagrams containing "holes" or "incomplete parts", sometimes known as comb diagrams. We…
Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on reasoning frameworks for path expressions where node…
Several different topoi have played an important role in the development and applications of synthetic guarded domain theory (SGDT), a new kind of synthetic domain theory that abstracts the concept of guarded recursion frequently employed…
We present a categorical construction for modelling causal structures within a general class of process theories that include the theory of classical probabilistic processes as well as quantum theory. Unlike prior constructions within…
We introduce the concept of dynamical score networks for the representation and analysis of tonal compositions: a score is interpreted as a dynamical network where every chord is a node and each progression links successive chords. This…
In this article a sequential theory in the category of spaces and proper maps is described and developed. As a natural extension a sequential theory for exterior spaces and maps is obtained.
Visibility algorithms transform time series into graphs and encode dynamical information in their topology, paving the way for graph-theoretical time series analysis as well as building a bridge between nonlinear dynamics and network…
We provide a complete set of identities for the symmetric monoidal category, TOF, generated by the Toffoli gate and computational ancillary bits. We do so by demonstrating that the functor which evaluates circuits on total points, is an…
In this paper, we identify some categorical structures in which one can model predicative formal systems: in other words, predicative analogues of the notion of a topos, with the aim of using sheaf models to interprete predicative formal…
Computational properties of networks that can undergo cascades are examined. It is shown that universal Boolean logic circuits can be computed by a global cascade having antagonistic interactions. Determinism and cascade frequency of this…
Isham's topos-theoretic perspective on the logic of the consistent-histories theory is extended in two ways. First, the presheaves of consistent sets of history propositions in the topos proposed by Isham are endowed with a Vietoris-type of…
In this chapter a general mathematical framework for probabilistic theories of operationally understood circuits is laid out. Circuits are comprised of operations and wires. An operation is one use of an apparatus and a wire is a…
The fundamental correspondence between quantum chaotic single-particle systems and random matrix theory is well-understood via periodic orbit theory. In contrast, we show that many-body systems with explicit subsystem structure possess…
This paper introduces the notion of referring forms as a new metric for analyzing sequential circuits from a functional perspective. Sequential circuits are modeled as causal stream functions, the outputs of which depend solely on the past…
This thesis details a project to define a fully compositional theory of synchronous sequential circuits built from primitive components, motivated by applying techniques successfully used in programming languages to hardware. The first part…
It is straightforward to give a sum-over-paths expression for the transition amplitudes of a quantum circuit as long as the gates in the circuit are balanced, where to be balanced is to have all nonzero transition amplitudes of equal…
Peculiar measurements can be obtained on systems that undergo both pre- and post-selection. We prove a conjecture from [1] on logical Pre- and Post-Selection (PPS) paradoxes for a restricted case. We prove that all of these paradoxes admit…
Models for quantum computation with circuit connections subject to the quantum superposition principle have been recently proposed. There, a control quantum system can coherently determine the order in which a target quantum system…
Discrete time (coined) quantum walks are produced by the repeated application of a constant unitary transformation to a quantum system. By recasting these walks into the setting of periodic perturbations to an otherwise freely evolving…
Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on frameworks for reasoning about path expressions…