# Realizability in the Unitary Sphere

## TL;DR

This paper introduces a realizability-based semantics for a linear algebraic lambda-calculus that characterizes unitarity, providing a foundation that extends to classical and quantum lambda-calculi.

## Contribution

It presents a novel realizability semantics for linear algebraic lambda-calculus that captures unitarity and extends to classical and quantum systems.

## Key findings

- Semantics characterizes unitarity in the system
- Derived typing rules for a simply-typed linear algebraic lambda-calculus
- Extension of the calculus to classical and quantum lambda-calculi

## Abstract

In this paper we present a semantics for a linear algebraic lambda-calculus based on realizability. This semantics characterizes a notion of unitarity in the system, answering a long standing issue. We derive from the semantics a set of typing rules for a simply-typed linear algebraic lambda-calculus, and show how it extends both to classical and quantum lambda-calculi.