Realizability in the Unitary Sphere

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.

A Categorical Construction for the Computational Definition of Vector Spaces

“…As an immediate future work, we are willing to pursue a complete semantics for quantum computing, for which we need to add back the measurement operator, and define a notion of a norm, maybe following [9].…”

Quantum Control in the Unitary Sphere: Lambda-S1 and its Categorical Model

“…Pure terms and term distributions are intended to be evaluated according to a call-bypure-values strategy 2 , which is a declination of the call-by-value strategy in a computing environment where all functions are linear by construction. In its original form [6,7], a superposition t(v + w) reduced to (tv + tw), while in our case following [14], the first term is not even in the grammar, but it is just a notation for the former. This notation extends the syntactic constructs of the language by linearity, proceeding as follows: for all value distributions v = n i=1 α i • v i and w = m j=1 β j • w j , and for all term distributions s 1 , s 2 , t = p k=1 γ k • t k , and s = q ℓ=1 δ ℓ • s ℓ we have the notations given in Table 3.…”

“…This is a subset of the grammar from in [14]. In particular, we do not include the construction ♭A, however we use the notation A ♭ (read: A is flat) for the following property: A does not contain any ♯, except, maybe, at the right of an arrow.…”

“…In [14] we have worked out another type system for Lineal, this time, ensuring superpositions to be in the unitary sphere S 1 (that is, norm-1 vectors), as it is required by quantum computing, also characterizing isometries via a specific type. This has been obtained by means of realizability techniques [15][16][17][18]: Instead of deriving a computational meaning of proofs once the type system is set up, the idea of realizability is to consider the type system as a by-product of the operational semantics-programs are then potential realizers of types.…”

“…In this new paper, we extract a (finite) fixed type system following the realizability semantics [14], a calculus we call Lambda-S 1 , ensuring norm-1 superpositions, and we give a categorical model for this calculus. The developed model has some common grounds with the concrete model of Lambda-S [10,11], however, the chosen categories this time are not Set and Vec, but categories that use the fact that values in our calculus form a semi-vector space (an algebraic structure similar to a vector space, where its additive structure is a semi-group).…”