Essential Unitarity for Higher-Order Quantum Computation

2026-06-02

AI Summary

This work addresses the challenge of generalizing unitarity in higher-order quantum computation by introducing the notion of *essential unitarity*—the unique higher-order unitarity condition compatible with involutive tensor structures, coherent reindexing, and currying, which subsumes standard unitarity as its first-order instance. Building upon the boundary-center representation in compact closed categories and leveraging techniques such as polarized boundary linking, unitless tensor sums, and pure compositional extensions, the authors develop a semantic framework that characterizes information preservation at boundary interfaces. This framework successfully models single-slot, proportion-preserving, purity-preserving supermaps—such as the quantum switch—as coherent pure compositional extensions, and ensures that all morphisms within the quantum core satisfy essential unitarity.

Abstract

We develop a semantic framework for higher-order quantum computation based on a boundary-centric presentation of compact closed categories, building on Kelly--Laplaza and Abramsky.Morphisms are polarized boundary linkings composed by execution, with a unit-free monoidal sum providing reversible control and branching. We identify a notion of \emph{essential unitarity} generalizing unitarity from first-order processes to higher-order interfaces;at first order it coincides with standard unitarity, and at higher order it characterizes when information is preserved relative tothe boundary. Essential unitarity is the unique predicate compatible with dagger-monoidal structure, coherence reindexing, and currying, and reducing to ordinary unitarity at first order. Every morphism of the quantum core is essentially unitary. The framework realizes the coherent quantum switch and other one-slot, equal-ratio, purity-preserving supermaps as coherent pure-comb dilations. Extended Abstract appears in QPL 2026

Problem

• higher-order quantum computation
• unitarity
• essential unitarity
• quantum information preservation
• compact closed categories

Innovation

• essential unitarity
• higher-order quantum computation
• compact closed categories
• coherent quantum switch
• pure-comb dilation