---
title: "Intuitionistic Unitary Linear Logic: A Proof-Theoretical Approach to Purely Quantum Higher-Order"
authors:
  - "Julien Lamiroy"
  - "Benoît Valiron"
  - "Renaud Vilmart"
year: 2026
venue: "arXiv:cs.LO"
arxiv: "2609.02661"
doi: null
source: "https://arxiv.org/abs/2609.02661"
topics:
  - "curiosity"
  - "maths-foundations"
seed_rank: 1648
seed_batch: "curiosity-2026-09-09"
reviewed: "2026-09-09"
pool: "maths-foundations"
relevance_score: 10
lineage: quantum-logic
cites:
  - title: "Intuitionistic Unitary Linear Logic: A Proof-Theoretical Approach to Purely Quantum Higher-Order"
    url: "https://arxiv.org/abs/2609.02661"
    year: 2026
    arxiv: "2609.02661"
    doi: null
see:
  - "1585-quantum-hopf-rigidity-in-symmetric-monoidal-categories"
  - "1025-on-the-sphere-spectrum-from-the-viewpoint-of-linear-logic"
---

# Intuitionistic Unitary Linear Logic: A Proof-Theoretical Approach to Purely Quantum Higher-Order

## One-sentence takeaway

A proof theory for intuitionistic unitary linear logic that can host non-causal higher-order quantum processes such as the quantum switch.

## Why it's lovely

Why you might love this: ordinary quantum circuits are causal; the quantum switch is not. Lamiroy–Valiron–Vilmart give a purely proof-theoretic unitary linear logic that aims at higher-order quantum without smuggling causality back in. Diagrammatic / linear-logic sparkle with a real physics target — and a sideways glance at Broadside's 'everything is a plugin' harness mindset.

## Key ideas

- Circuit model cannot represent non-causal higher-order processes (quantum switch).
- Intuitionistic unitary linear logic as a proof theory for purely quantum higher-order.
- Keeps unitarity as a first-class constraint rather than a semantic afterthought.
- Sits with linear-logic / diagrammatic / ZX-adjacent cards without being a ZX paper.

## Caveats

Proof-theoretic proposal — not a drop-in quantum compiler. Compare carefully to existing process theories and categorical quantum mechanics accounts.

## Links

- arXiv abs: https://arxiv.org/abs/2609.02661
- PDF: https://arxiv.org/pdf/2609.02661.pdf