The paper presents a new model of deterministic games (G) tailored to Intuitionist Multiplicative Additive Linear Logic (IMALL) to address the problem of modeling higher-order quantum computation. The key highlights and insights are:
The games in G are characterized by history-sensitive strategies that observe distributivity - specifically, tensor over plus and implication over product.
The authors introduce a category V that serves as a model for IMLL+L, which is universal in accommodating all linear operators on n-ary booleans.
They explore reversible and unitary computation, and define a category of unitaries (U) as a subcategory of V. The first-order fragment of U constitutes a rig-groupoid, complemented with additional structure.
The authors show how to interpret additives reversibly, overcoming the difficulties described in prior work, and extending the approach to the quantum realm.
The semantics of a program is modeled as a set of traces, and the authors generalize the notion of axiom links as perfect matchings on bipartite graphs, which correspond to unitary flows.
The model satisfies the key desiderata outlined by Selinger: universality, compatibility with base types, and realizability by quantum processes. It also adheres to the principle of conservativity, ensuring that new theorems do not emerge among old formulas.
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