Essential Unitarity for Higher-Order Quantum Computation (Extended Abstract)
Samson Abramsky, Radha Jagadeesan
We develop a boundary-centric semantic framework for higher-order quantum computation, building on the Kelly-Laplaza description of compact closure and Abramsky's execution account. In the semantic carrier Perm(C), morphisms are complex-linear combinations of polarized boundary linkings, composed by execution. Finite-family addresses provide coherent control over finite-level quantum registers (qudits) while retaining the multiplicative boundary structure. We identify essential unitarity, a boundary condition extending ordinary unitarity to higher-order interfaces. On positive qudit registers it coincides with ordinary matrix unitarity; at higher order it expresses preservation of information across the full polarized boundary. We define a unit-free coherent quantum core generated bymultiplicative wiring, unitary gates on positive qudit registers, and contextual coherent control, and prove that every one of its morphisms is essentially unitary. The framework realizes an applied coherent quantum switch and the unitary stages of equal-ratio one-slot supermap dilations with explicit memory. Complete proofs appear in the full version of this paper.
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