Uploaded April 2026 | Updated September 2026, 2 weeks ago

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Topos Institute Colloquium, 23rd of April 2026.
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We present a categorical axiomatisation of quantum computation based on free constructions. Starting from a PROP of base circuits, we introduce a theory of computational control given by eight natural equations, and show that adjoining control syntactically corresponds semantically to completing the PROP to its free rig category. Next, we show that two further generators and three further equations suffice to fully characterise quantum computing. The resulting free model replaces the usual linear-algebraic semantics with a purely symbolic, combinatorial one. We argue that this gives a conceptually simpler foundation for quantum computing that isolates quantum advantage in a precise categorical sense. (Based on joint work doi.org/10.1073/pnas.2510881123 and arxiv.org/abs/2510.05032 with Robin Kaarsgaard, Louis Lemonnier, Neil Julien Ross, Amr Sabry, and Jacques Carette.)