One rig to control them all
summary
A synthetic construction adds eight equations to any base prop of gates to produce controlled gates and their iterated controls, and the paper proves this syntactic extension equals taking the free rig category (tensor plus direct sum) on the base prop. Examples include reversible Boolean circuits and finite-dimensional quantum circuits; completeness for unitaries follows after one final quotient equation. The core claim is that direct sum = control, so control operations are exactly the rig (distributive) interaction used to model circuits.
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chapters (10)
- 0:00Quantum circuit control intro
- 1:10Quantum circuits and tensor products
- 3:03Equational theories for circuits
- 5:12Defining structural equations
- 9:29Category theory and circuit props
- 11:34Adding control to circuits
- 14:50Direct sums and tensor equivalence
- 17:03Control in rig categories
- 18:50Summary and implications
- 20:01Question and answer session
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