9 Jan. 2023, LMF Paris–Saclay
A box represents any process with systems as input and output.
A (monoidal) signature is given by:
Given two signatures and ,
a morphism is a pair of maps:
Given a signature , we can define (string) diagrams by recursion:
Diagrams are subject to three axioms.
Quantum gate sets are signatures!
Formal grammars are signatures!
Chomsky: Syntactic Structures (1957)
A (strict monoidal) category is a signature with three maps
such that associativity, unitality and naturality hold.
A (strict monoidal) functor is a morphism of signatures that preserves identity, composition and tensor.
Theorem (Joyal & Street, 1988): is the free monoidal category.
Intuition: The functors are uniquely determined by their image on boxes, i.e. a morphism of signatures .
Hadzihasanovic, Ng, Wang: Two complete axiomatisations of pure-state qubit quantum computing (2018)
Natural language semantics as a functor , defined using the lambda calculus and first-order logic.
DisCoCat models are functors , unifying Dis(tributional) and Co(mpositional) semantics with Cat(egories).
A QNLP model is a functor .
Coecke, Kissinger: The Compositional Structure of Multipartite Quantum Entanglement (2010)
Dur, Vidal, Cirac:
Three qubits can be entangled in two inequivalent ways. (2000)
If we assume GHZ and W as given, then we can solve
distributed consensus and leader election.
D’Hondt & Panangaden:
The Computational Power of the W and GHZ states (2003)
Tani, Kobayashi, Matsumoto:
Exact Quantum Algorithms for the Leader Election Problem (2007)