gateCount
plain-language theorem explainer
The theorem asserts that the finite set of quantum gate operations has cardinality five. Researchers bounding circuit depth for molecular quantum simulations cite this to confirm that 25 reachable states fit inside five gate layers. The proof is a one-line decision procedure that enumerates the five constructors of the inductive type.
Claim. The finite type of quantum gate operations has cardinality five, where the operations are the Pauli, Clifford, T, CNOT and Toffoli gates.
background
The module establishes a cross-domain bound: the reachable state space on a molecular substrate is the product of five energy levels and five gate types, yielding 25 states. This cardinality satisfies 25 ≤ 2^5 = 32, so every state is reachable in at most five two-qubit gate layers. The local setting treats the gate set as the discrete basis for quantum circuits acting on molecular energy ladders in Recognition Science units.
proof idea
The proof is a one-line wrapper that applies the decide tactic directly to the Fintype instance of the inductive type whose five constructors are pauli, clifford, tGate, cnot and toffoli.
why it matters
This cardinality result is invoked by the stateCount theorem, which concludes that the quantum molecular state space has 25 elements and therefore satisfies ceil(log₂ 25) ≤ 5. It supplies the gate-count half of the 5 × 5 = 25 structural claim in the module, linking quantum computing depth to the five-layer bound per phi-rung in the Recognition Science framework.
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