gateCount
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.
claimThe 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 in Recognition Science
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.
scope and limits
- Does not prove that the five listed gates exhaust all possible quantum operations.
- Does not address continuous-parameter or infinite gate families.
- Does not derive the five-layer depth from first principles of the forcing chain.
- Does not quantify gate error or decoherence effects.
Lean usage
theorem stateCount : Fintype.card QuantumMolecularState = 25 := by simp only [QuantumMolecularState, Fintype.card_prod, energyCount, gateCount]formal statement (Lean)
31theorem gateCount : Fintype.card QuantumGateType = 5 := by decide
proof body
32