quantumComputingDepthCert
This definition assembles a certificate showing that five quantum gate types, an eight-element Pauli group, and three universal gates match the Recognition Science parameters for spatial dimension D. Quantum information theorists would cite it when mapping standard gate sets onto the eight-tick octave and phi-ladder structure. The construction is a direct record update that inserts the three component theorems into the QuantumComputingDepthCert structure.
claimThe quantum computing depth certificate is the structure asserting that the cardinality of the set of quantum gate types equals 5, the size of the single-qubit Pauli group equals $2^3$, and the number of gates in a universal set equals 3.
background
The module models quantum computation as sequences of J-cost-minimizing recognition operations inside the Recognition Science framework. The structure QuantumComputingDepthCert packages three numerical equalities that link gate counts to the spatial dimension D: five canonical gate types, eight Pauli elements, and three universal gates. These rest on the upstream theorems that the cardinality of QuantumGateType is five, the Pauli group size is exactly eight, and the universal gate count is three.
proof idea
The definition is a direct structure construction that assigns the five_gates field to the theorem quantumGateTypeCount, the pauli_8 field to pauliGroupSize_2cubed, and the universal_D field to universalGates_eq_D. Each assignment is a one-line wrapper; the component theorems themselves are proved by decide or rfl.
why it matters in Recognition Science
The definition closes the module's demonstration that standard quantum gate counts reproduce the Recognition Science landmarks T7 (eight-tick octave giving $2^3$) and T8 (D = 3). It supplies a concrete witness for the claim that quantum computation depth follows from RS parameters such as J-uniqueness and the phi fixed point. No downstream theorems currently depend on it.
scope and limits
- Does not derive the listed gate types from the Recognition Composition Law.
- Does not compute explicit J-cost values for individual gates.
- Does not extend the Pauli count to multi-qubit operators.
- Does not prove that three gates suffice for universality inside RS.
formal statement (Lean)
40def quantumComputingDepthCert : QuantumComputingDepthCert where
41 five_gates := quantumGateTypeCount
proof body
Definition body.
42 pauli_8 := pauliGroupSize_2cubed
43 universal_D := universalGates_eq_D
44
45end IndisputableMonolith.Physics.QuantumComputingDepthFromRS