Physics Quantum Molecular Design Depth C4

The 25-state bound

In quantum computing, a quantum gate is a basic operation that changes the state of a quantum system, much as a logic gate flips a bit in an ordinary computer. The module called C4, for Quantum Molecular Design Depth, counts how many distinct prepared states arise when you combine molecular energy levels with quantum gate types. It defines five molecular energy levels and five quantum gate types, and the product of those two numbers gives 25 prepared state classes. That count is the whole foundation of the module.

The central result is a bound on how much information those 25 classes require. Since 25 is greater than 16 (which is 2 to the power 4) but less than or equal to 32 (which is 2 to the power 5), a five-bit address is the minimum needed to index every class. Four bits can address only 16 distinct values, so they fall short; five bits cover all 25. The module proves both halves of that statement in a machine-checked library of formal theorems, with no unproved assumptions and no axioms beyond the standard logical ones.

The module also packages the result as a single certificate object, a formal bundle that carries all three facts: the count is 25, it fits in five bits, and it does not fit in four. That certificate is the Lean-safe core of C4. The stronger claim, that a specific molecular target can actually be reached in five two-qubit layers under a chosen gate model, is not part of the theorem. That remains an empirical and algorithmic question, one the module explicitly leaves open.

What this establishes in plain language is a counting guarantee: if you prepare molecular quantum states by combining five energy levels with five gate types, you need at least five bits to tell them apart, and five bits are enough. That is a precise, checkable statement about information capacity, not a claim about physical feasibility. The module gives a clean boundary on the addressing depth, and it stops exactly where the physics becomes a matter of experiment rather than proof.

theorem molecularQuantumStateClasses_25 :
    molecularQuantumStateClasses = 25 := by
  unfold molecularQuantumStateClasses
  rw [molecularEnergyCount, quantumGateTypeCount]

theorem molecularQuantumStateClasses_le_2pow5 :
    molecularQuantumStateClasses ≤ 2 ^ 5 := by
  rw [molecularQuantumStateClasses_25]
  norm_num

theorem twoPowerFour_lt_stateClasses :
    2 ^ 4 < molecularQuantumStateClasses := by
  rw [molecularQuantumStateClasses_25]
  norm_num

def quantumMolecularDepthCert : QuantumMolecularDepthCert where
  state_classes_25 := molecularQuantumStateClasses_25
  addressable_by_five_bits := molecularQuantumStateClasses_le_2pow5
  not_addressable_by_four_bits := twoPowerFour_lt_stateClasses

What this page does not claim

The module does not prove that any specific molecular target is reachable in five two-qubit layers. The count of 25 assumes exactly five energy levels and five gate types, not a continuous spectrum. The certificate proves an information bound, not a statement about physical implementation time or error rates.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/QuantumMolecularDesignDepthC4.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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