| import OAI.NumberTheory.TwoPoint.Circuits.CircuitGateFailure |
|
|
| namespace OAI |
|
|
| /-! A single fixed polynomial approximates an arbitrarily wide AND gate |
| under any supplied finite input law. The sampling and its error bound are |
| explicit, and the polynomial degree is bounded by the number of sampled |
| linear factors. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| lemma FiniteLaw.probability_mono {α : Type*} [Fintype α] |
| (μ : FiniteLaw α) {E F : α → Prop} (h : ∀ x, E x → F x) : |
| μ.probability E ≤ μ.probability F := by |
| unfold FiniteLaw.probability |
| apply μ.average_mono |
| intro x |
| by_cases he : E x |
| · simp [he, h x he] |
| · simp [he] |
| split_ifs <;> norm_num |
|
|
| theorem gate_pointwise_failure (k s : ℕ) (b : Fin k → Bool) : |
| (gateSamplingLaw k k s).probability (fun x => |
| sampledAndPolynomial x (fun i => if b i then 1 else 0) ≠ boolAndValue b) ≤ |
| (7 / 8 : ℝ) ^ s := by |
| by_cases hb : ∀ i, b i = true |
| · have he (x : GateSamplingChoices k k s) : |
| sampledAndPolynomial x (fun i => if b i then 1 else 0) = boolAndValue b := by |
| rw [sampledAndPolynomial_all_true x b hb] |
| unfold boolAndValue |
| rw [ite_eq_left hb] |
| have hz : (gateSamplingLaw k k s).probability (fun x => |
| sampledAndPolynomial x (fun i => if b i then 1 else 0) ≠ boolAndValue b) = 0 := by |
| unfold FiniteLaw.probability |
| simp only [he, ne_eq, not_true_eq_false, ite_false] |
| exact FiniteLaw.average_const _ 0 |
| rw [hz] |
| positivity |
| · have hnonempty : (falseCoordinates b).Nonempty := by |
| push Not at hb |
| obtain ⟨i, hi⟩ := hb |
| refine ⟨i, ?_⟩ |
| simp only [falseCoordinates, mem_filter, mem_univ, true_and] |
| cases hbi : b i <;> simp_all |
| have hcard : (falseCoordinates b).card ≤ k := by |
| exact (card_le_card (filter_subset _ _)).trans_eq (card_fin k) |
| exact ((gateSamplingLaw k k s).probability_mono |
| (fun x hx => sampledAndPolynomial_failure x b hx)).trans |
| (gateSamplingLaw_failure k k s (falseCoordinates b) hnonempty hcard) |
|
|
| /-- The probabilistic method fixes all samples simultaneously for the input |
| law, without imposing a condition on that law. -/ |
| theorem exists_and_gate_approximation {α : Type*} [Fintype α] |
| (ν : FiniteLaw α) (k s : ℕ) (b : α → Fin k → Bool) : |
| ∃ x : GateSamplingChoices k k s, |
| ν.probability (fun y => |
| sampledAndPolynomial x (fun i => if b y i then 1 else 0) ≠ boolAndValue (b y)) ≤ |
| (7 / 8 : ℝ) ^ s := by |
| exact exists_choice_of_pointwise_failure (gateSamplingLaw k k s) ν _ |
| (fun y => gate_pointwise_failure k s (b y)) |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
