File size: 2,686 Bytes
8769036 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | 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
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