| import OAI.NumberTheory.TwoPoint.Circuits.CircuitGateApproximation |
| import OAI.NumberTheory.TwoPoint.Bounds.CrudeWordCounting |
|
|
| namespace OAI |
|
|
| /-! Recursive use of the sampled gate polynomial. Errors in the child |
| polynomials are charged by a finite union bound, in addition to the new |
| gate's sampling error. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| theorem exists_composed_and_approximation {n : ℕ} |
| (ν : FiniteLaw (BooleanCube n)) (k s d : ℕ) |
| (b : Fin k → BooleanCube n → Bool) (P : Fin k → BooleanCube n → ℝ) |
| (hP : ∀ i, WalshDegreeLE (P i) d) (δ : Fin k → ℝ) |
| (hδ : ∀ i, ν.probability (fun y => P i y ≠ if b i y then 1 else 0) ≤ δ i) : |
| ∃ Q : BooleanCube n → ℝ, |
| WalshDegreeLE Q (d * s * (Nat.log 2 k + 3)) ∧ |
| ν.probability (fun y => Q y ≠ boolAndValue (fun i => b i y)) ≤ |
| (7 / 8 : ℝ) ^ s + ∑ i, δ i := by |
| obtain ⟨x, hx⟩ := exists_and_gate_approximation ν k s (fun y i => b i y) |
| let Q := fun y => sampledAndPolynomial x (fun i => P i y) |
| refine ⟨Q, sampledAndPolynomial_degree x P d hP, ?_⟩ |
| let E : Option (Fin k) → BooleanCube n → Prop |
| | none => fun y => sampledAndPolynomial x (fun i => if b i y then 1 else 0) ≠ |
| boolAndValue (fun i => b i y) |
| | some i => fun y => P i y ≠ if b i y then 1 else 0 |
| have hcover : ∀ y, Q y ≠ boolAndValue (fun i => b i y) → ∃ i, E i y := by |
| intro y hy |
| by_cases hc : ∃ i, P i y ≠ if b i y then 1 else 0 |
| · obtain ⟨i, hi⟩ := hc |
| exact ⟨some i, hi⟩ |
| · have heq : (fun i => P i y) = fun i => if b i y then 1 else 0 := by |
| funext i |
| by_contra hi |
| exact hc ⟨i, hi⟩ |
| refine ⟨none, ?_⟩ |
| change sampledAndPolynomial x (fun i => if b i y then 1 else 0) ≠ _ |
| simpa only [Q, heq] using hy |
| calc |
| ν.probability (fun y => Q y ≠ boolAndValue (fun i => b i y)) ≤ |
| ν.probability (fun y => ∃ i, E i y) := ν.probability_mono hcover |
| _ ≤ ∑ i : Option (Fin k), ν.probability (E i) := ν.probability_exists_le E |
| _ = ν.probability (E none) + ∑ i : Fin k, ν.probability (E (some i)) := by |
| simp only [Fintype.sum_option] |
| _ ≤ (7 / 8 : ℝ) ^ s + ∑ i, δ i := |
| add_le_add hx (sum_le_sum (fun i _ => hδ i)) |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
