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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 | 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
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