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2f887db | 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 | import Mathlib
namespace OAI
section
open scoped BigOperators
namespace ExactQuantumFactoring
/-- Little-endian computational wires, with the same bit ordering as the
manuscript and input/output interface. -/
def bitsValue : {b : ℕ} → (Fin b → Bool) → BitVec b
| 0, _ => 0
| b+1, x => BitVec.cons (x (Fin.last b)) (bitsValue (fun i => x i.castSucc))
lemma bitsValue_bit {b : ℕ} (x : Fin b → Bool) (i : Fin b) :
(bitsValue x).getLsbD i.val = x i := by
induction b with
| zero => exact Fin.elim0 i
| succ b ih =>
refine Fin.lastCases ?_ (fun j => ?_) i
· simp only [bitsValue, Fin.val_last, BitVec.getLsbD_cons, ite_true]
· simp only [bitsValue, BitVec.getLsbD_cons, Fin.val_castSucc,
ite_eq_right (Nat.ne_of_lt j.isLt)]
exact ih _ j
lemma bitsValue_bits {b : ℕ} (x : BitVec b) :
bitsValue (fun i : Fin b => x.getLsbD i.val) = x := by
apply BitVec.eq_of_getLsbD_eq
intro i hi
exact bitsValue_bit _ ⟨i,hi⟩
def bitsEquiv (b : ℕ) : (Fin b → Bool) ≃ BitVec b where
toFun := bitsValue
invFun := fun x i => x.getLsbD i.val
left_inv := by intro x; funext i; exact bitsValue_bit x i
right_inv := bitsValue_bits
/-- The exact integer in the two most significant wires is the quarter-bin. -/
lemma bitsValue_quarter (b : ℕ) (x : Fin (b+2) → Bool) :
(bitsValue x).toNat / 2^b =
2*(x (Fin.last (b+1))).toNat + (x ((Fin.last b).castSucc)).toNat := by
have hlow := (bitsValue (fun i : Fin b => x i.castSucc.castSucc)).isLt
simp only [bitsValue, BitVec.toNat_cons', Nat.shiftLeft_eq]
have he : (x (Fin.last (b+1))).toNat * 2^(b+1) +
((x ((Fin.last b).castSucc)).toNat * 2^b +
(bitsValue (fun i : Fin b => x i.castSucc.castSucc)).toNat) =
(2*(x (Fin.last (b+1))).toNat + (x ((Fin.last b).castSucc)).toNat) * 2^b +
(bitsValue (fun i : Fin b => x i.castSucc.castSucc)).toNat := by
rw [pow_succ]
ring
rw [he, Nat.mul_comm _ (2^b), Nat.mul_add_div (by positivity),
Nat.div_eq_of_lt hlow, Nat.add_zero]
end ExactQuantumFactoring
end
end OAI
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