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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
import OAI.Computability.QuantumFactoring.PhasePreparationConnection
namespace OAI
section
open scoped BigOperators
namespace ExactQuantumFactoring
/-- The fixed little-endian physical representation of a modular residue. -/
def basisResidue (b : ℕ) : Basis b ≃ ZMod (2^b) :=
(basisNumber b).trans (ZMod.finEquiv (2^b)).toEquiv
lemma finEquiv_val {n : ℕ} [NeZero n] (z : Fin n) :
(ZMod.finEquiv n z).val = z.val := by
cases n with
| zero => exact False.elim (NeZero.ne 0 rfl)
| succ n => rfl
lemma basisResidue_val (b : ℕ) (x : Basis b) :
(basisResidue b x).val = (bitsValue x).toNat := by
exact finEquiv_val _
lemma basisResidue_eq (b : ℕ) (x : Basis b) :
basisResidue b x = ((bitsValue x).toNat : ZMod (2^b)) := by
rw [← basisResidue_val, ZMod.natCast_zmod_val]
lemma bitsValue_residue (b : ℕ) (z : ZMod (2^b)) :
(bitsValue ((basisResidue b).symm z)).toNat = z.val := by
rw [← basisResidue_val, Equiv.apply_symm_apply]
lemma basisResidue_bits (b : ℕ) (v : BitVec b) :
basisResidue b (fun i => v.getLsbD i.val) = (v.toNat : ZMod (2^b)) := by
rw [basisResidue_eq, bitsValue_bits]
lemma bitsValue_residue_add (b : ℕ) (z : ZMod (2^b)) (d : Basis b) :
(fun i : Fin b => (bitsValue ((basisResidue b).symm z)+bitsValue d).getLsbD i.val) =
(basisResidue b).symm (z+basisResidue b d) := by
apply (basisResidue b).injective
rw [basisResidue_bits, Equiv.apply_symm_apply, BitVec.toNat_add]
rw [ZMod.natCast_mod, Nat.cast_add, bitsValue_residue, ZMod.natCast_zmod_val,
basisResidue_eq]
lemma basisResidue_zero (b : ℕ) : basisResidue b (fun _ => false) = 0 := by
rw [basisResidue_eq]
have h : bitsValue (fun _ : Fin b => false) = (0 : BitVec b) := by
apply BitVec.eq_of_getLsbD_eq
intro i hi
rw [show (bitsValue (fun _ : Fin b => false)).getLsbD i = false from bitsValue_bit _ ⟨i,hi⟩]
simp
rw [h]
simp
end ExactQuantumFactoring
end
end OAI
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