File size: 2,851 Bytes
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 63 64 65 66 67 68 | import Mathlib
import OAI.Computability.QuantumFactoring.InitialOrderState
import OAI.Computability.QuantumFactoring.PreparedOracle
namespace OAI
section
open scoped BigOperators
open scoped BigOperators
open scoped BigOperators
open scoped BigOperators
open scoped BigOperators
namespace ExactQuantumFactoring
open scoped BigOperators
open BooleanNetwork
/-- A coherent fresh launch on an encoded old history. The local law refers
only to the actually computed Boolean initializer, never to a selected outcome. -/
lemma preparedOracle_encode {α β : Type*} [Fintype α] [Fintype β] {p q r : ℕ}
(e : α→Basis p) (f : β→Basis q) (ψ : α→ℂ) (F : α→β→ℂ)
(c : BooleanNetwork p q) (hc : c.net.count ≤ r) (ops : List (Instruction q))
(hlocal : ∀ a,(programMatrix ops).mulVec (basisVector (c.eval (e a)))=encodeState f (F a)) :
(programMatrix (preparedOracle c hc ops)).mulVec
(encodeState (fun a=>packed r (e a) (fun _=>false)) ψ)=
encodeState (fun ab : α×β=>packed r (e ab.1) (f ab.2)) (RecordedHistory.appendState ψ F) := by
rw [encodeState,Matrix.mulVec_sum]
simp only [Matrix.mulVec_smul,preparedOracle_basis,hlocal,encodeState_comp]
unfold encodeState RecordedHistory.appendState
rw [Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro a _
rw [Finset.smul_sum]
apply Finset.sum_congr rfl
intro b _
rw [smul_smul]
rfl
def appendPrepared {p q r : ℕ} (A : List (Instruction p)) (c : BooleanNetwork p q)
(hc : c.net.count ≤ r) (ops : List (Instruction q)) : List (Instruction (p+q+r)) :=
firstProgram q r A++preparedOracle c hc ops
lemma appendPrepared_length {p q r : ℕ} (A : List (Instruction p)) (c : BooleanNetwork p q)
(hc : c.net.count ≤ r) (ops : List (Instruction q)) :
(appendPrepared A c hc ops).length ≤ A.length+4*c.net.count+2*q+ops.length := by
have h := preparedOracle_length c hc ops
simp only [appendPrepared,List.length_append,firstProgram_length]
omega
lemma appendPrepared_state {α β : Type*} [Fintype α] [Fintype β] {p q r : ℕ}
(e : α→Basis p) (f : β→Basis q) (ψ : α→ℂ) (F : α→β→ℂ)
(A : List (Instruction p)) (v : Basis p)
(hA : (programMatrix A).mulVec (basisVector v)=encodeState e ψ)
(c : BooleanNetwork p q) (hc : c.net.count ≤ r) (ops : List (Instruction q))
(hlocal : ∀ a,(programMatrix ops).mulVec (basisVector (c.eval (e a)))=encodeState f (F a)) :
(programMatrix (appendPrepared A c hc ops)).mulVec (basisVector (packed r v (fun _=>false)))=
encodeState (fun ab : α×β=>packed r (e ab.1) (f ab.2)) (RecordedHistory.appendState ψ F) := by
rw [appendPrepared,programMatrix_append,←Matrix.mulVec_mulVec,firstProgram_basis,hA,encodeState_comp]
exact preparedOracle_encode e f ψ F c hc ops hlocal
end ExactQuantumFactoring
end
end OAI
|