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f205175 | 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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import OAI.Probability.CubeShuffle.ExceptionalOperators
import OAI.RepresentationTheory.FiniteUnitary.KernelBounds
import OAI.RepresentationTheory.YoungSymmetry.UnitaryModel
import OAI.Probability.CubeShuffle.DimensionEstimates
namespace OAI
namespace CubeShuffle.UnitaryFinite
open scoped BigOperators Classical
lemma tupleAction_kernel {α ι : Type*} [Fintype ι] [DecidableEq α]
{Ω : Type*} [Fintype Ω] (P : Ω → Equiv.Perm α)
(e f : ι ↪ α) : actionKernel tupleAction P e f=PairRouting.tupleProbability P e f := by
unfold actionKernel PairRouting.tupleProbability
congr 1
funext ω
congr 1
exact propext Function.Embedding.ext_iff
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
omit [FiniteDimensional ℂ V] in
lemma full_tuple_fixed (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
fixedSpace tupleAction (Function.Embedding.refl (Card d)) ρ=⊤ := by
apply Submodule.eq_top_iff'.mpr
intro v g hg
have he : g=1 := by
apply Equiv.ext
intro x
exact congrArg (fun f : Card d ↪ Card d => f x) hg
rw [he,map_one]
rfl
omit [FiniteDimensional ℂ V] in
lemma full_tuple_finrank (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
Module.finrank ℂ (fixedSpace tupleAction (Function.Embedding.refl (Card d)) ρ)=Module.finrank ℂ V := by
rw [full_tuple_fixed,finrank_top]
theorem palindrome_irrep_amplification (d : ℕ) {ι : Type*} [Fintype ι]
(f : ι ↪ Card d) (hk : 0<Fintype.card ι)
(ρ : Representation ℂ (Equiv.Perm (Card d)) V) [Representation.IsIrreducible ρ]
(hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ (fixedSpace tupleAction f ρ))
(z : ℝ) (hz : 0<z) :
‖sampleOperator ρ (palindromePerm d)‖ ≤
Real.exp (densityConstant*Fintype.card ι*((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*
z^(-(1/1000:ℝ))+Real.sqrt (z/(Module.finrank ℂ (fixedSpace tupleAction f ρ):ℝ)) := by
apply sampleOperator_amplification tupleAction f (tupleAction_transitive f) ρ hρ
(palindromePerm d) _ _ (1/1000) z _ (by norm_num) hz (by positivity) _ hm
· intro e f
simp only [tupleAction_kernel]
exact palindrome_kernel_symmetric d e f
· intro e
simp only [tupleAction_kernel]
exact PairRouting.tupleProbability_row_sum _ e
· intro e
simp only [tupleAction_kernel]
exact palindrome_row_moment_bound d e hk
theorem modified_irrep_amplification (L r : ℕ) (E : Finset (Card r))
(ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
(hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
(u z : ℝ) (hz : 0<z)
(hM : ∀ e : Card (L+r) ↪ Card (L+r), modifiedRowMoment L r E e (1/1000)≤Real.exp u) :
‖sampleOperator ρ (modifiedPerm L r E)‖ ≤
Real.exp u*z^(-(1/1000:ℝ))+Real.sqrt (z/(Module.finrank ℂ V:ℝ)) := by
let f := Function.Embedding.refl (Card (L+r))
have h := sampleOperator_amplification tupleAction f (tupleAction_transitive f) ρ hρ
(modifiedPerm L r E) (fun e f => by simpa only [tupleAction_kernel] using modified_kernel_symmetric L r E e f)
(fun e => by simpa only [tupleAction_kernel] using PairRouting.tupleProbability_row_sum (modifiedPerm L r E) e)
(1/1000) z (Real.exp u) (by norm_num) hz (by positivity)
(fun e => by simpa only [tupleAction_kernel,modifiedRowMoment] using hM e)
(by simpa only [f,full_tuple_finrank] using hm)
simpa only [f,full_tuple_finrank] using h
end CubeShuffle.UnitaryFinite
namespace CubeShuffle.UnitaryFinite
open scoped BigOperators Classical
open DimensionEstimates
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
theorem full_palindrome_decay (d : ℕ)
(ρ : Representation ℂ (Equiv.Perm (Card d)) V) [Representation.IsIrreducible ρ]
(hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
(hden : densityConstant*(2:ℝ)^d≤Real.log (Module.finrank ℂ V)/4000)
(hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V)) :
‖sampleOperator ρ (palindromePerm d)‖≤Real.exp (-Real.log (Module.finrank ℂ V)/8000) := by
have h := palindrome_irrep_amplification d (Function.Embedding.refl (Card d))
(Fintype.card_pos) ρ hρ (by simpa only [full_tuple_finrank] using hm)
(Real.exp (Real.log (Module.finrank ℂ V)/2)) (Real.exp_pos _)
simp only [full_tuple_finrank,card_positions,Nat.cast_pow,Nat.cast_ofNat,
div_self (by positivity : (2:ℝ)^d≠0),Real.one_rpow,mul_one] at h
exact h.trans (amplification_decay _ _ (by exact_mod_cast hm) hden hl)
theorem full_modified_decay (L r : ℕ) (E : Finset (Card r))
(ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
(hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V) (u : ℝ)
(hM : ∀ e : Card (L+r) ↪ Card (L+r), modifiedRowMoment L r E e (1/1000)≤Real.exp u)
(hden : u≤Real.log (Module.finrank ℂ V)/4000)
(hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V)) :
‖sampleOperator ρ (modifiedPerm L r E)‖≤Real.exp (-Real.log (Module.finrank ℂ V)/8000) := by
exact (modified_irrep_amplification L r E ρ hρ hm u
(Real.exp (Real.log (Module.finrank ℂ V)/2)) (Real.exp_pos _) hM).trans
(amplification_decay _ _ (by exact_mod_cast hm) hden hl)
lemma palindrome_exception_cards (r L : ℕ)
(ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) (hρ : IsUnitary ρ) (n : ℕ) :
‖sampleOperator ρ (palindromePerm (L+r))‖ ≤
(blockGap L)^n+∑ E : Finset (Card r), if E.card<n then
‖sampleOperator ρ (modifiedPerm L r E)‖ else 0 := by
let σ := ρ.comp (splitPerm r L).symm.toMonoidHom
have he : σ.comp (splitPerm r L).toMonoidHom=ρ := by
ext g v
change ρ ((splitPerm r L).symm ((splitPerm r L) g)) v=ρ g v
rw [MulEquiv.symm_apply_apply]
have h := palindrome_exception_bound r L σ (unitary_comp ρ hρ _) n
simpa only [he] using h
end CubeShuffle.UnitaryFinite
end OAI
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