| import Mathlib |
| import OAI.MathematicalPhysics.PEPSFilters.LocalOperators |
| import OAI.MathematicalPhysics.PEPSSubvolume.Marginal |
| import OAI.MathematicalPhysics.PEPSSubvolume.PhysicalModular |
|
|
| namespace OAI |
|
|
| /-! Quadratic crossing bounds for a global filter optimizer. -/ |
|
|
| noncomputable section |
| open scoped BigOperators ComplexOrder |
| open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator |
| open scoped BigOperators |
| open scoped Topology |
| open Filter |
| open scoped MatrixOrder |
| open scoped BigOperators Matrix.Norms.L2Operator |
| open scoped ComplexOrder BigOperators Matrix.Norms.L2Operator |
| open Matrix |
| open PolynomialPEPS.PinnedEntropy |
|
|
| namespace PolynomialPEPS.Subvolume.OptimizerEnergy |
| open scoped BigOperators Matrix.Norms.L2Operator |
| open Matrix PolynomialPEPS.Subvolume.SpectralCurve PolynomialPEPS.Subvolume.CrossingIdentity |
| open PolynomialPEPS.Subvolume.PhysicalModular PolynomialPEPS.Subvolume.OptimizerMarginal PolynomialPEPS.Subvolume.ActualDiagonal |
| variable {L q : ℕ} |
|
|
| theorem filter_spectral_sum (X : Finset (Vertex L)) (F : LocalPositiveFilter q X) |
| (p : ℝ) (hF : filterTracePower F p=1) |
| (U : unitary (Matrix (RegionConfiguration q X) (RegionConfiguration q X) ℂ)) |
| (e : RegionConfiguration q X → ℝ) (he : ∀ i, 0 ≤ e i) |
| (hrep : F.matrix = spectralHom U (fun i => (e i:ℂ))) : |
| ∑ i, (e i)^p=1 := by |
| unfold filterTracePower NestedFilter.tracePower at hF |
| have hs' := sum_eigenvalues_spectralHom U e (spectral_positive U e he).isHermitian |
| (fun t => t^p) |
| have heig := F.positive.isHermitian.eigenvalues_eq_eigenvalues_iff |
| (spectral_positive U e he).isHermitian |
| have heq := heig.mpr (congrArg Matrix.charpoly hrep) |
| rw [heq] at hF |
| exact hs'.symm.trans hF |
|
|
| theorem spectral_real_error_le {ι : Type*} [Fintype ι] |
| (X : Finset (Vertex L)) (v : State L q) |
| (F : LocalPositiveFilter q X) |
| (U : unitary (Matrix (RegionConfiguration q X) (RegionConfiguration q X) ℂ)) |
| (e : RegionConfiguration q X → ℝ) (he : ∀ i, 0 ≤ e i) |
| (a : ℝ) (ha : 0<a) (ha' : a ≤ 1/2) (hs : ∑ i, (e i)^(2/a)=1) |
| (hrep : F.matrix = spectralHom U (fun i => (e i:ℂ))) |
| (hρ : reducedDensity v X = (‖v‖^2:ℂ) • spectralHom U (fun i => (((e i)^(2/a):ℝ):ℂ))) |
| (H : Operator L q) (hH : H.IsHermitian) |
| (A : ι → Matrix (RegionConfiguration q X) (RegionConfiguration q X) ℂ) |
| (B : ι → Matrix (RegionConfiguration q Xᶜ) (RegionConfiguration q Xᶜ) ℂ) |
| (hdecomp : H = ∑ i, tensorAcross X (A i) (B i)) : |
| |(inner ℂ v (asMap H v)).re - (inner ℂ v (asMap |
| (liftLocal X F.matrix * H * liftLocal X (inverseOnSupport U e)) v)).re| ≤ |
| 8*a^2*‖v‖^2*(∑ i, ‖A i‖*‖B i‖) := by |
| classical |
| let R := liftLocal X F.matrix |
| let D := liftLocal X (inverseOnSupport U e) |
| have hR : R.IsHermitian := F.positive.isHermitian.isSelfAdjoint.map (localLiftHom X) |
| have hD : D.IsHermitian := by |
| have hf : IsSelfAdjoint (fun i => ((e i:ℂ)⁻¹)) := by |
| rw [isSelfAdjoint_iff] |
| ext i |
| simp |
| exact ((hf.map (spectralHom U)).map (localLiftHom X)).isHermitian |
| have hb := physical_sum_error_le X v U e he a ha ha' hs hρ A B |
| change ‖inner ℂ v (asMap (modularError (liftLocal X (spectralHom U (fun i => (e i:ℂ)))) D |
| (∑ i, tensorAcross X (A i) (B i))) v)‖ ≤ _ at hb |
| rw [← hrep,← hdecomp] at hb |
| have hr := modularError_real R D H hR hD hH v |
| change |(inner ℂ v (asMap H v)).re-(inner ℂ v (asMap (R*H*D) v)).re| ≤ _ |
| rw [abs_sub_comm,← hr] |
| exact (Complex.abs_re_le_norm _).trans hb |
|
|
| theorem crossing_error_le {ι : Type*} [Fintype ι] (hq : 0 < q) |
| (X : ℕ → Finset (Vertex L)) (hX : Monotone X) |
| (a : ℕ → ℝ) (ψ : State L q) (n : ℕ) |
| (F : (j : ℕ) → LocalPositiveFilter q (X j)) |
| (hF : IsFilterOptimizer ψ (fun j : Fin n => a j.val) (fun j : Fin n => F j.val)) |
| (k : ℕ) (hk : k<n) (ha : 0<a k) (ha' : a k ≤ 1/2) |
| (H : Operator L q) (hH : H.IsHermitian) |
| (hHs : ∀ j, k<j → j<n → SupportedOn H (X j)) |
| (hc : ∀ j, j<k → Commute H (liftLocal (X j) (F j).matrix)) |
| (A : ι → Matrix (RegionConfiguration q (X k)) (RegionConfiguration q (X k)) ℂ) |
| (B : ι → Matrix (RegionConfiguration q (X k)ᶜ) (RegionConfiguration q (X k)ᶜ) ℂ) |
| (hdecomp : H = ∑ i, tensorAcross (X k) (A i) (B i)) : |
| let P := orderedPrefix (fun j => liftLocal (X j) (F j).matrix) n |
| let v := asMap (L := L) (q := q) P ψ |
| |(inner ℂ v (asMap H v)).re - (inner ℂ v (asMap (P*H) ψ)).re| ≤ |
| 8*(a k)^2*‖v‖^2*(∑ i, ‖A i‖*‖B i‖) := by |
| classical |
| dsimp only |
| let P := orderedPrefix (fun j => liftLocal (X j) (F j).matrix) n |
| let v := asMap (L := L) (q := q) P ψ |
| have hp : 2<2/a k := (lt_div_iff₀ ha).mpr (by linarith) |
| obtain ⟨U,e,he,hrep,hρ⟩ := optimizer_marginal_spectral hq X hX a ψ n F hF k hk |
| (ne_of_gt (lt_trans (by norm_num) hp)) |
| have hs := filter_spectral_sum (X k) (F k) (2/a k) (hF.1 ⟨k,hk⟩) U e he hrep |
| have hsub := optimizer_modular_substitution hq X hX a ψ n F hF k hk hp U e he hrep hs H hHs hc |
| change inner ℂ v (asMap (P*H) ψ) = inner ℂ v (asMap |
| (liftLocal (X k) (F k).matrix * H * liftLocal (X k) (inverseOnSupport U e)) v) at hsub |
| change |(inner ℂ v (asMap H v)).re-(inner ℂ v (asMap (P*H) ψ)).re| ≤ _ |
| rw [hsub] |
| exact spectral_real_error_le (X k) v (F k) U e he (a k) ha ha' hs hrep hρ H hH A B hdecomp |
|
|
| end PolynomialPEPS.Subvolume.OptimizerEnergy |
|
|
| end |
|
|
| end OAI |
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|
