| import Mathlib |
| import OAI.MathematicalPhysics.PEPSFilters.LocalOperators |
| import OAI.MathematicalPhysics.PEPSSubvolume.Crossing |
|
|
| namespace OAI |
|
|
| /-! Neutral local terms, ground energy and the full-system gap. -/ |
|
|
| 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.OptimizerNeutral |
| open scoped BigOperators |
| open PolynomialPEPS.Subvolume.TraceStripping PolynomialPEPS.Subvolume.CrossingIdentity |
| variable {L q : ℕ} |
|
|
| lemma inserted_eq_right (A : ℕ → Operator L q) (H : Operator L q) |
| (r n : ℕ) (hr : r≤n) (hc : ∀ j, j<r → Commute H (A j)) : |
| insertedProduct A H r n = orderedPrefix A n * H := by |
| have hcomm := commute_prefix A H r hc |
| unfold insertedProduct |
| rw [mul_assoc,hcomm.eq,← mul_assoc,← orderedPrefix_split] |
| rw [Nat.add_sub_of_le hr] |
|
|
| theorem neutral_identity (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)) |
| (r : ℕ) (hr : r≤n) (H : Operator L q) |
| (hH : ∀ j, r≤j → j<n → SupportedOn H (X j)) |
| (hc : ∀ j, j<r → Commute H (liftLocal (X j) (F j).matrix)) : |
| 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) = inner ℂ v (asMap (P*H) ψ) := by |
| have ht := optimizer_strip_outer hq X hX a ψ n F hF r hr H hH |
| dsimp only at ht ⊢ |
| rw [inserted_eq_right _ H r n hr hc] at ht |
| exact ht |
|
|
| theorem excitation_of_transition_errors {ι : Type*} [Fintype ι] |
| (h : ι → Operator L q) (Ω : State L q) (E₀ : ℝ) |
| (hg : asMap (∑ i, h i) Ω = (E₀:ℂ) • Ω) |
| (P : Operator L q) (ε : ι → ℝ) |
| (he : ∀ i, |(inner ℂ (asMap P Ω) (asMap (h i) (asMap P Ω))).re - |
| (inner ℂ (asMap P Ω) (asMap (P*h i) Ω)).re| ≤ ε i) : |
| (inner ℂ (asMap P Ω) (asMap (∑ i, h i) (asMap P Ω))).re - |
| E₀*‖asMap P Ω‖^2 ≤ ∑ i, ε i := by |
| let v := asMap (L := L) (q := q) P Ω |
| have hg' : asMap (P*(∑ i, h i)) Ω = (E₀:ℂ) • v := by |
| change Matrix.toEuclideanCLM (𝕜 := ℂ) (P*(∑ i, h i)) Ω = _ |
| rw [map_mul] |
| change asMap P (asMap (∑ i, h i) Ω) = _ |
| rw [hg,map_smul] |
| have htrans : (inner ℂ v (asMap (P*(∑ i, h i)) Ω)).re = E₀*‖v‖^2 := by |
| rw [hg',inner_smul_right,inner_self_eq_norm_sq_to_K] |
| simp [← Complex.ofReal_pow] |
| change (inner ℂ v (asMap (∑ i, h i) v)).re-E₀*‖v‖^2 ≤ _ |
| rw [← htrans,Matrix.mul_sum] |
| simp only [asMap,map_sum,_root_.sum_apply,inner_sum,Complex.re_sum, |
| ← Finset.sum_sub_distrib] |
| apply Finset.sum_le_sum |
| intro i hi |
| exact (le_abs_self _).trans (he i) |
|
|
| theorem gap_fidelity (H : Operator L q) (Ω v : State L q) (E₀ Δ ε : ℝ) |
| (hg : FullSystemGap H Ω E₀ Δ) (hΔ : 0<Δ) |
| (he : (inner ℂ v (asMap H v)).re-E₀*‖v‖^2 ≤ ε*‖v‖^2) : |
| (1-ε/Δ)*‖v‖^2 ≤ ‖inner ℂ Ω v‖^2 := by |
| have h := (hg v).trans he |
| apply (le_of_mul_le_mul_left (a := Δ)) |
| · calc |
| Δ*((1-ε/Δ)*‖v‖^2) = (Δ-ε)*‖v‖^2 := by field_simp |
| _ ≤ Δ*‖inner ℂ Ω v‖^2 := by nlinarith |
| · exact hΔ |
|
|
| end PolynomialPEPS.Subvolume.OptimizerNeutral |
|
|
| end |
|
|
| end OAI |
|
|
