6.5. QuantumBlockEncoding/ConstructiveIsometryLocal.lean🔗

5 explicit public declarations, in source order.

Definition6.5.1

uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “coordinates”. Low bond wires, highest emitted bit, with an explicit finite coordinate map.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Circuit and register semantics, reusable tensor-train and matrix constructions, and explicit exact-real storage-cost refinements. Each declaration's hypotheses and conclusion fix its certified scope.

Technical source note. Low bond wires, highest emitted bit, with an explicit finite coordinate map.

Declaration kind. def.

Source: QuantumBlockEncoding/ConstructiveIsometryLocal.lean:15. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition6.5.1●1 definition

defdefined in QuantumBlockEncoding/ConstructiveIsometryLocal.lean

complete

def QuantumBlockEncoding.ConstructiveIsometryLocal.coordinates (q : ℕ) :
  QuantumBlockEncoding.PrimitiveBasis (q + 1) ≃ Fin (2 * 2 ^ q)

def QuantumBlockEncoding.ConstructiveIsometryLocal.coordinates
  (q : ℕ) :
  QuantumBlockEncoding.PrimitiveBasis
      (q + 1) ≃
    Fin (2 * 2 ^ q)

Low bond wires, highest emitted bit, with an explicit finite coordinate map.

Definition6.5.2

uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “local completion”.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. def.

Source: QuantumBlockEncoding/ConstructiveIsometryLocal.lean:18. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition6.5.2●1 definition

defdefined in QuantumBlockEncoding/ConstructiveIsometryLocal.lean

complete

def QuantumBlockEncoding.ConstructiveIsometryLocal.localCompletion {q r : ℕ}
  (hr : r ≤ 2 ^ q)
  (V : Matrix (QuantumBlockEncoding.PrimitiveBasis (q + 1)) (Fin r) ℝ) :
  Matrix (QuantumBlockEncoding.PrimitiveBasis (q + 1))
    (QuantumBlockEncoding.PrimitiveBasis (q + 1)) ℝ

def QuantumBlockEncoding.ConstructiveIsometryLocal.localCompletion
  {q r : ℕ} (hr : r ≤ 2 ^ q)
  (V :
    Matrix
      (QuantumBlockEncoding.PrimitiveBasis
        (q + 1))
      (Fin r) ℝ) :
  Matrix
    (QuantumBlockEncoding.PrimitiveBasis
      (q + 1))
    (QuantumBlockEncoding.PrimitiveBasis
      (q + 1))
    ℝ

Theorem6.5.3

uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “local completion spec”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/ConstructiveIsometryLocal.lean:26. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem6.5.3●1 theorem

theoremdefined in QuantumBlockEncoding/ConstructiveIsometryLocal.lean

complete

theorem QuantumBlockEncoding.ConstructiveIsometryLocal.localCompletion_spec
  {q r : ℕ} (hr : r ≤ 2 ^ q)
  (V : Matrix (QuantumBlockEncoding.PrimitiveBasis (q + 1)) (Fin r) ℝ)
  (hV : V.transpose * V = 1) :
  (QuantumBlockEncoding.ConstructiveIsometryLocal.localCompletion hr
            V).transpose *
        QuantumBlockEncoding.ConstructiveIsometryLocal.localCompletion
          hr V =
      1 ∧
    (QuantumBlockEncoding.ConstructiveIsometryLocal.localCompletion hr
            V).det =
        1 ∧
      ∀ (row : QuantumBlockEncoding.PrimitiveBasis (q + 1)) (a : Fin r),
        QuantumBlockEncoding.ConstructiveIsometryLocal.localCompletion
            hr V row
            ((QuantumBlockEncoding.TensorTrainLocalCompiler.activePositions
                hr)
              a) =
          V row a

theorem QuantumBlockEncoding.ConstructiveIsometryLocal.localCompletion_spec
  {q r : ℕ} (hr : r ≤ 2 ^ q)
  (V :
    Matrix
      (QuantumBlockEncoding.PrimitiveBasis
        (q + 1))
      (Fin r) ℝ)
  (hV : V.transpose * V = 1) :
  (QuantumBlockEncoding.ConstructiveIsometryLocal.localCompletion
            hr V).transpose *
        QuantumBlockEncoding.ConstructiveIsometryLocal.localCompletion
          hr V =
      1 ∧
    (QuantumBlockEncoding.ConstructiveIsometryLocal.localCompletion
            hr V).det =
        1 ∧
      ∀
        (row :
          QuantumBlockEncoding.PrimitiveBasis
            (q + 1))
        (a : Fin r),
        QuantumBlockEncoding.ConstructiveIsometryLocal.localCompletion
            hr V row
            ((QuantumBlockEncoding.TensorTrainLocalCompiler.activePositions
                hr)
              a) =
          V row a

Definition6.5.4

uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “complete stage”. Actual stage matrix derived from the chain's real occupied columns.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Technical source note. Actual stage matrix derived from the chain's real occupied columns.

Declaration kind. def.

Source: QuantumBlockEncoding/ConstructiveIsometryLocal.lean:34. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition6.5.4●1 definition

defdefined in QuantumBlockEncoding/ConstructiveIsometryLocal.lean

complete

def QuantumBlockEncoding.ConstructiveIsometryLocal.completeStage
  {n l r q : ℕ}
  (C : QuantumBlockEncoding.TensorTrainCanonical.Chain n l r)
  (hB : QuantumBlockEncoding.TensorTrainCanonical.maxBond C ≤ 2 ^ q)
  (t : ℕ) :
  Matrix (QuantumBlockEncoding.PrimitiveBasis (q + 1))
    (QuantumBlockEncoding.PrimitiveBasis (q + 1)) ℝ

def QuantumBlockEncoding.ConstructiveIsometryLocal.completeStage
  {n l r q : ℕ}
  (C :
    QuantumBlockEncoding.TensorTrainCanonical.Chain
      n l r)
  (hB :
    QuantumBlockEncoding.TensorTrainCanonical.maxBond
        C ≤
      2 ^ q)
  (t : ℕ) :
  Matrix
    (QuantumBlockEncoding.PrimitiveBasis
      (q + 1))
    (QuantumBlockEncoding.PrimitiveBasis
      (q + 1))
    ℝ

Actual stage matrix derived from the chain's real occupied columns.

Theorem6.5.5

uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “complete stage spec”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/ConstructiveIsometryLocal.lean:39. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem6.5.5●1 theorem

theoremdefined in QuantumBlockEncoding/ConstructiveIsometryLocal.lean

complete

theorem QuantumBlockEncoding.ConstructiveIsometryLocal.completeStage_spec
  {n l r q : ℕ}
  (C : QuantumBlockEncoding.TensorTrainCanonical.Chain n l r)
  (hC : QuantumBlockEncoding.TensorTrainCanonical.RightCanonical C)
  (hB : QuantumBlockEncoding.TensorTrainCanonical.maxBond C ≤ 2 ^ q)
  (t : ℕ) (ht : t < n) :
  (QuantumBlockEncoding.ConstructiveIsometryLocal.completeStage C hB
            t).transpose *
        QuantumBlockEncoding.ConstructiveIsometryLocal.completeStage C
          hB t =
      1 ∧
    (QuantumBlockEncoding.ConstructiveIsometryLocal.completeStage C hB
            t).det =
        1 ∧
      ∀ (bit : Fin 2) (b a : QuantumBlockEncoding.PrimitiveBasis q),
        ↑((QuantumBlockEncoding.primitiveBasisLEEquiv q) a) <
            QuantumBlockEncoding.TensorTrainSchedule.rankAt C t →
          ↑(QuantumBlockEncoding.ConstructiveIsometryLocal.completeStage
                C hB t (Fin.snoc b bit) (Fin.snoc a 0)) =
            QuantumBlockEncoding.TensorTrainSchedule.paddedAt C t
              (bit, (QuantumBlockEncoding.primitiveBasisLEEquiv q) b)
              ((QuantumBlockEncoding.primitiveBasisLEEquiv q) a)

theorem QuantumBlockEncoding.ConstructiveIsometryLocal.completeStage_spec
  {n l r q : ℕ}
  (C :
    QuantumBlockEncoding.TensorTrainCanonical.Chain
      n l r)
  (hC :
    QuantumBlockEncoding.TensorTrainCanonical.RightCanonical
      C)
  (hB :
    QuantumBlockEncoding.TensorTrainCanonical.maxBond
        C ≤
      2 ^ q)
  (t : ℕ) (ht : t < n) :
  (QuantumBlockEncoding.ConstructiveIsometryLocal.completeStage
            C hB t).transpose *
        QuantumBlockEncoding.ConstructiveIsometryLocal.completeStage
          C hB t =
      1 ∧
    (QuantumBlockEncoding.ConstructiveIsometryLocal.completeStage
            C hB t).det =
        1 ∧
      ∀ (bit : Fin 2)
        (b a :
          QuantumBlockEncoding.PrimitiveBasis
            q),
        ↑((QuantumBlockEncoding.primitiveBasisLEEquiv
                  q)
                a) <
            QuantumBlockEncoding.TensorTrainSchedule.rankAt
              C t →
          ↑(QuantumBlockEncoding.ConstructiveIsometryLocal.completeStage
                C hB t (Fin.snoc b bit)
                (Fin.snoc a 0)) =
            QuantumBlockEncoding.TensorTrainSchedule.paddedAt
              C t
              (bit,
                (QuantumBlockEncoding.primitiveBasisLEEquiv
                    q)
                  b)
              ((QuantumBlockEncoding.primitiveBasisLEEquiv
                  q)
                a)