6.46. QuantumBlockEncoding/TensorTrainWord.lean🔗

12 explicit public declarations, in source order.

Definition6.46.1

uses 0used by 0✓L∃∀N

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

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. 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/TensorTrainWord.lean:10. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition6.46.1●1 definition

defdefined in QuantumBlockEncoding/TensorTrainWord.lean

complete

def QuantumBlockEncoding.TensorTrainWord.toBasis {n : ℕ} :
  QuantumBlockEncoding.TensorTrainCanonical.Word n →
    QuantumBlockEncoding.PrimitiveBasis n

def QuantumBlockEncoding.TensorTrainWord.toBasis
  {n : ℕ} :
  QuantumBlockEncoding.TensorTrainCanonical.Word
      n →
    QuantumBlockEncoding.PrimitiveBasis n

Theorem6.46.2

uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “to basis word of basis”; 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/TensorTrainWord.lean:14. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem6.46.2●1 theorem

theoremdefined in QuantumBlockEncoding/TensorTrainWord.lean

complete

theorem QuantumBlockEncoding.TensorTrainWord.toBasis_wordOfBasis {n : ℕ}
  (x : QuantumBlockEncoding.PrimitiveBasis n) :
  QuantumBlockEncoding.TensorTrainWord.toBasis
      (QuantumBlockEncoding.TensorTrainSchedule.wordOfBasis x) =
    x

theorem QuantumBlockEncoding.TensorTrainWord.toBasis_wordOfBasis
  {n : ℕ}
  (x :
    QuantumBlockEncoding.PrimitiveBasis
      n) :
  QuantumBlockEncoding.TensorTrainWord.toBasis
      (QuantumBlockEncoding.TensorTrainSchedule.wordOfBasis
        x) =
    x

Theorem6.46.3

uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “word of basis to basis”; 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/TensorTrainWord.lean:20. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem6.46.3●1 theorem

theoremdefined in QuantumBlockEncoding/TensorTrainWord.lean

complete

theorem QuantumBlockEncoding.TensorTrainWord.wordOfBasis_toBasis {n : ℕ}
  (x : QuantumBlockEncoding.TensorTrainCanonical.Word n) :
  QuantumBlockEncoding.TensorTrainSchedule.wordOfBasis
      (QuantumBlockEncoding.TensorTrainWord.toBasis x) =
    x

theorem QuantumBlockEncoding.TensorTrainWord.wordOfBasis_toBasis
  {n : ℕ}
  (x :
    QuantumBlockEncoding.TensorTrainCanonical.Word
      n) :
  QuantumBlockEncoding.TensorTrainSchedule.wordOfBasis
      (QuantumBlockEncoding.TensorTrainWord.toBasis
        x) =
    x

Definition6.46.4

uses 0used by 0✓L∃∀N

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

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

Lean code for Definition6.46.4●1 definition

defdefined in QuantumBlockEncoding/TensorTrainWord.lean

complete

def QuantumBlockEncoding.TensorTrainWord.basisEquiv (n : ℕ) :
  QuantumBlockEncoding.TensorTrainCanonical.Word n ≃
    QuantumBlockEncoding.PrimitiveBasis n

def QuantumBlockEncoding.TensorTrainWord.basisEquiv
  (n : ℕ) :
  QuantumBlockEncoding.TensorTrainCanonical.Word
      n ≃
    QuantumBlockEncoding.PrimitiveBasis n

Definition6.46.5

uses 0used by 0✓L∃∀N

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

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/TensorTrainWord.lean:32. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition6.46.5●1 definition

defdefined in QuantumBlockEncoding/TensorTrainWord.lean

complete

def QuantumBlockEncoding.TensorTrainWord.reverseBasis (n : ℕ) :
  QuantumBlockEncoding.PrimitiveBasis n ≃
    QuantumBlockEncoding.PrimitiveBasis n

def QuantumBlockEncoding.TensorTrainWord.reverseBasis
  (n : ℕ) :
  QuantumBlockEncoding.PrimitiveBasis n ≃
    QuantumBlockEncoding.PrimitiveBasis n

Definition6.46.6

uses 0used by 0✓L∃∀N

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

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/TensorTrainWord.lean:38. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition6.46.6●1 definition

defdefined in QuantumBlockEncoding/TensorTrainWord.lean

complete

def QuantumBlockEncoding.TensorTrainWord.sampleEquiv (n : ℕ) :
  QuantumBlockEncoding.TensorTrainCanonical.Word n ≃
    Fin (QuantumBlockEncoding.gridSize n)

def QuantumBlockEncoding.TensorTrainWord.sampleEquiv
  (n : ℕ) :
  QuantumBlockEncoding.TensorTrainCanonical.Word
      n ≃
    Fin (QuantumBlockEncoding.gridSize n)

Definition6.46.7

uses 0used by 0✓L∃∀N

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

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/TensorTrainWord.lean:41. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition6.46.7●1 definition

defdefined in QuantumBlockEncoding/TensorTrainWord.lean

complete

def QuantumBlockEncoding.TensorTrainWord.toBits {n : ℕ} :
  QuantumBlockEncoding.TensorTrainCanonical.Word n → List Bool

def QuantumBlockEncoding.TensorTrainWord.toBits
  {n : ℕ} :
  QuantumBlockEncoding.TensorTrainCanonical.Word
      n →
    List Bool

Theorem6.46.8

uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “to bits length”; 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/TensorTrainWord.lean:45. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem6.46.8●1 theorem

theoremdefined in QuantumBlockEncoding/TensorTrainWord.lean

complete

theorem QuantumBlockEncoding.TensorTrainWord.toBits_length {n : ℕ}
  (x : QuantumBlockEncoding.TensorTrainCanonical.Word n) :
  (QuantumBlockEncoding.TensorTrainWord.toBits x).length = n

theorem QuantumBlockEncoding.TensorTrainWord.toBits_length
  {n : ℕ}
  (x :
    QuantumBlockEncoding.TensorTrainCanonical.Word
      n) :
  (QuantumBlockEncoding.TensorTrainWord.toBits
        x).length =
    n

Theorem6.46.9

uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “primitive snoc value”; 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/TensorTrainWord.lean:50. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem6.46.9●1 theorem

theoremdefined in QuantumBlockEncoding/TensorTrainWord.lean

complete

theorem QuantumBlockEncoding.TensorTrainWord.primitive_snoc_value (n : ℕ)
  (x : QuantumBlockEncoding.PrimitiveBasis n) (bit : Fin 2) :
  ↑((QuantumBlockEncoding.primitiveBasisLEEquiv (n + 1))
        (Fin.snoc x bit)) =
    ↑((QuantumBlockEncoding.primitiveBasisLEEquiv n) x) + 2 ^ n * ↑bit

theorem QuantumBlockEncoding.TensorTrainWord.primitive_snoc_value
  (n : ℕ)
  (x :
    QuantumBlockEncoding.PrimitiveBasis n)
  (bit : Fin 2) :
  ↑((QuantumBlockEncoding.primitiveBasisLEEquiv
          (n + 1))
        (Fin.snoc x bit)) =
    ↑((QuantumBlockEncoding.primitiveBasisLEEquiv
            n)
          x) +
      2 ^ n * ↑bit

Theorem6.46.10

uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “sample equiv value”; 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/TensorTrainWord.lean:73. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem6.46.10●1 theorem

theoremdefined in QuantumBlockEncoding/TensorTrainWord.lean

complete

theorem QuantumBlockEncoding.TensorTrainWord.sampleEquiv_value {n : ℕ}
  (x : QuantumBlockEncoding.TensorTrainCanonical.Word n) :
  ↑((QuantumBlockEncoding.TensorTrainWord.sampleEquiv n) x) =
    QuantumBlockEncoding.HermiteBoundaryInjection.wordValue
      (QuantumBlockEncoding.TensorTrainWord.toBits x)

theorem QuantumBlockEncoding.TensorTrainWord.sampleEquiv_value
  {n : ℕ}
  (x :
    QuantumBlockEncoding.TensorTrainCanonical.Word
      n) :
  ↑((QuantumBlockEncoding.TensorTrainWord.sampleEquiv
          n)
        x) =
    QuantumBlockEncoding.HermiteBoundaryInjection.wordValue
      (QuantumBlockEncoding.TensorTrainWord.toBits
        x)

Theorem6.46.11

uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “word sample index eq”; 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/TensorTrainWord.lean:90. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem6.46.11●1 theorem

theoremdefined in QuantumBlockEncoding/TensorTrainWord.lean

complete

theorem QuantumBlockEncoding.TensorTrainWord.wordSampleIndex_eq (n : ℕ)
  (x : QuantumBlockEncoding.TensorTrainCanonical.Word (n + 1)) :
  QuantumBlockEncoding.HermiteBoundaryInjection.wordSampleIndex n
      (QuantumBlockEncoding.TensorTrainWord.toBits x) ⋯ =
    (QuantumBlockEncoding.TensorTrainWord.sampleEquiv (n + 1)) x

theorem QuantumBlockEncoding.TensorTrainWord.wordSampleIndex_eq
  (n : ℕ)
  (x :
    QuantumBlockEncoding.TensorTrainCanonical.Word
      (n + 1)) :
  QuantumBlockEncoding.HermiteBoundaryInjection.wordSampleIndex
      n
      (QuantumBlockEncoding.TensorTrainWord.toBits
        x)
      ⋯ =
    (QuantumBlockEncoding.TensorTrainWord.sampleEquiv
        (n + 1))
      x

Theorem6.46.12

uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “sample equiv public”; 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/TensorTrainWord.lean:96. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem6.46.12●1 theorem

theoremdefined in QuantumBlockEncoding/TensorTrainWord.lean

complete

theorem QuantumBlockEncoding.TensorTrainWord.sampleEquiv_public {n : ℕ}
  (x : QuantumBlockEncoding.PrimitiveBasis n) :
  (QuantumBlockEncoding.TensorTrainWord.sampleEquiv n)
      (QuantumBlockEncoding.TensorTrainSchedule.wordOfBasis fun i =>
        x i.rev) =
    (QuantumBlockEncoding.primitiveBasisLEEquiv n) x

theorem QuantumBlockEncoding.TensorTrainWord.sampleEquiv_public
  {n : ℕ}
  (x :
    QuantumBlockEncoding.PrimitiveBasis
      n) :
  (QuantumBlockEncoding.TensorTrainWord.sampleEquiv
        n)
      (QuantumBlockEncoding.TensorTrainSchedule.wordOfBasis
        fun i => x i.rev) =
    (QuantumBlockEncoding.primitiveBasisLEEquiv
        n)
      x