ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Cancelling identity padding fixes the LCU scale

Example

For the truncated Taylor LCU in Finite Taylor LCU polynomial, suppose its listed coefficient norm is $s_{K} = \sum_{k = 0}^{K} \frac{(Λ ∣ h ∣)^{k}}{k!} \leq 2.$ Set $q = (2 - s_{K}) / 2$ and append two distinct identity entries with coefficients $+ q$ and $- q$ . The represented polynomial is unchanged while its listed coefficient norm becomes exactly $2$ .

For example, take the one-term Hamiltonian $H = 1 \cdot Z$ , $h = \log 2$ , and $K = 0$ . Then $M_{0} = I$ , $s_{0} = 1$ , and the padded list $(1 I, \frac{1}{2} I, - \frac{1}{2} I)$ still sums to $I$ and has absolute coefficient sum $2$ .

Facts & Assumptions

Given: A truncated Taylor LCU whose listed coefficient norm satisfies $s_{K} \leq 2$ .

[F1]

Finite Taylor LCU polynomial: the $k = 0$ term is the empty tuple with coefficient $1$ and unitary $I$ , and the listed coefficient norm is $s_{K} (h) = \sum_{k = 0}^{K} (Λ ∣ h ∣)^{k} / k!$ .

[F2]

The Pauli matrices: in the ordered qubit basis, $Z = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ .

[F3]

The natural logarithm as the inverse of the exponential function: for $x > 0$ , $\log x$ is the real number whose exponential is $x$ .

Verification

Given: As in Facts & Assumptions.

1.1givenF1algebra

The hypothesis $s_{K} \leq 2$ gives $q = (2 - s_{K}) / 2 \geq 0$ .

2.1givenF1step 1.1algebra

The two appended terms add $q I - q I = 0$ to the matrix and add $∣ q ∣ + ∣ - q ∣ = 2 q = 2 - s_{K}$ to the coefficient norm. The new norm is therefore $s_{K} + (2 - s_{K}) = 2$ , including $q = 0$ .

3.1givenF1F2F3step 2.1algebra∎

In the stated example, [F2] gives $Z^{†} = Z$ and $Z^{†} Z = I$ , so $H = 1 \cdot Z$ is a valid one-term Hamiltonian with $Λ = 1$ ; [F3] makes $h = \log 2$ a real parameter. Since $K = 0$ , [F1] gives $M_{0} = I$ and $s_{0} = 1$ . Hence $q = 1 / 2$ ; the three coefficients sum to $1$ and their absolute values sum to $1 + 1 / 2 + 1 / 2 = 2$ .

Depends on

Used by

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Sources

Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 28 §28.3 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, §9.3.1 (standard reference, not scraped)