ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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.

Amplitude-Table State Preparation Input Boundary

Example

Take $n = 2$ and the ordered amplitude table

$a_{00} = 1 / 2, a_{01} = i / 2, a_{10} = 1 / 2, a_{11} = 1 / 2.$

It has $D = 4$ entries. For $b \geq 2$ , each real and imaginary coordinate is exactly representable by a signed binary fixed-point code with one sign bit and $b - 1$ fractional bits; thus the dense classical table contains $2 D b = 8 b$ bits, separate from coherent gates. The prefix procedure uses three rotation nodes, each with angle $π / 2$ , and one nontrivial relative leaf phase. No QRAM query is assumed.

Facts & Assumptions

Given: The explicit two-qubit table, the lexicographic basis order $00, 01, 10, 11$ , and the exact prefix-rotation preparation statement.

[F1]

For a normalized table, the prefix procedure exactly prepares its ray with the stated gate and clean-work bounds (QC24 State Preparation Rotation Level Upper Bound).

[F2]

The procedure defines $r_{p} = \sqrt{\sum_{x extends p} ∣ a_{x} ∣^{2}}$ and, for $r_{p} > 0$ , $θ_{p} = 2 atan2 (r_{p 1}, r_{p 0})$ . Its $R_{y} (θ)$ has first column $(\cos (θ / 2), \sin (θ / 2))^{T}$ . Leaf phases are $\arg (a_{x}) - \arg (a_{x_{*}})$ , with $x_{*}$ the first nonzero leaf; zero relative phases are skipped (QC24 State Preparation).

Verification

1.1givenalgebra

The norm is $∣ a_{00} ∣^{2} + ∣ a_{01} ∣^{2} + ∣ a_{10} ∣^{2} + ∣ a_{11} ∣^{2} = 1 / 4 + 1 / 4 + 1 / 4 + 1 / 4 = 1$ , so the table is a valid normalized input. Four complex entries with $b$ bits for each of their two real coordinates occupy $8 b$ bits.

2.1F2step 1.1algebra

The subtree norms are $r_{\emptyset} = 1$ , $r_{0} = r_{1} = 1 / \sqrt{2}$ , and $r_{00} = r_{01} = r_{10} = r_{11} = 1 / 2$ . Hence $θ_{\emptyset} = θ_{0} = θ_{1} = π / 2$ by the prefix formula in [F2].

3.1F2step 2.1algebra

The first rotation gives amplitudes $1 / \sqrt{2}$ on prefixes $0$ and $1$ . The controlled $p = 0$ rotation splits its branch equally between $∣ 00 ⟩$ and $∣ 01 ⟩$ , and the $p = 1$ rotation splits its branch equally between $∣ 10 ⟩$ and $∣ 11 ⟩$ . The resulting vector is $(1 / 2, 1 / 2, 1 / 2, 1 / 2)$ .

4.1F1F2step 3.1givenalgebra∎

Applying the phase $e^{i \arg (a_{01})} = i$ to the $∣ 01 ⟩$ branch yields $(1 / 2, i / 2, 1 / 2, 1 / 2)$ exactly. The table's classical bit scan and these controlled gates are separate costs; none is a QRAM access.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Preskill, Quantum Computation, Chapter 5, Classical and Quantum Circuits (standard reference, not scraped)
Vivek V. Shende, Stephen S. Bullock, and Igor L. Markov, Synthesis of Quantum Logic Circuits (standard reference, not scraped)