TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

QSP signal-query and phase-rotation count

Statement

A length- $d$ sequence from QSP signal matrix and ordered phase sequence has exactly $d$ signal factors $W_{R} (x)$ and exactly $d + 1$ phase factors $Z (ϕ_{j})$ , for $0 \leq j \leq d$ . In the scalar QSP model the signal acts on one two-dimensional signal register. If each $W_{R}$ is provided by a named query oracle, the sequence uses exactly $d$ calls to that oracle; it makes no controlled or inverse signal calls. The oracle's work registers, implementation, and query/gate charges are those explicitly supplied with it. The phase-rotation, ordinary-gate, circuit-description, bit, and numerical-precision coordinates remain separate as in Circuit-family and computation cost model.

Facts & Assumptions

Given: A nonnegative integer $d$ , the ordered QSP product, and the declared signal-oracle and circuit-cost interfaces.

[F1]

QSP signal matrix and ordered phase sequence: $U_{Φ} = Z (ϕ_{0}) \prod_{j = 1}^{d} (W_{R} (x) Z (ϕ_{j}))$ ; the empty product at $d = 0$ is $Z (ϕ_{0})$ .

[F2]

Real bounded-polynomial QSP phase synthesis: the output list contains the $d + 1$ phase values and the target length $d$ ; phase generation is classical preprocessing.

[F3]

Circuit-family and computation cost model: named oracle calls, ordinary gates, phase precision, circuit-description size, and classical computation costs are distinct coordinates; a matrix description alone gives no oracle call.

[F4]

Hybrid bound for imperfect QSP phase rotations: any compiled phase or signal operator errors are bounded separately by the sum of their per-factor operator errors.

Proof

1.1F1givenalgebra

The product in [F1] contains one factor $W_{R}$ for each integer $j = 1, \dots, d$ , hence exactly $d$ signal occurrences. It contains $Z (ϕ_{0})$ once and one $Z (ϕ_{j})$ for each $j = 1, \dots, d$ , hence exactly $d + 1$ phase rotations. The case $d = 0$ has zero signal factors and the single endpoint phase $Z (ϕ_{0})$ .

2.1F1F2F3F4step 1.1∎

A named signal query is invoked once for each displayed $W_{R}$ factor and no other signal call appears in the sequence; therefore the query count is exactly $d$ . The one-qubit signal matrix acts on $C^{2}$ , while any implementation work space is included only if the supplied signal interface declares it. The phase and compilation coordinates remain distinct by [F3]–[F4]; no controlled or inverse call is inferred.

Depends on

Used by

Qubitization query and gate resources Theorem

Dependency tree · two levels

42 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

András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, Quantum singular value transformation and beyond (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 29 (standard reference, not scraped)