TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Completion of a bounded real parity polynomial

Statement

Let $d \geq 0$ and let $p \in R [x]$ have degree at most $d$ , parity $d m o d 2$ , and $∣ p (x) ∣ \leq 1$ for every $x \in [- 1, 1]$ . Then there is a QSP-admissible pair $P, Q$ of degree bound $d$ such that $Re P (x) = p (x)$ on $[- 1, 1]$ . In fact, the completion is exact, so for every $δ > 0$ it also satisfies $∣ Re P (x) - p (x) ∣ \leq δ$ .

Facts & Assumptions

Given: An integer $d \geq 0$ and a real polynomial $p$ with the stated degree, parity, and interval bound.

[F1]

A nonnegative even polynomial has a bounded parity complement: an even real polynomial $R$ of degree at most $2 d$ that is nonnegative on $[- 1, 1]$ has $R = B^{2} + (1 - x^{2}) C^{2}$ with the required degree bounds and parities.

[F2]

Admissible polynomial pair for Hermitian-reflection QSP: admissibility requires the stated degree/parity bounds and $P P^{*} + (1 - x^{2}) Q Q^{*} = 1$ .

[F3]

Real and imaginary parts, complex conjugation, and modulus: conjugation fixes real coefficients and sends $i$ to $- i$ .

Proof

1.1givenalgebra

Put $R = 1 - p^{2}$ . Since $p (- x) = (- 1)^{d} p (x)$ , $p^{2}$ and hence $R$ are even. Its degree is at most $2 d$ , and the interval bound on $p$ gives $R (x) \geq 0$ for every $x \in [- 1, 1]$ .

2.1F1step 1.1

Apply [F1] to obtain real polynomials $B, C$ with $R = B^{2} + (1 - x^{2}) C^{2}$ , degree bounds $d, d - 1$ , and parities $d, d - 1$ .

3.1F2F3givenstep 2.1algebra

Define $P = p + i B$ and $Q = C$ . By [F3], $P^{*} = p - i B$ and $Q^{*} = C$ , so $P P^{*} = p^{2} + B^{2}$ and $Q Q^{*} = C^{2}$ . Their sum in the admissibility identity is therefore $p^{2} + B^{2} + (1 - x^{2}) C^{2} = p^{2} + R = 1$ . The degree bounds and parities of $p, B, C$ also give the required bounds and parities of $P, Q$ ; hence the pair is admissible by [F2].

4.1step 3.1algebra∎

For real $x$ , the real part of $P (x) = p (x) + i B (x)$ is exactly $p (x)$ . Thus the uniform error is zero and is at most every $δ > 0$ .

Depends on

Used by

Correctness of QSP phase synthesis Theorem
QSVT action on singular subspaces Theorem
Real bounded polynomial extraction in QSVT Theorem

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

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)