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

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Admissible polynomial pair for Hermitian-reflection QSP

Definition

For an integer $d \geq 0$ , a pair $P, Q \in C [x]$ is QSP-admissible of degree bound $d$ when:

$P = 0$ or $\deg P \leq d$ , and every exponent with nonzero coefficient in $P$ is congruent to $d$ modulo $2$ ;
$Q = 0$ or $\deg Q \leq d - 1$ , and every exponent with nonzero coefficient in $Q$ is congruent to $d - 1$ modulo $2$ ;
the formal polynomial identity $P P^{*} + (1 - x^{2}) Q Q^{*} = 1$ holds, where $P^{*} (x) = \sum_{k} \overline{p_{k}} x^{k}$ if $P (x) = \sum_{k} p_{k} x^{k}$ .

The zero polynomial satisfies every degree bound and both parity conditions. On real $x$ , coefficient conjugation gives $P^{*} (x) = \overline{P (x)}$ . The bound $d$ is nominal; a polynomial may have smaller degree of the same parity.

Facts & Assumptions

Given: An integer $d \geq 0$ and an admissible pair $P, Q$ as defined above.

[F1]

Real and imaginary parts, complex conjugation, and modulus: complex conjugation fixes real numbers and $∣ z ∣^{2} = z \overline{z} \geq 0$ .

Verification

1.1givenF1algebra

For real $x$ , conjugating each coefficient of $P$ and using $x = \overline{x}$ gives $P^{*} (x) = \overline{P (x)}$ ; the same calculation gives $Q^{*} (x) = \overline{Q (x)}$ . Hence the polynomial identity implies $∣ P (x) ∣^{2} + (1 - x^{2}) ∣ Q (x) ∣^{2} = 1$ for every real $x$ .

2.1step 1.1algebra∎

If $∣ x ∣ \leq 1$ , then both summands are nonnegative, so $∣ P (x) ∣^{2} \leq 1$ and $(1 - x^{2}) ∣ Q (x) ∣^{2} \leq 1$ . Taking nonnegative square roots yields $∣ P (x) ∣ \leq 1$ and $\sqrt{1 - x^{2}} ∣ Q (x) ∣ \leq 1$ . At $d = 0$ the degree convention forces $Q = 0$ , so the second bound also holds at both endpoints.

Depends on

Used by

Quantum singular-value transformation circuit Algorithm
Real bounded-polynomial QSP phase synthesis Algorithm
Hamiltonian-simulation polynomial target Definition
Qubitization access model Definition
Degree-one QSP phase identity Example
A nonnegative even polynomial has a bounded parity complement Lemma
Completion of a bounded real parity polynomial Theorem
Constructive factorization of an admissible QSP polynomial pair Theorem
Correctness of QSP phase synthesis Theorem
Real bounded polynomial extraction in QSVT Theorem

Dependency tree · two levels

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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)