LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

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.

Hermitian-reflection and SU(2) QSP convention conversion

Statement

Put $s = \sqrt{1 - x^{2}}$ , $W_{R} = (\begin{array}{cc} x & s \\ s & - x \end{array})$ , $W_{S} = (\begin{array}{cc} x & i s \\ i s & x \end{array})$ , and $C = Z (π / 4)$ . Then $C W_{R} C = i W_{S}^{†}, W_{S}^{†} = Z (π / 2) W_{S} Z (- π / 2) .$ For $d \geq 1$ , define $U_{S} (Ψ) = Z (ψ_{0}) \prod_{j = 1}^{d} (W_{S} Z (ψ_{j}))$ . If $Ψ$ is the SU(2)-convention phase list with $ψ_{0} = ϕ_{0} + π / 4$ , $ψ_{j} = ϕ_{j} - π / 2$ for $1 \leq j < d$ , and $ψ_{d} = ϕ_{d} - 3 π / 4$ , then $U_{R} (Φ) = i^{d} U_{S} (Ψ) .$ The affine phase-list map is invertible. The scalar $i^{d}$ is part of the identity: in a control-first block $diag (I, U)$ it acts on the active control branch and cannot be discarded as a global phase.

Facts & Assumptions

Given: $x \in [- 1, 1]$ , real phases, and the definitions of $W_{R}$ and $Z$ in QSP signal matrix and ordered phase sequence.

[F1]

QSP signal matrix and ordered phase sequence fixes $W_{R}$ , $Z (ϕ)$ , the ordered product, and the control-relevant phase convention.

[F2]

Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ gives $e^{i θ} = \cos θ + i \sin θ$ for real $θ$ .

[F3]

$\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential gives $e^{z + w} = e^{z} e^{w}$ and hence $Z (a) Z (b) = Z (a + b)$ .

[F4]

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

[F5]

Controlled Unitary Access: control-first access to a two-dimensional target has block matrix $diag (I_{2}, U)$ when that controlled call is explicitly supplied.

[F6]

Quarter-turn values and shifts by pi/2 and pi gives $\sin (π / 2) = 1$ and $\cos (π / 2) = 0$ ; together with [F2], these give $e^{i π / 2} = i$ and $e^{- i π / 2} = - i$ .

Proof

1.1F1F2F3F4F6algebra

By [F2, F6], $e^{i π / 2} = i$ , and [F3] gives $e^{i π / 4} e^{- i π / 4} = 1$ . Multiplying the three matrices entry by entry therefore gives $C W_{R} C = (\begin{array}{cc} i x & s \\ s & i x \end{array}) = i W_{S}^{†}$ , since [F4] gives $W_{S}^{†} = (\begin{array}{cc} x & - i s \\ - i s & x \end{array})$ .

1.2F1F2F3F4F6algebra

By [F2, F6], the diagonal entries of $Z (π / 2)$ and $Z (- π / 2)$ are $i, - i$ and $- i, i$ , respectively. Multiplication gives $Z (π / 2) W_{S} Z (- π / 2) = (\begin{array}{cc} x & - i s \\ - i s & x \end{array}) = W_{S}^{†}$ .

2.1F1F3step 1.1step 1.2algebra

Rearranging step 1.1 yields $W_{R} = i Z (- π / 4) W_{S}^{†} Z (- π / 4)$ . Substitution in each signal factor and combination of adjacent diagonal phases by [F3] gives $U_{R} (Φ) = i^{d} Z (ϕ_{0} - π / 4) \prod_{j = 1}^{d - 1} (W_{S}^{†} Z (ϕ_{j} - π / 2)) W_{S}^{†} Z (ϕ_{d} - π / 4)$ .

3.1step 1.2step 2.1algebra

Replacing each $W_{S}^{†}$ by the right side of step 1.2 adds $π / 2$ to the initial phase, leaves each intermediate phase unchanged after its two adjacent shifts cancel, and subtracts $π / 2$ from the final phase. The resulting SU(2) list is exactly the stated $Ψ$ , and the scalar remains $i^{d}$ . Each phase shift is affine with coefficient $1$ , so the map and its inverse are bijections on $R^{d + 1}$ .

4.1F1F5step 3.1algebra∎

When the operation model explicitly supplies the controlled call on a control-first qubit and two-dimensional target, [F5] gives $C (U_{S}) = diag (I_{2}, U_{S})$ . Substituting $i^{d} U_{S}$ on the active branch gives $diag (I_{2}, i^{d} U_{S})$ , which differs from $i^{d} diag (I_{2}, U_{S})$ unless $i^{d} = 1$ , because the inactive block remains $I_{2}$ . Thus $i^{d}$ is a relative phase under control and must be retained.

Depends on

Used by

Real bounded-polynomial QSP phase synthesis Algorithm
Arithmetic cost of exact tower completion and QSP phase peeling Lemma
Constructive factorization of an admissible QSP polynomial pair Theorem
QSVT action on singular subspaces Theorem
Qubitization lifts scalar QSP Theorem

Dependency tree · two levels

35 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)