LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Sine-window normalization and positive prefix weights

Statement

For a power of two $T \geq 2$ and integers $0 \leq a \leq b \leq T$ , $W_{T} (a, b) = \frac{b - a}{2} - \frac{\sin (2 π b / T) - \sin (2 π a / T)}{4 \sin (π / T)} .$ In particular, $W_{T} (0, T) = T / 2$ , so $∣ Ψ_{T} ⟩$ is normalized, and every nonempty interval has $W_{T} (a, b) \geq T^{- 2}$ .

Facts & Assumptions

Given: The sine window and interval mass in Finite sine-window phase state.

[F1]

Finite geometric sum for phase-estimation amplitudes: finite geometric sums satisfy $\sum_{j = 0}^{n - 1} z^{j} = (1 - z^{n}) / (1 - z)$ when $z \neq 1$ .

[F2]

Finite sine-window phase state: $W_{T} (a, b)$ is the sum of the indicated nonnegative sine squares.

[F3]

Double-angle and quadratic power-reduction identities: $\sin^{2} x = (1 - \cos (2 x)) / 2$ .

[F4]

$\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential, and The addition formulas for sine and cosine: $e^{i x} = \cos x + i \sin x$ , $e^{i π} = - 1$ , the exponential is multiplicative, and sine/cosine satisfy their addition formulas.

[F5]

Sine chord and linear bounds: $\sin (π u) \geq 2 u$ for $0 \leq u \leq 1 / 2$ .

[F6]

Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions: $\sin (π - x) = \sin x$ .

Proof

1.1F1F2F3F4F5algebra

Put $z = e^{2 π i / T}$ . By [F4], $z^{T} = e^{2 π i} = 1$ . Also [F5] gives $\sin (π / T) \geq 2 / T > 0$ , so $1 - z \neq 0$ . Reindexing [F1] gives $\sum_{τ = a}^{b - 1} e^{2 π i (τ + 1 / 2) / T} = e^{2 π i (a + 1 / 2) / T} \frac{1 - z^{b - a}}{1 - z} = e^{i π (a + b) / T} \frac{\sin (π (b - a) / T)}{\sin (π / T)} .$ The real part of the last expression is $[\sin (2 π b / T) - \sin (2 π a / T)] / [2 \sin (π / T)]$ by the sine addition formula [F4]. Applying [F3] to the defining sum [F2] proves the displayed formula for $W_{T} (a, b)$ .

2.1step 1.1algebra

Setting $a = 0, b = T$ in the formula of step 1.1 makes its sine difference zero, so $W_{T} (0, T) = T / 2$ . The squared norm of $∣ Ψ_{T} ⟩$ is $(2 / T) W_{T} (0, T) = 1$ .

3.1F2F5F6algebra∎

If $a < b$ , choose an integer $τ$ in $[a, b)$ . Then $τ + 1 / 2$ lies between $1 / 2$ and $T - 1 / 2$ , so its angle lies in $[π / (2 T), π - π / (2 T)]$ . The chord bound [F5] gives $\sin (π / (2 T)) \geq 1 / T$ , and [F6] reflects this lower bound to the upper half of the interval. That summand is at least $T^{- 2}$ and all other summands are nonnegative by [F2].

Depends on

Used by

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Sources

Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, Quantum algorithm for linear systems of equations (standard reference, not scraped)
Andrew M. Childs, Robin Kothari, and Rolando D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)