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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.
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A bounded polynomial approximation to a scaled exponential

Example

For $H = diag (- 1 / 2, 1 / 3)$ , take $α = 1$ and target $f (x) = e^{- i x}$ on $[- 1, 1]$ . A one-ancilla exact dilation has top-left block $H$ . Decompose $f (x) = \cos x - i \sin x$ into the even component $g_{0} (x) = \cos x$ and odd component $g_{3} (x) = - \sin x$ ; the other two components are identically zero and are omitted using their exact zero certificates, so $J = {0, 3}$ . The bounded parity polynomials $p_{R} (x) = 1$ (even, degree $0$ ) and $p_{I} (x) = - x$ (odd, degree $1$ ) use exact QSP phase lists $(0)$ and $(π / 2, π / 2)$ , respectively, and give an LCU block $(I - i H) / 2$ , a $2$ -block encoding of the linear approximation to $e^{- i H}$ . In exact phase and LCU models its normalized top-block error is at most $7 / 96$ .

Facts & Assumptions

Given: $H = diag (- 1 / 2, 1 / 3)$ , $α = 1$ , exact ideal phase operations and LCU, and the functions and polynomials above.

[F1]

Finite block encoding: a unitary whose designated compression is $H$ with normalization one is an exact block encoding of $H$ .

[F2]

QSVT implementation of a bounded matrix function, Matrix-function transformation error and resource bound, QSVT action on singular subspaces, QSP signal matrix and ordered phase sequence, LCU projected-block correctness, and Uniform polynomial approximation transfers to a Hermitian matrix: parity polynomial phase lists give their stated singular transforms, the exact LCU compression is the normalized weighted sum, and scalar error on the spectrum transfers to operator norm.

[F3]

The exponential definitions of complex sine, cosine, hyperbolic sine, and hyperbolic cosine equal their entire power series and $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ : $e^{- i x} = \cos x - i \sin x$ , with the usual convergent Taylor series for sine and cosine.

[F4]

The alternating series test: if $(b_{k})$ is nonincreasing with $b_{k} \to 0$ then $\sum_{k} (- 1)^{k} b_{k}$ converges, the sum lies between any two consecutive partial sums, and the error after $n$ terms is at most $b_{n}$ : for an alternating series whose nonnegative term magnitudes decrease to zero, the remainder is bounded by the first omitted term.

Verification

1.1F1givenalgebra

Let $S = \sqrt{I - H^{2}} = diag (\sqrt{3} / 2, 2 \sqrt{2} / 3)$ and set $U = (\begin{array}{cc} H & S \\ S & - H \end{array})$ . Since $S = S^{*}$ , $H = H^{*}$ , and $H S = S H$ , $U^{*} = U$ and $U^{2} = (\begin{array}{cc} H^{2} + S^{2} & H S - S H \\ S H - H S & S^{2} + H^{2} \end{array}) = I$ . Thus $U$ is unitary and its top-left block is $H$ , proving the exact dilation assertion by [F1].

2.1F2F3step 1.1givenalgebra

The real part of $e^{- i x}$ is even and the imaginary part is $- \sin x$ , which is odd, by [F3]. For phase list $(0)$ the degree-zero signal sequence has upper-left entry $1$ . For $(π / 2, π / 2)$ , the upper-left entry of $Z (π / 2) W_{R} (x) Z (π / 2)$ is $x e^{i π} = - x$ , by the sequence definition [F2]. Using the exact dilation from step 1.1, the polynomials have matching parity and satisfy $∣ p_{R} (x) ∣ = 1$ , $∣ p_{I} (x) ∣ \leq 1$ on $[- 1, 1]$ . The coefficient weights are both one, so the LCU scale is $B = 2$ ; by [F2] its normalized weighted sum is $(I + i (- H)) / 2 = (I - i H) / 2$ , with the stated normalization.

3.1F2F3F4step 2.1algebra∎

The sine and cosine series converge by [F3], so their terms tend to zero. For $∣ x ∣ \leq 1$ , their term magnitudes are nonincreasing, so [F4] gives $∣ 1 - \cos x ∣ \leq x^{2} / 2$ ; the sine series similarly gives $∣ x - \sin x ∣ \leq ∣ x ∣^{3} / 6$ . On the two eigenvalues these imply component errors $τ_{0} \leq 1 / 8$ and $τ_{3} \leq 1 / 48$ . With $ε_{B E} = 0$ , exact phases and exact LCU, the error formula in [F2] gives normalized block error at most $(1 / 8 + 1 / 48) / 2 = 7 / 96$ . Equivalently, $∣ (1 - i x) / 2 - e^{- i x} / 2 ∣ \leq (∣ 1 - \cos x ∣ + ∣ x - \sin x ∣) / 2 \leq (x^{2} / 2 + ∣ x ∣^{3} / 6) / 2$ , which is $7 / 96$ at $x = - 1 / 2$ and $5 / 162$ at $x = 1 / 3$ . The scalar error is at most $7 / 96$ on $σ (H)$ ; applying the polynomial matrix-function transfer theorem [F2] to $p (x) = (1 - i x) / 2$ and $f (x) = e^{- i x} / 2$ gives the same operator-norm bound.

Depends on

Used by

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Sources

András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, Quantum singular value transformation and beyond, §§2–3 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 29 §29.3 (standard reference, not scraped)