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Low-degree Chebyshev simulation polynomial

Example

For real $τ$ , truncate the Jacobi–Anger expansion at degree $d = 2$ . The resulting complex polynomial is $P_{2} (x) = J_{0} (τ) - 2 J_{2} (τ) T_{2} (x) - 2 i J_{1} (τ) T_{1} (x) = J_{0} (τ) - 2 J_{2} (τ) (2 x^{2} - 1) - 2 i J_{1} (τ) x .$ Its real part is even and its imaginary part is odd. Its uniform remainder is bounded by $2 \sum_{k > 2} ∣ J_{k} (τ) ∣$ , and by the displayed factorial bound of Jacobi–Anger Chebyshev truncation when $∣ τ ∣ / 8 < 1$ .

Facts & Assumptions

Given: The Bessel series, Chebyshev recurrence, and real parameter $τ$ of Hamiltonian-simulation polynomial target.

[F1]

Hamiltonian-simulation polynomial target: $J_{k} (τ) = \sum_{j \geq 0} (- 1)^{j} (τ / 2)^{2 j + k} / (j! (j + k)!)$ , $T_{0} (x) = 1$ , $T_{1} (x) = x$ , and $T_{k + 1} = 2 x T_{k} - T_{k - 1}$ for $k \geq 1$ .

[F2]

Jacobi–Anger Chebyshev truncation: truncating at degree $d$ has error at most $2 \sum_{k > d} ∣ J_{k} (τ) ∣$ , with the factorial bound when its ratio condition holds.

Proof

Given: As in Facts & Assumptions.

1.1F1givenalgebra

The recurrence gives $T_{2} (x) = 2 x T_{1} (x) - T_{0} (x) = 2 x^{2} - 1$ . The $k = 0$ term of the Chebyshev expansion is $J_{0}$ ; the $k = 1$ coefficient is $2 (- i) J_{1} = - 2 i J_{1}$ ; and the $k = 2$ coefficient is $2 (- i)^{2} J_{2} = - 2 J_{2}$ . Therefore the degree-two truncation is exactly the displayed $P_{2}$ .

2.1F1F2step 1.1algebra

The series in [F1] is real for real $τ$ , so $J_{0}, J_{1}, J_{2}$ are real. Since $T_{2}$ is even and $T_{1}$ is odd, $Re P_{2} = J_{0} - 2 J_{2} T_{2}$ is even and $Im P_{2} = - 2 J_{1} T_{1}$ is odd. Applying [F2] with $d = 2$ gives the uniform tail $2 \sum_{k > 2} ∣ J_{k} (τ) ∣$ ; its factorial denominator condition is $∣ τ ∣ / [2 (2 + 2)] = ∣ τ ∣ / 8 < 1$ .

3.1F1F2step 1.1algebra∎

At $τ = 0$ , the series gives $J_{0} (0) = 1$ and $J_{1} (0) = J_{2} (0) = 0$ , so $P_{2} (x) = 1 = e^{0}$ . At $x = \pm 1$ , $T_{1} (x) = x$ and $T_{2} (x) = 1$ , and the same tail bound from [F2] applies without an endpoint exception.

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, Lemma 57 (standard reference, not scraped)