Chebyshev expansions

Polynomials as Chebyshev expansions¶

In some QSP variants (including QSVT) generated polynomials are better expressed in terms of Chebyshev expansions $$ P(x) = \sum_{k = 0}^n c_k T_k(x) $$ where $ T_k(\cos \theta) = \cos(k \theta) $ are the Chebyshev polynomials of the first kind.

nlft-qsp provides a class ChebyshevTExpansion, as well as methods to convert Laurent polynomials into the corresponding Chebyshev expansions.

P = ChebyshevTExpansion([2, 6, 3])

\[ P(x) = 2 + 6 T_1(x) + 3 T_2(x) = 2 + 6x + 3 (2x^2 - 1) = -1 + 6x + 6x^2 \]

Converting to/from Laurent polynomials¶

By passing a Laurent polynomial $P(z)$ to a ChebyshevTExpansion you get $T(x)$ such that $T(\cos \theta) = P(e^{i\theta})$ for any $\theta \in [0, 2\pi]$.

from nlft_qsp import *
import numpy as np

P = Polynomial([2, 1, 5, 1, 2], support_start=-2)
print("P(z) =", P)

T = ChebyshevTExpansion.from_laurent_polynomial(P)

print(P(np.exp(1j*np.pi/3)), T(np.cos(np.pi/3)))

P(z) = (2+0j) z^-2 + (1+0j) z^-1 + (5+0j) z^0 + (1+0j) z^1 + (2+0j) z^2
(4+2.220446049250313e-16j) (4.000000000000001+0j)

Vice versa, it is possible to convert a Chebyshev expansion into a Laurent polynomial:

from nlft_qsp import *
import numpy as np

T = ChebyshevTExpansion([1, 5, 7])

P = T.to_laurent()
print("P(z) =", P)

print(P(np.exp(1j*np.pi/4)), T(np.cos(np.pi/4)))

P(z) = (3.5+0j) z^-2 + (2.5+0j) z^-1 + (1+0j) z^0 + (2.5+0j) z^1 + (3.5+0j) z^2
(4.5355339059327395+1.1891241456679403e-16j) (4.535533905932738+0j)

Converting to/from polynomials¶

The package also provides methods to convert a Polynomial $P(x)$ into a Chebyshev expansion $T(x)$ such that $P(x) = T(x)$ and vice versa, i.e., they perform a change of basis between the monomial basis $\{ x^k \}_k$ and the Chebyshev basis $\{ T_k(x) \}_k$.

from nlft_qsp import *

T = ChebyshevTExpansion([2, 6, 3])

P = T.to_polynomial()

print("P(z) =", P)
print(P(0.75), T(0.75))

P(z) = (-1+0j) z^0 + (6+0j) z^1 + (6+0j) z^2
(6.875+0j) (6.875+0j)

from nlft_qsp import *

P = Polynomial([-1, 6, 6])

T = ChebyshevTExpansion.from_polynomial(P)

print("T(x) =", T)
print(P(0.75), T(0.75))

T(x) = (2+0j) T_0(x) + (6+0j) T_1(x) + (3+0j) T_2(x)
(6.875+0j) (6.875+0j)