qml.poly_to_angles¶

poly_to_angles(poly, routine, angle_solver='root-finding')[source]¶

Computes the angles needed to implement a polynomial transformation with quantum signal processing (QSP) or quantum singular value transformation (QSVT).

The polynomial $P(x) = \sum_n a_n x^n$ must satisfy $|P(x)| \leq 1$ for $x \in [-1, 1]$, the coefficients $a_n$ must be real and the exponents $n$ must be either all even or all odd. For more details see arXiv:2105.02859.

Parameters

• poly (tensor-like) – coefficients of the polynomial ordered from lowest to highest power

• routine (str) – the routine for which the angles are computed. Must be either "QSP" or "QSVT".

• angle_solver (str) – the method used to calculate the angles. Default is "root-finding". "root-finding" is currently the only supported method.

Returns

computed angles for the specified routine

Return type

(tensor-like)

Raises

• AssertionError – if poly is not valid

• AssertionError – if routine or angle_solver is not supported

Example

This example generates the QSVT angles for the polynomial $P(x) = x - \frac{x^3}{2} + \frac{x^5}{3}$.

>>> poly = np.array([0, 1.0, 0, -1/2, 0, 1/3])
>>> qsvt_angles = qml.poly_to_angles(poly, "QSVT")
>>> print(qsvt_angles)
[-5.49778714  1.57079633  1.57079633  0.5833829   1.61095884  0.74753829]

Usage Details

This example applies the polynomial $P(x) = x - \frac{x^3}{2} + \frac{x^5}{3}$ to a block-encoding of $x = 0.2$.

poly = np.array([0, 1.0, 0, -1/2, 0, 1/3])

qsvt_angles = qml.poly_to_angles(poly, "QSVT")

x = 0.2

# Encode x in the top left of the matrix
block_encoding = qml.RX(2 * np.arccos(x), wires=0)
projectors = [qml.PCPhase(angle, dim=1, wires=0) for angle in qsvt_angles]

@qml.qnode(qml.device("default.qubit"))
def circuit_qsvt():
    qml.QSVT(block_encoding, projectors)
    return qml.state()

output = qml.matrix(circuit_qsvt, wire_order=[0])()[0, 0]
expected = sum(coef * (x**i) for i, coef in enumerate(poly))

print("output qsvt: ", output.real)
print("P(x) =       ", expected)

output qsvt:  0.19610666666647059
P(x) =        0.19610666666666668