qml.poly_to_angles

poly_to_angles(poly, routine, angle_solver='root-finding')

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

The polynomial \(P(x) = \sum_n a_n x^n\) must satisfy \(|P(x)| \leq 1\) for \(x \in [-1, 1]\). In QSP and QSVT, 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", "QSVT" or "GQSP".

• angle_solver (str) –

  Specifies the method used to calculate the angles. Options include:

  • "root-finding": effective for polynomials of degree up to \(\sim 1000\)

  • "iterative": effective for polynomials of degree higher than \(\sim 1000\) for the "QSP" and "QSVT" routines.

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

code/api/pennylane.poly_to_angles

 

Download Python script

 

Download Notebook

 

View on GitHub