BlockEncoding.poly#

BlockEncoding.poly(p: ArrayLike, kind: Literal['Polynomial', 'Chebyshev'] = 'Polynomial') → BlockEncoding[source]#

Returns a BlockEncoding representing a polynomial transformation of the operator.

For a block-encoded Hermitian matrix \(A\) and a (complex) polynomial \(p(z)\), this method returns a BlockEncoding of the operator \(p(A)\). This is achieved using Generalized Quantum Eigenvalue Transformation (GQET).

Parameters:

pArrayLike

1-D array containing the polynomial coefficients, ordered from lowest order term to highest.

kind{“Polynomial”, “Chebyshev”}

The basis in which the coefficients are defined.

"Polynomial": \(p(x) = \sum c_i x^i\)
"Chebyshev": \(p(x) = \sum c_i T_i(x)\), where \(T_i\) are Chebyshev polynomials of the first kind.

Default is "Polynomial".

Returns:

BlockEncoding: A new Block-Encoding instance representing the transformed operator \(p(A)\).

Examples

Define a Hermitian matrix \(A\) and a vector \(\vec{b}\).

import numpy as np

A = np.array([[0.73255474, 0.14516978, -0.14510851, -0.0391581],
            [0.14516978, 0.68701415, -0.04929867, -0.00999921],
            [-0.14510851, -0.04929867, 0.76587818, -0.03420339],
            [-0.0391581, -0.00999921, -0.03420339, 0.58862043]])

b = np.array([0, 1, 1, 1])

Generate a block-encoding \(A\) of and use poly() to find a block-encoding of \(p(A)\).

from qrisp import *
from qrisp.block_encodings import BlockEncoding

BA = BlockEncoding.from_array(A)

BA_poly = BA.poly(np.array([1.,2.,1.]))

# Prepares operand variable in state |b>
def prep_b():
    operand = QuantumVariable(2)
    prepare(operand, b)
    return operand

@terminal_sampling
def main():
    operand = BA_poly.apply_rus(prep_b)()
    return operand

res_dict = main()
amps = np.sqrt([res_dict.get(i, 0) for i in range(len(b))])

Finally, compare the quantum simulation result with the classical solution:

c = (np.eye(4) + 2 * A + A @ A) @ b
c = c / np.linalg.norm(c)

print("QUANTUM SIMULATION\n", amps, "\nCLASSICAL SOLUTION\n", c)
# QUANTUM SIMULATION
# [0.02986315 0.57992481 0.62416743 0.52269535]
# CLASSICAL SOLUTION
# [-0.02986321  0.57992485  0.6241675   0.52269522]